1,1

Referring to his proof of Bertrand's postulate, Ramanujan states a generalization: "From this we easily deduce that pi(x) - pi(x/2) >= 1, 2, 3, 4, 5, ..., if x >= 2, 11, 17, 29, 41, ..., respectively." Since the a(n) are prime (by their minimality), I call them "Ramanujan primes."

See the additional references and links mentioned in A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

2n log 2n < a(n) < 4n log 4n for n >= 1, and Prime(2n) < a(n) < Prime(4n) if n > 1. Also, a(n) ~ Prime(2n) as n -> infinity. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

Shanta Laishram has proved that a(n) < Prime(3n) for all n >= 1. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2009]

a(n) - 3n log 3n is sometimes positive, but negative with increasing frequency as n grows since a(n) ~ 2n log 2n. There should be a constant m s.t. for n >= m we have a(n) < 3n log 3n.

A good approximation to a(n) = R_n for n in [1..1000] is A162996(n) = Round(kn * (ln(kn)+1)), with k = 2.216 determined empirically from the first 1000 Ramanujan primes, which approximates the {kn}_th prime number which in turn approximates the n_th Ramanujan prime and where Abs(A162996(n) - R_n) < 2 * Sqrt(A162996(n)) for n in [1..1000]. Since R_n ~ Prime(2n) ~ 2n * (ln(2n)+1) ~ 2n * ln(2n), while A162996(n) ~ Prime(kn) ~ kn * (ln(kn)+1) ~ kn * ln(kn), A162996(n) / R_n ~ k/2 = 2.216/2 = 1.108 which implies an asymptotic overestimate of about 10% (a better approximation would need k to depend on n and be asymptotic to 2.) [From Daniel Forgues (squid(AT)zensearch.com), Jul 29 2009]

Let p_n be the n-th prime. If p_n>=3 is in the sequence, then all integers (p_n+1)/2, (p_n+3)/2, ... , (p_(n+1)-1)/2 are composite numbers. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 12 2009]

Denote by q(n) the prime which is the nearest from the right to a(n)/2. Then there exists a prime between a(n) and 2q(n). Converse, generally speaking, is not true, i.e. there exist primes outside the sequence, but possess such property (e.g., 109) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]

The Mathematica program FasterRamanujanPrimeList uses Laishram's result that a(n) < Prime(3n). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 15 2009]

A generalization. For k>1 (not necessarily integer), we call a Ramanujan k-prime R_n^(k) the prime a_k(n) which is the smallest number such that if x >= a_k(n), then pi(x)- pi(x/k) >= n. Note that, the sequence of all primes corresponds to the case of "k=oo". These numbers possess the following properties: R_n^(k)~p_((k/(k-1))n) as n tends to the infinity; if A_k(x) is the counting function of the Ramanujan k-primes not exceeding x, then A_k(x)~(1-1/k)\pi(x); let p be a k-Ramanujan prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the nth prime. Then the interval (p, k*p_(m+1)) contains a prime. Conjecture. For every k>=2 there exist non-k-Ramanujan primes, which possess the latter property. For example, for k=2, the smallest such prime is 109. Problem. For every k>2 to estimate the smallest non-k-Ramanujan prime,which possess the latter property. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Sep 01 2009]

Shanta Laishram, On a conjecture on Ramanujan primes, preprint, 2009. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2009]

S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc. 11 (1919), 181-182.

S. Ramanujan, Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson), Amer. Math. Soc., Providence, 2000, pp. 208-209.

J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630-635.

T. D. Noe, Table of n, a(n) for n=1..1000

S. Ramanujan, A Proof Of Bertrand's Postulate

V. Shevelev, On critical small intervals containing primes [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

J. Sondow, Ramanujan primes and Bertrand's postulate [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

Eric Weisstein's World of Mathematics, Bertrand's Postulate

Eric Weisstein's World of Mathematics, Ramanujan Prime

Wikipedia, Ramanujan prime

a(n) = 1 + max{k: pi(k) - pi(k/2) = n - 1}.

a(n) = A080360(n-1) + 1 for n > 1 [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 11 2008]

a(n)>=A080359(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

a(1) = 2 is Bertrand's postulate: pi(x) - pi(x/2) >= 1 for all x >= 2.

a(2) = 11 because a(2) < 8 log 8 < 17 and pi(n) - pi(n/2) > 1 for n = 16, 15, ..., 11 but pi(10) - pi(5) = 1. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 26 2009]

