1,2

An interesting general discussion of the phenomenon of 'random primes' (generalizing the lucky numbers) occurs in Hawkins (1958). Heyde (1978) proves that Hawkins' random primes do not only almost always satisfy the Prime Number Theorem but also the Riemann Hypothesis. - Alf van der Poorten, Jun 27 2002

A145649(a(n)) = 1; complement of A050505. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 15 2008]

Bui and Keating establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59 - 81]. We here extend this to k => 2 and generalize it to all l-tuples of Hawkins primes. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 24 2009]

M. Gardner, Lucky numbers and 2187, Math. Intellig., 19 (No. 2, 1997), 26-29.

M. Gardner, Gardner's Workout, Chapter 21 "Lucky Numbers and 2187" pp. 149-156 A. K. Peters MA 2002.

V. Gardiner, R. Lazarus, N. Metropolis and S. Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1955), 117-119.

R. K. Guy, Unsolved Problems in Number Theory, C3.

D. Hawkins, The random sieve, Math. Mag. 31 (1958), 1-3.

D. Hawkins and W. E. Briggs, The lucky number theorem. Math. Mag. 31 1958 81-84.

C. C. Heyde, Ann. Probability, 6 (1978), 850-875.

C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 99.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 114.

R. J. Mathar, Table of n, a(n) for n = 1..30981

H. M. Bui, J. P. Keating, On twin primes associated with the Hawkins random sieve, version 2, Mar 24, 2009. J. Number Theory 119 (2006), 284-296. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 24 2009]

I. Peterson, MathTrek, Martin Gardner's Lucky Numbers

I. Peterson, See also

W. Schneider, Lucky Numbers [Broken link?]

T. Sillke, S.M.Ulam's Lucky Numbers

G. Villemin's Almanach of Numbers, Nombre Chanceux

Eric Weisstein's World of Mathematics, Lucky number.

Wikipedia, Lucky number

Index entries for "core" sequences

Index entries for sequences generated by sieves [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 15 2008]

Start with the natural numbers. Delete every 2nd number, leaving 1 3 5 7 ...; the 2nd number remaining is 3, so delete every 3rd number, leaving 1 3 7 9 13 15 ...; now delete every 7th number, leaving 1 3 7 9 13 ...; now delete every 9th number; etc.

## luckynumbers(n) returns all lucky numbers from 1 to n. ## Try n=10^5 just for fun. luckynumbers:=proc(L) local k, Lnext, Lprev; Lprev:=[$1..n]; for k from 1 do if k=1 or k=2 then Lnext:= map(w-> Lprev[w], remove(z -> z mod Lprev[2] = 0, [$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) then break fi; Lprev:=Lnext; else Lnext:= map(w-> Lprev[w], remove(z -> z mod Lprev[k] = 0, [$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) then break fi; Lprev:=Lnext; fi; od; return Lnext; end: - Walter A. Kehowski (wkehowski(AT)cox.net), Jun 05 2008

t = 2Range@200 - 1; f[n_] := Block[{k = t[[n]]}, t = Delete[t, Table[{k}, {k, k, Length@t, k}]]]; Do[f@n, {n, 2, 30}]; t (from Robert G. Wilson v (rgwv(at)rgwv.com), May 09 2006)

Cf. A137164-A137185.

Sequence in context: A032678 A073671 A024901 this_sequence A120226 A137310 A118567

Adjacent sequences: A000956 A000957 A000958 this_sequence A000960 A000961 A000962

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com). Entry updated Mar 07 2008

1,1

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.

Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.

In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. Keep in mind that 12 is a canonical base for mathematics in general since any prime greater than 3 is of the form 6k+-1, any prime of the form 4k+1 is a sum of squares while any prime of the form 4k+3 is never a sum of squares and lcm(6,4)=12. - Walter A. Kehowski (wkehowski(AT)cox.net), May 31 2008

John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

f[x_, y_]:=7*x^2+4*x*y+52*y^2; lst={}; Do[Do[p=f[x, y]; If[PrimeQ[p], AppendTo[lst, p]], {y, -4!, 3*4!}], {x, -4!, 3*4!}]; Take[Union[lst], 90] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009]

Union[QuadPrimes[7, 4, 52, 10000], QuadPrimes[7, -4, 52, 10000]] (* see A106856 *)

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.

Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.

Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.

Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.

Cf. A107007, A107003, A139830, A107169, A139831, A107154, A139847.

Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.

Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.

Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Sequence in context: A165878 A142358 A020477 this_sequence A142400 A032460 A101746

Adjacent sequences: A140630 A140631 A140632 this_sequence A140634 A140635 A140636

nonn,easy

T. D. Noe (noe(AT)sspectra.com), May 19 2008

1,1

It appears that the only square in this sequence is 4. Checked 10^11 terms. a(10^11)=247754953701579144582110673365391267. - T. D. Noe (noe(AT)sspectra.com), Sep 06 2005

a(2n-1) is divisible by 2, a(3n+1) is divisible by 3, a(4n-3) is divisible by 4, a(6n+1) is divisible by 6, a(8n-3) is divisible by 8, a(12n+1) is divisible by 12, a(24n-11) is divisible by 24. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 15 2006

The sequence is best looked at in base 12, with X for 10 and E for 11: 4, 11, 32, 73, 154, 275, 476, 717, X98, 1479, 1E3X, 289E, 3860, 4941, 6082, 7823, 9844, EX25, 12546, 15447, 18548, 20089, 2406X, 2876E, 320E0, 37E91, 42152, 488E3, 53754, 5E015, 68416, 76337, 85178, 94399, X51EX, E643E, 108760, 120001. Since the squares of all primes greater than 3 are always 1 mod 12, the sequence obeys the rule a(n) mod 12 = (n-1) mod 12 for n>=2. The rule gives a(2n-1) = (2n-2) mod 12 and so a(2n-1) must be divisible by 2. a(3n+1) = (3n) mod 12 so a(3n+1) is divisible by 3. The other rules are proved similarly. Remember: base 12 is a research tool! - Walter Kehowski (wkehowski(AT)cox.net), Jun 24 2006

For all primes p > 3, we have p^2 = 1 (mod m) for m=2,3,4,6,8,12,24 (and only these m). Using a covering argument, it is not hard to show that all terms except a(24k+13) are nonsquares. Hence in the search for square a(n), only 1 out of every 24 terms needs to be checked. - T. D. Noe (noe(AT)sspectra.com), Jan 23 2008

N. J. A. Sloane, Table of n, a(n) for n = 1..5000

Carlos Rivera, Puzzle 128: Sum of consecutive squared primes a square

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics: Prime Sums

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics: Prime Zeta Function

a(n) = A007504(n)^2 - 2*A024447(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 15 2006

a(n) = sum(prime(i)^2,i=1..n) - Walter Kehowski (wkehowski(AT)cox.net), Jun 24 2006

a(n) ~ 0.40 n^3 log(n)^2 - T. D. Noe (noe(AT)sspectra.com), Jan 23 2008

Table[ Sum[ Prime[k]^2, {k, 1, n} ], {n, 1, 40} ]

Partial sums of A001248. Cf. A033632.

Cf. A007504 (sum of the first n primes).

Sequence in context: A155344 A155418 A155235 this_sequence A047094 A145128 A089092

Adjacent sequences: A024447 A024448 A024449 this_sequence A024451 A024452 A024453

nonn

Clark Kimberling (ck6(AT)evansville.edu)

0,1

Union of A081874 and A081925. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 28 2006

Ron Knott, Pythagorean Triples and Online Calculators

P. Alfeld, Pythagorean Triples

N. Exner, Generating Pythagorean Triples(Applet)

W. A. Kehowski, Pythagorean Triples

lst={}; amx=99; Do[For[b=a+1, b<(a^2/2), c=(a^2+b^2)^(1/2); If[c==IntegerPart[c]&&GCD[a, b, c]==1, AppendTo[lst, a]]; b=b+2], {a, 3, amx}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 07 2008]

Cf. A020882-A020886. Different from A024352.

Cf. A009004.

Sequence in context: A160238 A025050 A025051 this_sequence A024352 A134407 A144724

Adjacent sequences: A020881 A020882 A020883 this_sequence A020885 A020886 A020887

nonn,easy,nice

Clark Kimberling (ck6(AT)evansville.edu)

Extended and corrected by David W. Wilson (davidwwilson(AT)comcast.net)

1,1

In other words, some subset of the numbers { 1 <= d < n : d divides n } adds up to n. - N. J. A. Sloane (njas(AT)research.att.com), Apr 06 2008

Also, numbers n such that A033630(n) > 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 02 2007

By definition, does not include the weird numbers A006037.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Unsolved Problems in Number Theory, B2.

Problem E2308, Amer. Math. Monthly, 79 (1972), 774.

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

Anonymous, Semiperfect Numbers: Definition

David Eppstein, Title?

