

A000961


Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).
(Formerly M0517 N0185)


891



1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227
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OFFSET

1,2


COMMENTS

The term "prime power" is ambiguous. To a mathematician it means any number p^k, p prime, k >= 0, including p^0 = 1.
Any nonzero integer is a product of primes and units, where the units are +1 and 1. This is tied to the Fundamental Theorem of Arithmetic which proves that the factorizations are unique up to order and units. (So, since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power. See A246655 for the sequence without 1.)
These numbers are (apart from 1) the numbers of elements in finite fields.  Franz Vrabec, Aug 11 2004
Numbers whose divisors form a geometrical progression. The divisors of p^k are 1, p, p^2, p^3, ..., p^k.  Amarnath Murthy, Jan 09 2002
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, Nov 18 2007
These are precisely the numbers such that lcm(1,...,m1) < 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 if a(n) is a Mersenne prime or a(n)+1 is a Fermat prime; the converse is true except for n=7 (from Catalan's conjecture) and n=1, since 2^11 and 2^0+1 are not considered as Mersenne resp. Fermat prime.  M. F. Hasler, Jan 18 2007, Apr 18 2010
Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] yields a prime at x=1, cf. A020500.  M. F. Hasler, 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, Apr 04 2008
Numbers n such that Sum_{p1p is prime and divisor of n} = Product_{p1p is prime and divisor of n}. A055631(n) = A173557(n1).  JuriStepan Gerasimov, Dec 09 2009, Mar 10 2010
The positive integers n such that every element of the symmetric group S_n which has order n is an ncycle.  W. Edwin Clark, Aug 05 2014
Numbers whose (increasingly ordered) divisors are alternatingly squares and nonsquares.  Michel Marcus, Jan 16 2019
Possible numbers of elements in a finite vector space.  Jianing Song, Apr 22 2021


REFERENCES

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).


LINKS

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].


FORMULA

Panaitopol (2001) gives many properties, inequalities and asymptotics (including a(n) ~ pi(n)).  N. J. A. Sloane, Oct 31 2014
m=a(n) for some n <=> lcm(1,...,m1) < 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), except for n=1 and n=7.  M. F. Hasler, Jan 18 2007, Apr 18 2010


MAPLE

readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d, `, n) fi: od:
# second Maple program:
a:= proc(n) option remember; local k; for k from
1+a(n1) while nops(ifactors(k)[2])>1 do od; k


MATHEMATICA

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 *)


PROG

(Magma) [1] cat [ n : n in [2..250]  IsPrimePower(n) ]; // corrected by Arkadiusz Wesolowski, Jul 20 2012
(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, Jan 18 2007
(PARI) nextA000961(n)=my(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 \\ Michael B. Porter, Nov 02 2009
(PARI) list(lim)=my(v=primes(primepi(lim)), u=List([1])); forprime(p=2, sqrtint(lim\1), for(e=2, log(lim+.5)\log(p), listput(u, p^e))); vecsort(concat(v, Vec(u))) \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
import Data.Set (singleton, deleteFindMin, insert)
a000961 n = a000961_list !! (n1)
a000961_list = 1 : g (singleton 2) (tail a000040_list) where
g s (p:ps) = m : g (insert (m * a020639 m) $ insert p s') ps
where (m, s') = deleteFindMin s
(Sage)
R = [1]
for i in (2..n):
if i.is_prime_power(): R.append(i)
return R
(Python)
from sympy import primerange
def A000961_list(limit): # following Python style, list terms < limit
L = [1]
for p in primerange(1, limit):
pe = p
while pe < limit:
L.append(pe)
pe *= p


CROSSREFS

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2.  N. J. A. Sloane, Mar 24 2018


KEYWORD

nonn,easy,core,nice


AUTHOR



EXTENSIONS



STATUS

approved



