cp's OEIS Frontend

A117657 Number of solutions to x^(k+3)=x^3 mod n for some k>=1.

%I A117657 #19 Sep 08 2020 05:50:23
%S A117657 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,12,17,18,19,20,21,22,23,24,25,26,
%T A117657 27,28,29,30,31,24,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,36,49,
%U A117657 50,51,52,53,54,55,56,57,58,59,60,61,62,63,48,65,66,67,68,69,70,71,72
%N A117657 Number of solutions to x^(k+3)=x^3 mod n for some k>=1.
%H A117657 Amiram Eldar, <a href="/A117657/b117657.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%H A117657 Steven R. Finch, <a href="https://arxiv.org/abs/math/0605019">Idempotents and Nilpotents Modulo n</a>, arXiv:math/0605019 [math.NT], 2006-2017.
%F A117657 Multiplicative with a(p^e) = p^floor(2*e/3) + (p-1)*p^(e-1) for prime p. - _Andrew Howroyd_, Jul 17 2018
%F A117657 Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 - 1/(p^2*(p^3 + p^2 + p + 1))) = 0.49014556800434888957... - _Amiram Eldar_, Sep 08 2020
%t A117657 a[n_] := Product[{p, e} = pe; p^Floor[2e/3] + (p-1) p^(e-1), {pe, FactorInteger[n]}];
%t A117657 Array[a, 72] (* _Jean-François Alcover_, Nov 11 2018, after _Andrew Howroyd_ *)
%o A117657 (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); p^(2*e\3) + (p-1)*p^(e-1))} \\ _Andrew Howroyd_, Jul 17 2018
%Y A117657 Cf. A117656, A117660.
%K A117657 mult,nonn
%O A117657 1,2
%A A117657 _Steven Finch_, Apr 11 2006