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 A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1). (Formerly M0517 N0185) 873
 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 (list; graph; refs; listen; history; text; internal format)
 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 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,...,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 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 2^1-1 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. - Juri-Stepan Gerasimov, Oct 30 2009 Numbers n such that Sum_{p-1|p is prime and divisor of n} = Product_{p-1|p is prime and divisor of n}. A055631(n) = A173557(n-1). - Juri-Stepan 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 n-cycle. - W. Edwin Clark, Aug 05 2014 Conjecture: these are numbers m such that Sum_{k=0..m-1} 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 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 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 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,...,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), except for n=1 and n=7. - M. F. Hasler, Jan 18 2007, Apr 18 2010 A001221(a(n)) < 2. - Juri-Stepan 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))=1 then printf(`%d, `, n) fi: od: # second Maple program: a:= proc(n) option remember; local k; for k from       1+a(n-1) while nops(ifactors(k))>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]][]^m[[k]][], {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]] (* Jean-François Alcover, Jul 07 2015 *) PROG (Magma)  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= 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. - Juri-Stepan 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 EXTENSIONS Description modified by Ralf Stephan, Aug 29 2014 STATUS approved

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Last modified September 30 20:49 EDT 2022. Contains 357106 sequences. (Running on oeis4.)