1,2

Since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power.

These numbers are (apart from 1) the numbers of elements in finite fields. - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Aug 11 2004

Numbers whose divisors form a geometrical progresion. The divisors of p^k are 1, p, p^2, p^3, ...p^k. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 09 2002

a(n) = A025473(n)^A025474(n). - David Wasserman (wasserma(AT)spawar.navy.mil), Feb 16 2006

a(n) = A117331(A117333(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 08 2006

These are also precisely the orders of those finite affine planes that are known to exist as of today. (The order of a finite affine plane is the number of points in an arbitrarily chosen line of that plane. This number is unique for all lines comprise the same number of points.) - Peter C. Heinig (algorithms(AT)gmx.de), Aug 09 2006

Except for first term, the index of the second number divisible by n in A002378, if the index equals n. - Mats Granvik (mgranvik(AT)abo.fi), Nov 18 2007

These are precisely the numbers such that lcm(1,...,m-1)<lcm(1,...,m) (=A003418(m) for m>0; here for m=1, the l.h.s. is taken to be 0). We have a(n+1)=a(n)+1 iff n is a Mersenne prime or n+1 is a Fermat prime. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 18 2007

The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. - Zak Seidov (zakseidov(AT)yahoo.com), Feb 06 2008

Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] yields a prime at x=1, cf. A020500. - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 04 2008

Also, {A138929(k) ; k>1} = {2*A000961(k) ; k>1} = {4,6,8,10,14,16,18,22,26,32,34,38,46,50,54,58,62,64,74,82,86,94,98,...} are exactly the indices for which Phi[k](-1) is prime. - M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 04 2008

A143201(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 12 2008]

Number of distinct primes dividing n=omega(n)<2. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 30 2009]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993

R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.

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).

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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.

Index entries for "core" sequences

m=a(n) for some n <=> lcm(1,...,m-1)<lcm(1,...,m), where lcm(1...0):=0 as to include a(1)=1. a(n+1)=a(n)+1 <=> a(n+1)=A019434(k) or a(n)=A000668(k) for some k (by Catalan's conjecture). - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 18 2007

A001221(a(n))<2. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 30 2009]

readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d, `, n) fi: od:

Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]

Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]

max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]^m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 1000}]; a(*Artur Jasinski*)

(MAGMA) [ n : n in [1..1000] | IsPrimePower(n) ];

(PARI) A000961(n, l=-1, k=0)=until(n--<1, until(l<lcm(l, k++), ); l=lcm(l, k)); k print_A000961(lim=999, l=-1)=for(k=1, lim, l==lcm(l, k)&next; l=lcm(l, k); print1(k, ", ")) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 18 2007

(PARI) isA000961(n) = (omega(n) == 1 | n == 1) [From Michael Porter (michael_b_porter(AT)yahoo.com), Sep 23 2009]

(PARI) nextA000961(n) = {local(m, r, p); m=2*n; for(e=1, ceil(log(n+0.01)/log(2)), r=(n+0.01)^(1/e); p=prime(primepi(r)+1); m=min(m, p^e)); m} [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 02 2009]

Cf. A010055, A065515, A095874, A025473.

Cf. indices of record values of A003418; A000668 and A019434 give a member of twin pairs a(n+1)=a(n)+1.

A138929(n) = 2*a(n).

Sequence in context: A059046 A144711 A036116 this_sequence A128603 A096165 A164336

Adjacent sequences: A000958 A000959 A000960 this_sequence A000962 A000963 A000964

nonn,easy,core,nice,new

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