Consider a(9)=71. Then the nearest prime>71/2 is q(9)=37, and between a(9) and 2q(9), i.e. between 71 and 74 there exists a prime (73). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]

(RamanujanPrimeList[n_] := With[{T=Table[{k, PrimePi[k]-PrimePi[k/2]}, {k, Ceiling[N[4*n*Log[4*n]]]}]}, Table[1+First[Last[Select[T, Last[ # ]==i-1&]]], {i, 1, n}]]; RamanujanPrimeList[54]) [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 15 2009]

(FasterRamanujanPrimeList[n_] := With[{T=Table[{k, PrimePi[k]-PrimePi[k/2]}, {k, Prime[3*n]}]}, Table[1+First[Last[Select[T, Last[ # ]==i-1&]]], {i, 1, n}]]; FasterRamanujanPrimeList[54]) [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 15 2009]

Cf. A006992 Bertrand primes, A056171 pi(n) - pi(n/2).

Cf. A000720, A014085, A060715, A143223, A143224, A143225, A143226, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

Cf. A080360. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 11 2008]

Contribution from Daniel Forgues (squid(AT)zensearch.com), Jul 21 2009: (Start)

Cf. A162996 Round(kn * (ln(kn)+1)), with k = 2.216 as an approximation of R_n = n_th Ramanujan Prime.

Cf. A163160 Round(kn * (ln(kn)+1)) - R_n, where k = 2.216 and R_n = n_th Ramanujan prime. (End)

A080359 A164368 A164288 A164554 A164333 A164294 A164371 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]

Sequence in context: A087379 A019364 A164368 this_sequence A117155 A141176 A118839

Adjacent sequences: A104269 A104270 A104271 this_sequence A104273 A104274 A104275

nonn,nice

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Feb 27 2005

Link corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2009

0,4

Same as n!/A061355(n) and (1+n+n(n-1)+n(n-1)(n-2)+...+n!)/A061354(n). Relatively prime to n.

GCD(a(n),a(n+1)) = 1.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality

GCD(n!, 1+n+n(n-1)+n(n-1)(n-2)+...+n!)

GCD(n!, A(n)) where A(0) = 1, A(n) = n*A(n-1)+1

E.g. 1/0!+1/1!+1/2!+1/3!=16/6=(2*8)/(2*3) so a(3)=2.

f[n_] := n! / Denominator[ Sum[1/k!, {k, 0, n}]]; Table[ f[n], {n, 0, 74}] (from Robert G. Wilson v)

(A[n_] := If[n==0, 1, n*A[n-1]+1]; Table[GCD[A[n], n! ], {n, 0, 74}])

Cf. A093647, A093651.

(n+1)!/(a(n)*a(n+1)) = A123899(n). (n+3)!/(a(n)*a(n+1)*a(n+2)) = A123900(n). (n+3)/GCD(a(n), a(n+2)) = A123901(n). Cf. also A000522, A061354, A061355.

Sequence in context: A134304 A134569 A072883 this_sequence A082469 A088151 A159906

Adjacent sequences: A093098 A093099 A093100 this_sequence A093102 A093103 A093104

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), May 10 2004, Oct 18 2006

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 14 2004

0,2

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebychev) says there is always a prime between n and 2n.

Hashimoto's plot of (1 - a(n)) shows that |a(n)| is small compared to n for n < 30,000.

Contribution from Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 07 2008: (Start)

It appears that there are only a finite number of negative terms (see A143226).

If the negative terms are bounded, then Legendre's conjecture is true, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes). (End)

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate

M. Hassani, Counting primes in the interval (n^2,(n+1)^2)

J. Pintz, Landau's problems on primes

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

T. D. Noe, Plot of A143223(n) for n to 10^6 [From T. D. Noe (noe(AT)sspectra.com), Aug 04 2008]

S. Ramanujan, A Proof Of Bertrand's Postulate [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] - [pi(2n) - pi(n)] (for n > 1)

There are 4 primes between 6^2 and 7^2 and 2 primes between 6 and 2*6, so a(6) = 4 - 2 = 2.

a(1) = 2 because there are two primes between 1^2 and 2^2 (namely, 2 and 3) and none between 1 and 2. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 07 2008]

L={0, 2}; Do[L=Append[L, (PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], {n, 2, 100}]; L

See A000720, A014085, A060715, A143224, A143225, A143226.

