cp's OEIS Frontend

A097706 Part of n composed of prime factors of form 4k+3.

%I A097706 #35 Jun 28 2022 17:16:46
%S A097706 1,1,3,1,1,3,7,1,9,1,11,3,1,7,3,1,1,9,19,1,21,11,23,3,1,1,27,7,1,3,31,
%T A097706 1,33,1,7,9,1,19,3,1,1,21,43,11,9,23,47,3,49,1,3,1,1,27,11,7,57,1,59,
%U A097706 3,1,31,63,1,1,33,67,1,69,7,71,9,1,1,3,19,77,3,79,1,81,1,83,21
%N A097706 Part of n composed of prime factors of form 4k+3.
%C A097706 Largest term of A004614 that divides n. - _Peter Munn_, Apr 15 2021
%H A097706 Alois P. Heinz, <a href="/A097706/b097706.txt">Table of n, a(n) for n = 1..10000</a>
%F A097706 a(n) = n/A072436(n).
%F A097706 a(A004614(n)) = A004614(n).
%F A097706 a(A072437(n)) = 1.
%F A097706 a(n) = A000265(n)/A170818(n). - _Peter Munn_, Apr 15 2021
%p A097706 a:= n-> mul(`if`(irem(i[1], 4)=3, i[1]^i[2], 1), i=ifactors(n)[2]):
%p A097706 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 09 2014
%t A097706 a[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}]; Array[a, 100] (* _Jean-François Alcover_, Jun 16 2015, updated May 29 2019 *)
%o A097706 (PARI) a(n)=local(f); f=factor(n); prod(k=1, matsize(f)[1], if(f[k, 1]%4<>3, 1, f[k, 1]^f[k, 2]))
%o A097706 (Python)
%o A097706 from sympy import factorint
%o A097706 from operator import mul
%o A097706 def a072436(n):
%o A097706     f=factorint(n)
%o A097706     return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])
%o A097706 def a(n): return n/a072436(n) # _Indranil Ghosh_, May 08 2017
%o A097706 (Python)
%o A097706 from math import prod
%o A097706 from sympy import factorint
%o A097706 def A097706(n): return prod(p**e for p, e in factorint(n).items() if p & 3 == 3) # _Chai Wah Wu_, Jun 28 2022
%Y A097706 Equivalent sequence for distinct prime factors: A170819.
%Y A097706 Equivalent sequences for prime factors of other forms: A000265 (2k+1), A170818 (4k+1), A072436 (not 4k+3), A248909 (6k+1), A343431 (6k+5).
%Y A097706 Range of values: A004614.
%Y A097706 Positions of 1's: A072437.
%Y A097706 Cf. also A065338, A065339, A260728, A286363.
%K A097706 nonn,mult,easy
%O A097706 1,3
%A A097706 _Ralf Stephan_, Aug 30 2004