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Powerful numbers whose prime factors are all of the form 4*k + 1.
5

%I #10 Mar 06 2024 17:25:08

%S 1,25,125,169,289,625,841,1369,1681,2197,2809,3125,3721,4225,4913,

%T 5329,7225,7921,9409,10201,11881,12769,15625,18769,21025,21125,22201,

%U 24389,24649,28561,29929,32761,34225,36125,37249,38809,42025,48841,50653,52441,54289,54925

%N Powerful numbers whose prime factors are all of the form 4*k + 1.

%C Closed under multiplication.

%H Amiram Eldar, <a href="/A369563/b369563.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F Sum_{n>=1} 1/a(n) = Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) = A175647 * A334424 = 1.0654356335... .

%t q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 1 && Last[#] > 1 &]; Select[Range[50000], q]

%o (PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 1 || f[i, 2] == 1, return(0))); 1;}

%Y Intersection of A001694 and A004613.

%Y Subsequence: A146945.

%Y Similar sequence: A352492, A369564, A369565, A369566.

%Y Cf. A002144, A008846, A175647, A334424.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jan 26 2024