Negative terms are A143227. Cf. A104272 (Ramanujan primes). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

Sequence in context: A083896 A084115 A080028 this_sequence A063993 A115722 A115721

Adjacent sequences: A143220 A143221 A143222 this_sequence A143224 A143225 A143226

sign

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008

Corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 07 2008, Aug 09 2008

1,2

The sequence gives the zeros in A143223. The number of primes in question is A143225(n).

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebychev) says there is always a prime between n and 2n.

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate

M. Hassani, Counting primes in the interval (n^2,(n+1)^2)

J. Pintz, Landau's problems on primes

J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

S. Ramanujan, A Proof Of Bertrand's Postulate [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

A143223(n) = 0

There are the same number of primes (namely 3) between 9^2 and 10^2 as between 9 and 2*9, so 9 is a member.

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L, n]], {n, 0, 2000}]; L

See A000720, A014085, A060715, A143223, A143225, A143226.

Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

Sequence in context: A118414 A137628 A020297 this_sequence A068810 A077115 A073946

Adjacent sequences: A143221 A143222 A143223 this_sequence A143225 A143226 A143227

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008

1,2

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebychev) says there is always a prime between n and 2n.

See the additional reference and link to Ramanujan's work mentioned in A143223. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate

M. Hassani, Counting primes in the interval (n^2,(n+1)^2)

J. Pintz, Landau's problems on primes

J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0

There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L, PrimePi[2n]-PrimePi[n]]], {n, 0, 2000}]; L

See A000720, A014085, A060715, A143223, A143224, A143226.

Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

Sequence in context: A004166 A110759 A063750 this_sequence A099720 A162349 A072404

Adjacent sequences: A143222 A143223 A143224 this_sequence A143226 A143227 A143228

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008

1,1

Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebychev) says there is always a prime between n and 2n.

It appears that this sequence is finite; searching up to 10^5, the last n appears to be 48717. [From T. D. Noe (noe(AT)sspectra.com), Aug 01 2008]

If the sequence is finite, then, by Bertrand's postulate, Legendre's conjecture is true, at least for all sufficiently large n. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

No other n <= 10^6. The plot of A143223 shows that it is quite likely that there are no additional terms. [From T. D. Noe (noe(AT)sspectra.com), Aug 04 2008]

See the additional reference and link to Ramanujan's work mentioned in A143223. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

T. D. Noe, Table of n, a(n) for n=1..413

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate

M. Hassani, Counting primes in the interval (n^2,(n+1)^2)

J. Pintz, Landau's problems on primes

J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]

A143223(n) < 0

There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 and 43^2, so 42 is a member.

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L, n]], {n, 0, 500}]; L

See A000720, A014085, A060715, A143223, A143224, A143225.

Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]

Sequence in context: A125009 A008886 A029695 this_sequence A043136 A039313 A043916

Adjacent sequences: A143223 A143224 A143225 this_sequence A143227 A143228 A143229

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008

1,2

If the sequence is bounded (e.g., if it is finite), then Legendre's conjecture is true: there is always a prime between n^2 and (n+1)^2, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes).

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1989, p. 19.

S. Ramanujan, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.

S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209.

T. D. Noe, Table of n, a(n) for n=1..413

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate

M. Hassani, Counting primes in the interval (n^2,(n+1)^2)

T. D. Noe, Plot of the points (A143226(n), A143227(n))

J. Pintz, Landau's problems on primes

J. Sondow, Ramanujan Prime in MathWorld

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld

E. W. Weisstein, Legendre's Conjecture in MathWorld

a(n) = |A143223(A143226(n))|

The 1st positive value of ((pi(2n) - pi(n)) - (pi((n+1)^2) - pi(n^2))) is 1 (at n = 42), the 2nd is 2 (at n = 55) and the 3rd is 1 (at n = 56), so a(1) = 1, a(2) = 2, a(3) = 1.

L={}; Do[ With[ {d=(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2])}, If[d>0, L=Append[L, d]]], {n, 0, 1000}]; L

Cf. A000720, A014085, A060715, A104272, A143223, A143224, A143225, A143226 = corresponding values of n.

Sequence in context: A064823 A140225 A104758 this_sequence A026791 A080576 A083671

Adjacent sequences: A143224 A143225 A143226 this_sequence A143228 A143229 A143230

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008

0,3

A cubic analog of Somos's quadratic recurrence sequence A052129.

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. (to appear).

