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%I #51 Feb 02 2023 12:46:50
%S 1,3,5,7,9,11,13,17,19,23,25,27,29,31,37,41,43,47,49,53,59,61,67,71,
%T 73,79,81,83,89,97,101,103,107,109,113,121,125,127,131,137,139,149,
%U 151,157,163,167,169,173,179,181,191,193,197,199,211,223,227,229,233,239
%N Odd prime powers.
%C Let a(n)=p^e, then tau(a(n)^2) = tau(p^(2*e)) = 2*e+1 = 2*(e+1)-1 = tau(2*a(n))-1 where tau=A000005. - _Juri-Stepan Gerasimov_, Jul 14 2011
%H Robert Israel, <a href="/A061345/b061345.txt">Table of n, a(n) for n = 0..10000</a>
%H L. J. Corwin, <a href="/A033948/a033948.pdf">Irreducible polynomials over the integers which factor mod p for every p</a>, Unpublished Bell Labs Memo, Sep 07 1967 [Annotated scanned copy]
%F a(n) = A061344(n)-1.
%F Intersection of A000961 (prime powers) and A005408 (odd numbers). - _Robert Israel_, Jun 11 2014
%p select(t -> nops(ifactors(t)[2])<=1, [seq(2*i+1,i=0..1000)]); # _Robert Israel_, Jun 11 2014
%p # alternative:
%p A061345 := proc(n)
%p option remember;
%p local k ;
%p if n = 0 then
%p 1;
%p else
%p for k from procname(n-1)+2 by 2 do
%p if nops(numtheory[factorset](k)) = 1 then
%p return k ;
%p end if;
%p end do:
%p end if;
%p end proc: # _R. J. Mathar_, Jun 25 2016
%p isOddPrimepower := n -> type(n, 'primepower') and not type(n, 'even'):
%p A061345List := up_to -> select(isOddPrimepower, [`$`(1..up_to)]):
%p A061345List(240); # _Peter Luschny_, Feb 02 2023
%t t={1};k=0;Do[If[k==1,AppendTo[t,a1]];k=0;Do[c=Sqrt[a^2+b^2];If[IntegerQ[c]&&GCD[a,b,c]==1,k++;a1=a;b1=b;c1=c;],{b,4,a^2/2,2}],{a,3,260,2}];t (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)
%t Select[2 Range@ 130 - 1, PrimeNu@# < 2 &] (* _Robert G. Wilson v_, Jun 12 2014 *)
%o (Magma) [1] cat [n: n in [3..300 by 2] | IsPrimePower(n)]; // _Bruno Berselli_, Feb 25 2016
%o (PARI) is(n)=my(p); if(isprimepower(n,&p), p>2, n==1) \\ _Charles R Greathouse IV_, Jun 08 2016
%Y Cf. A061346, A000961, A005408, A061344, A075109, A075110.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 08 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2001
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