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

6 = 1+2+3, 12 = 1+2+3+6, 18 = 3+6+9, etc.

70 is not a member since the proper divisors of 70 are {1, 2, 5, 7, 10, 14, 35} and no subset adds to 70.

with(combinat); issemiperfect := proc(n) local b, S; b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if convert(S[nextvalue](), `+`)=n then b:=true; break fi od; return b end: select(proc(z) issemiperfect(z) end, [$1..1000]); - Walter A. Kehowski (wkehowski(AT)cox.net), Aug 12 2005

Flatten[ Position[ A033630, q_/; q>1 ] ] - from wouter.meeussen(at)pandora.be

(PARI from M. F. hasler, Apr 06 2008) isA005835(n, d=0)={ local(t); /* Return nonzero iff n is the sum of a subset of d which defaults to the set of proper divisors of n */

if( !d, /* Initialize d */ d=vecextract(divisors(n), "^-1"), /*else check if n equals one element of d */ setsearch( Set(d), n) & return(1));

/* Remove terms > n */ while( #d>1 & d[ #d]>n, d=vecextract(d, "^-1"));

/* If n is not smaller than the sum of all terms, we're done */ n >= (t = sum(i=1, #d, d[i])) & return( n==t );

/* If n is larger than M=max(d), then try to write n-M in terms of d \ { M } */ n > d[ #d ] & isA005835( n - d[ #d ], vecextract( d, "^-1") ) & return(1); /* else only d \ {M} is needed */ isA005835( n, vecextract( d, "^-1" ))}

for(n=1, 1000, isA005835(n)&print1(n", "))

The complement is A136447.

See A136446 for another version.

Cf. A006036, A005100, A033630.

Sequence in context: A119357 A097216 A023196 this_sequence A007620 A100715 A094519

Adjacent sequences: A005832 A005833 A005834 this_sequence A005836 A005837 A005838

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Better description and more terms from Jud McCranie (j.mccranie(AT)comcast.net) Oct 15 1997

1,1

If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - Farideh Firoozbakht (mymontain(AT)yahoo.com), Jan 28 2006

T. Trotter, Admirable Numbers.

12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.

with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); (Walter A. Kehowski (wkehowski(AT)cox.net), Aug 12 2005)

fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (from Robert G. Wilson v (rgwv(at)rgwv.com), Aug 13 2005)

(PARI) for(n=1, 10^3, ap=sigma(n)-2*n; if(ap>0 && (ap%2)==0, d=ap/2; if(d!=n && (n%d)==0, print1(n", ")))) - Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

Cf. A000396, A005101, A005100, A000203, A061645.

Sequence in context: A136724 A112769 A097320 this_sequence A111947 A109396 A055598

Adjacent sequences: A111589 A111590 A111591 this_sequence A111593 A111594 A111595

easy,nonn

Jason Earls (zevi_35711(AT)yahoo.com), Aug 09 2005

Better definition from Walter A. Kehowski (wkehowski(AT)cox.net), Aug 12 2005

1,1

Write down even numbers: 2 4 6 8 ...; first term > 2 is 4 so starting from 2 remove every 4th number: 2 4 6 10 12 14 18...; next number is 6 so remove every 6th term starting from 2: 2 4 6 10 12 18 20 22 26 etc.

Index entries for sequences generated by sieves

## Finds all Even Lucky Numbers up to n from the list 2...n. ## Try n=10^5 or 10^6 just for fun! evenluckynumbers:=proc(n) local k, Lnext, Lprev; Lprev:=[$2..n]; for k from 1 do Lnext:= map(w-> Lprev[w], remove(z -> z mod Lprev[k] = 0, [$1..nops(Lprev)])); if nops(Lnext)=nops(Lprev) then return Lnext fi; Lprev:=Lnext; od; end: - Walter A. Kehowski (wkehowski(AT)cox.net), Jun 06 2008

lst = Range[2, 308, 2]; i = 2; While[ i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; lst (from Robert G. Wilson v (rgwv(at)rgwv.com), May 11 2006)

A000959, A039672.

Sequence in context: A024907 A033098 A033868 this_sequence A072542 A167856 A162578

Adjacent sequences: A045951 A045952 A045953 this_sequence A045955 A045956 A045957

nice,nonn,easy

Felice Russo (felice.russo(AT)katamail.com)

1,1

Any prime number contains in its digits at least one of the term of this sequence and there is no smaller set. There are 26 terms in the sequence.

J.-P. Delahaye, "Pour la science", (French edition of Scientific American), Juin 2002, p. 99.

J. Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

C. K. Caldwell, The Prime Glossary, minimal prime

J. Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

Cf. A071070, A071071, A071072, A071073.

Sequence in context: A118985 A089769 A033274 this_sequence A002231 A087769 A084736

Adjacent sequences: A071059 A071060 A071061 this_sequence A071063 A071064 A071065

fini,full,nice,nonn,base

Benoit Cloitre (benoit7848c(AT)orange.fr), May 26 2002

Typo corrected by T. D. Noe (noe(AT)sspectra.com), Nov 15 2006

Typo corrected by Walter A. Kehowski (wkehowski(AT)cox.net), Feb 22 2007

1,2

Or, numbers n such that product of aliquot divisors of n is n^2.

A084110(a(n)) = 1, see also A084116. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 12 2003

If M(n) denotes the product of the divisors of n, then n is said to be k-multiplicatively perfect if M(n)=n^k. All such numbers are of the form p q^(k-1) or p^(2k-1). This statement is in Sandor's paper. Therfore all 2-multiplicatively perfect numbers are semiprime p*q or cubes p^3. - Walter A. Kehowski (wkehowski(AT)cox.net), Sep 13 2005

All 2-multiplicatively perfect numbers except 1 have 4 divisors (as implied by Kehowski) and the converse is also true that all numbers with 4 divisors are 2-multiplicatively perfect. [From Howard Berman (howard_berman(AT)hotmail.com), Oct 24 2008]

Also numbers n such that A000005(n) is a square , sigma_0(n)= c^2. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Aug 05 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. Springer-Verlag, NY, 1982, p. 19.

J. Sandor, On multiplicatively e-perfect numbers, J.Ineq.Pure Applied Math(JIPAM), 5(2004), no.4, article 114.

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

J. Sandor, Multiplicatively perfect numbers, J. Ineq. Pure Appl. Math. 2(2001), no. 1, article 3, 6 pp.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

1 together with A030513.

Divisors of 10 are 1,2,5,10 with product 100=10^2.

k:=2: MPL:=[]: for z from 1 to 1 do for n from 1 to 5000 do if convert(divisors(n), `*`) = n^k then MPL:=[op(MPL), n] fi od; od; MPL; (Kehowski)

Sequence in context: A120497 A036436 A036455 this_sequence A030513 A161918 A152126

Adjacent sequences: A007419 A007420 A007421 this_sequence A007423 A007424 A007425

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Some numbers were omitted - thanks to Erich Friedman (erich.friedman(AT)stetson.edu) for pointing this out.

1,2

Is the sequence infinite? Is each prime a(i)+a(j)+1, i<>j, always distinct?

Except for a(1), a(n) == 3 (mod 6). - Robert G. Wilson v Jun 02 2006.

The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n>2, a(n)+2 and a(n)+4 are both primes, so a proof that this sequence is infinite would also show that there are infinitely many twin primes. - N. J. A. Sloane (njas(AT)research.att.com), Apr 21 2007

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio Numerorum' III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.

a(1)=1, a(2)=3, but 5+1+1=7, 5+3+1=9; 7+1+1=9, 7+3+1=11; 9+1+1=11, 9+3+1=13 so a(3)=9.

EP:=[]: for w to 1 do for n from 1 to 8*10^6 do s:=2*n-1; Q:=map(z->z+s+1, EP); if andmap(isprime, Q) then EP:=[op(EP), s]; print(nops(EP), s); fi od od; EP;

a[1] = 1; a[2] = 3; a[n_] := a[n] = Block[{k = a[n - 1] + 6, t = Table[ a[i], {i, n - 1}] + 1}, While[ First@ Union@ PrimeQ[k + t] == False, k += 6]; k]; Do[ Print[ a[n]], {n, 15}] - Robert G. Wilson v (rgwv(at)rgwv.com), Jun 03 2006

Cf. A118818, A093483, A128933 (a(n)+1), A115760 (2*a(n)+1). Primes arising from this sequence are in A115782.

Sequence in context: A161712 A137368 A036215 this_sequence A110740 A042938 A084707

Adjacent sequences: A103825 A103826 A103827 this_sequence A103829 A103830 A103831

easy,nonn

Walter Kehowski (wkehowski(AT)cox.net), May 29 2006

a(12) from Robert G. Wilson v (rgwv(at)rgwv.com), Jun 03 2006

a(13) from Walter Kehowski (wkehowski(AT)cox.net), Jun 03 2006

Corrected definition. - walter kehowski (wkehowski(AT)cox.net), Nov 03 2008