A061345 - OEIS

%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, Unpublished Bell Labs Memo, Sep 07 1967 [Annotated scanned copy]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