A117657 Number of solutions to x^(k+3)=x^3 mod n for some k>=1.
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, 27, 28, 29, 30, 31, 24, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 36, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 48, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
- Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.
Programs
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Mathematica
a[n_] := Product[{p, e} = pe; p^Floor[2e/3] + (p-1) p^(e-1), {pe, FactorInteger[n]}]; Array[a, 72] (* Jean-François Alcover, Nov 11 2018, after Andrew Howroyd *) -
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
Formula
Multiplicative with a(p^e) = p^floor(2*e/3) + (p-1)*p^(e-1) for prime p. - Andrew Howroyd, Jul 17 2018
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