A235137 a(n) = Sum_{k = 1..n} k^phi(n), where phi(n) = A000010(n).

1, 3, 14, 30, 979, 91, 184820, 8772, 978405, 25333, 40851766526, 60710, 36720042483591, 19092295, 5666482312, 9961449608, 76762718946972480009, 105409929, 164309788542828686799730, 70540730666, 15909231318568907, 67403375450475, 1433191209985108404653810959324, 351625763020, 15975648280734359596251725645

Offset: 1

Author

Jonathan Sondow and Emmanuel Tsukerman, Jan 03 2014

Keywords

• nonn

Comments

a(n) == -1 (mod n) if and only if n is prime or is a Giuga number A007850.

a(n) == 1 (mod n) if (and probably only if) n is a primary pseudoperfect number A054377.

Examples

			a(4) = 30 since 1^(phi(4)) + 2^(phi(4)) + 3^(phi(4)) + 4^(phi(4))= 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30.
a(5) = 979, since phi(5) = 4 and 1^4 + 2^4 + 3^4 + 4^4 + 5^4 = 1 + 16 + 81 + 256 + 625 = 979.
a(6) = 91, since phi(6) = 2 and 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..388
• J. Sondow and K. MacMillan, Reducing the Erdős-Moser equation 1^n + 2^n + ... + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.
• J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
• Wikipedia, Giuga number
• Wikipedia, Primary pseudoperfect number

Crossrefs

Cf. A000010, A007850, A054377, A235138.

Programs

Mathematica

a[n_] := Sum[PowerMod[i, EulerPhi@n, n], {i, n}]

PARI

a(n) = sum(k=1, n , k^eulerphi(n)); \\ Michel Marcus, Oct 21 2015

Formula

a(n) (mod n) = A235138(n).