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant

Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant

a(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/4n - 15/32n^2 + 113/128n^3 + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

a(3) = 3*a(2)^3 = 3*(2*a(1)^3)^3 = 3*(2*(1*a(0)^3)^3)^3 = 3*(2*(1*1^3)^3)^3 = 3*(2*1)^3 = 3*8 = 24.

(a[n_] := If[n==0, 1, n*a[n-1]^3]; Table[a[n], {n, 0, 7}])

Cf. A052129, A112302, A116603, A123852, A123853, A123854.

Sequence in context: A108349 A000722 A098679 this_sequence A120122 A068943 A100815

Adjacent sequences: A123848 A123849 A123850 this_sequence A123852 A123853 A123854

easy,nonn

Petros Hadjicostas and Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Oct 15 2006

0,2

The next term > 1 is a(460) = 463. The primes 2, 5, 13, 37, 463 are the only terms > 1 up to n = 600000. If a(n) > 1 with n > 1, then a(n) = n+3 is prime. This uses A(n+2) = (n+2)(n+1)*A(n) + n+3. The terms > 1 are A064384 = primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. The proof uses (n-1)!/(n-k-1)! = (n-1)(n-2)...(n-k) == (-1)^k k! (mod n). Cf. Cloitre's comment in A064383.

An integer p > 1 is in the sequence if and only if p is prime and p|A(p-1), where A(0) = 1 and A(n) = n*A(n-1)+1 for n > 0. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 22 2006

Michael Mossinghoff has calculated that there are only five primes in the sequence up to 150 million. Heuristics suggest it contains infinitely many. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2007

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 3rd edition, 2004, B43.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Index entries for sequences related to factorial numbers

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection

J. Sondow, Which Partial Sums of the Taylor Series for e Are Convergents to e? (and a Link to the Primes $2, 5, 13, 37, 463, ...$) with an Appendix "Periodic Behaviour of Some Recurrence Sequences Related to $e$, Modulo Powers of 2" by Kyle Schalm

Eric Weisstein's World of Mathematics, Alternating Factorial

a(n) = A124780(n)/A124781(n) = A124782(n)/A123901(n)

a(n) = GCD(A(n), A(n+2))/GCD(A(n), A(n+2), n!) where A(n)=1+n+n(n-1)+...+n! - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Nov 10 2006

a(n) = GCD(N(n), N(n+2)), where N(n) = A061354(n) = numerator of Sum[1/k!,{k,0,n}]. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2007

a(2) = GCD(A(2), A(4))/GCD(d(2), d(4)) = GCD(5, 65)/GCD(1, 1) =

5/1 = 5

(A[n_] := Sum[n!/k!, {k, 0, n}]; d[n_] := GCD[A[n], n! ]; Table[GCD[A[n], A[n+2]]/GCD[d[n], d[n+2]], {n, 0, 100}])

A(n) = A000522, d(n) = A093101, GCD(A(n), A(n+2)) = A124780, GCD(d(n), d(n+2)) = A124781, (n+3)/GCD(A(n), A(n+2)) = A124782, (n+3)/GCD(d(n), d(n+2)) = A123901. Cf. A061354, A061355, A123899, A123900.

Cf. A129924.

Sequence in context: A062627 A011217 A078506 this_sequence A092134 A024548 A091772

Adjacent sequences: A124776 A124777 A124778 this_sequence A124780 A124781 A124782

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Nov 07 2006

0,1

Decimal expansion of Integrate[(x - 1)/((1 + x y) Log[x y]),{y,0,1},{x,0,1}]. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 27 2005

G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.

J. Borwein and P. Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.

D. Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly 108 (2001) 222-231.

J. Sondow, Double Integrals for Euler's Constant and ln(4/Pi) and an Analog of Hadjicostas's Formula, Amer. Math. Monthly 112 (2005) 61-65.

J. Sondow, Double Integrals for Euler's Constant and ln(4/Pi).

Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant

Eric Weisstein's World of Mathematics, Hadjicostas's Formula

J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)

Eric Weisstein's World of Mathematics, Digit Count

log(4/Pi) = 0.24156447527...

RealDigits[ Log[4/Pi], 10, 111][[1]]

Cf. A094641. See also A103130.

Cf. A110625, A110626.

Sequence in context: A060370 A165064 A021418 this_sequence A070937 A059573 A080427

Adjacent sequences: A094637 A094638 A094639 this_sequence A094641 A094642 A094643

cons,easy,nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu) and Robert G. Wilson v (rgwv(AT)rgwv.com), May 18 2004