

A000961


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


774



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

Of course 1 = p^0 for any prime p, so 1 is definitely the power of a prime.
The term "prime power" is ambiguous. To a mathematician it means any number p^k, p prime, k >= 0, including 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
a(n) = A025473(n)^A025474(n).  David Wasserman, Feb 16 2006
a(n) = A117331(A117333(n)).  Reinhard Zumkeller, 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, 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
The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015].  Zak Seidov, 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, 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
A143201(a(n)) = 1.  Reinhard Zumkeller, Aug 12 2008
Number of distinct primes dividing n=omega(n) < 2.  JuriStepan Gerasimov, Oct 30 2009
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
Numbers n such that A028236(n) = 1. Klaus Brockhaus, Nov 06 2010
A188666(k) = a(k+1) for k: 2*a(k) <= k < 2*a(k+1), k > 0; notably a(n+1) = A188666(2*a(n)).  Reinhard Zumkeller, Apr 25 2011
A003415(a(n)) = A192015(n); A068346(a(n)) = A192016(n); a(n)=A192134(n) + A192015(n).  Reinhard Zumkeller, Jun 26 2011
A089233(a(n)) = 0.  Reinhard Zumkeller, Sep 04 2013
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
Conjecture: these are numbers m such that Sum_{k=0..m1} k^phi(m) == phi(m) (mod m), where phi(m) = A000010(m).  Thomas Ordowski and Giovanni Resta, Jul 25 2018
Numbers whose (increasingly ordered) divisors are alternatingly squares and nonsquares.  Michel Marcus, Jan 16 2019


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

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].
Brady Haran and Günter Ziegler, Cannons and Sparrows, Numberphile video (2018).
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Projective Plane
Index entries for "core" sequences


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
A001221(a(n)) < 2.  JuriStepan Gerasimov, Oct 30 2009
A008480(a(n)) = 1 for all n >= 1.  Alois P. Heinz, May 26 2018


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
end: a(1):=1: A000961:= a:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 08 2013


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 *)
Join[{1}, Select[Range[2, 250], PrimePowerQ]] (* JeanFrançois Alcover, Jul 07 2015 *)


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) isA000961(n) = (omega(n) == 1  n == 1) \\ Michael B. Porter, Sep 23 2009
(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) is(n)=isprimepower(n)  n==1 \\ Charles R Greathouse IV, Nov 20 2012
(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
 Reinhard Zumkeller, May 01 2012, Apr 25 2011
(Sage)
def A000961_list(n):
R = [1]
for i in (2..n):
if i.is_prime_power(): R.append(i)
return R
A000961_list(227) # Peter Luschny, Feb 07 2012
(Python)
from sympy import primerange
A000961_list = [1]
for p in primerange(1, m):
pe = p
while pe < m:
A000961_list.append(pe)
pe *= p
A000961_list = sorted(A000961_list) # Chai Wah Wu, Sep 08 2014


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
Cf. A008480, 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).
Cf. A000040, A001221, A001477.  JuriStepan Gerasimov, Dec 09 2009
A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j).  Klaus Brockhaus, Nov 06 2010
A000015(n) = Min{term : >= n}; A031218(n) = Max{term : <= n}.
Complementary (in the positive integers) to sequence A024619.  Jason Kimberley, Nov 10 2015
Sequence in context: A337935 A036116 A246655 * A128603 A195943 A096165
Adjacent sequences: A000958 A000959 A000960 * A000962 A000963 A000964


KEYWORD

nonn,easy,core,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Description modified by Ralf Stephan, Aug 29 2014


STATUS

approved



