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%I M0517 N0185
%S 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,
%T 59,61,64,67,71,73,79,81,83,89,97,101,103,107,109,113,121,125,127,128,
%U 131,137,139,149,151,157,163,167,169,173,179,181,191,193,197,199,211,223,227
%N Prime powers p^k (p prime, k >= 0).
%C Since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power.
%C These numbers are (apart from 1) the numbers of elements in finite fields. - _Franz Vrabec_, Aug 11 2004
%C Numbers whose divisors form a geometrical progression. The divisors of p^k are 1, p, p^2, p^3, ...p^k. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 09 2002
%C a(n) = A025473(n)^A025474(n). - _David Wasserman_, Feb 16 2006
%C a(n) = A117331(A117333(n)). - _Reinhard Zumkeller_, Mar 08 2006
%C 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
%C 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
%C 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
%C The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. - Zak Seidov, Feb 06 2008
%C 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
%C 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
%C A143201(a(n)) = 1. [From _Reinhard Zumkeller_, Aug 12 2008]
%C Number of distinct primes dividing n=omega(n)<2. [From Juri-Stepan Gerasimov, Oct 30 2009]
%C Or, prime numbers^nonnegative numbers (without repetition). 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). [From Juri-Stepan Gerasimov, Dec 09 2009, Mar 10 2010]
%C Numbers n such that A028236(n) = 1. [From Klaus Brockhaus, Nov 06 2010]
%C 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]
%C A003415(a(n)) = A192015(n); A068346(a(n)) = A192016(n); a(n)=A192134(n)+A192015(n). [_Reinhard Zumkeller_, Jun 26 2011]
%D 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.
%D M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993
%D R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000961/b000961.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProjectivePlane.html">Projective Plane</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F 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
%F A001221(a(n))<2. [From Juri-Stepan Gerasimov, Oct 30 2009]
%p readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d,`,n) fi: od:
%p A000961 := proc(n)
%p option remember;
%p local k ;
%p if n = 1 then
%p 1;
%p else
%p for k from procname(n-1)+1 do
%p if nops(numtheory[factorset](k)) = 1 then
%p return k ;
%p end if;
%p end do:
%p end if;
%p end proc: # _Alois P. Heinz_, Apr 08 2013
%t Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
%t Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]
%t 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*)
%o (MAGMA) [1] cat [ n : n in [2..250] | IsPrimePower(n) ]; // corrected by Arkadiusz Wesolowski, Jul 20 2012
%o (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
%o (PARI) isA000961(n) = (omega(n) == 1 | n == 1) [From _Michael B. Porter_, Sep 23 2009]
%o (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 B. Porter_, Nov 02 2009]
%o (PARI) is(n)=isprimepower(n) || n==1 \\ _Charles R Greathouse IV_, Nov 20 2012
%o (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
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, insert)
%o a000961 n = a000961_list !! (n-1)
%o a000961_list = 1 : g (singleton 2) (tail a000040_list) where
%o g s (p:ps) = m : g (insert (m * a020639 m) $ insert p s') ps
%o where (m, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, May 01 2012, Apr 25 2011
%o (Sage)
%o def A000961_list(n) :
%o R = [1]
%o for i in (2..n) :
%o if i.is_prime_power() : R.append(i)
%o return R
%o A000961_list(227) # Peter Luschny, Feb 07 2012
%Y Cf. A010055, A065515, A095874, A025473.
%Y Cf. indices of record values of A003418; A000668 and A019434 give a member of twin pairs a(n+1)=a(n)+1.
%Y A138929(n) = 2*a(n).
%Y Cf. A000040, A001221, A001477. [From Juri-Stepan Gerasimov, Dec 09 2009]
%Y A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). [From Klaus Brockhaus, Nov 06 2010]
%Y A000015(n) = Min{term : >= n}; A031218(n) = Max{term : <= n}.
%K nonn,easy,core,nice
%O 1,2
%A _N. J. A. Sloane_.
%E Corrected comment and formula referring to Catalan's conjecture _M. F. Hasler_, Apr 18 2010
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