A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

1, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 6, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 18, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 24, 140

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..16384

Index entries for sequences related to sums of divisors.

Formula

a(n) is multiplicative with a(0) = 1, a(5^e) = 6 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.

G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = 25 (t/i)^2 f(t) where q = exp(2 Pi i t).

G.f.: 1 + Sum_{k>0} k * x^k / (1 - x^k) - Sum_{k>0} 25 * k * x^(25*k) / (1 - x^(25*k)).

Sum_{k=1..n} a(k) ~ (2*Pi^2/25) * n^2. - Amiram Eldar, Oct 04 2022

Example

G.f. = 1 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + ...
75 has six divisors: 1, 3, 5, 15, 25, 75, but both 25 and 75 are divisible by 25, thus not counted, and we have a(75) = 1+3+5+15 = 24. - Antti Karttunen, Nov 23 2017

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ If[ Mod[ d, 25] > 0, d, 0], {d, Divisors @ n}]];

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, if( d%25, d)))};
(PARI) {a(n) = if( n<1, n==0, 1 * (sigma(n) - if( n%25==0, 25 * sigma( n / 25))))};
(Sage) A = ModularForms( Gamma0(25), 2, prec=66) . basis(); A[0] + A[1] + 3*A[2] + 4*A[3] + 7*A[4];
(Magma) A := Basis( ModularForms( Gamma0(25), 2), 66); A[1] + A[2] + 3*A[3] + 4*A[4] + 7*A[5]; /* Michael Somos, Jun 12 2014 */

Crossrefs

Cf. A000118, A004011, A008653, A028594, A028887, A096726, A107505.

Sequence in context: A365174 A325317 A325316 * A365209 A097863 A287926

Adjacent sequences: A227128 A227129 A227130 * A227132 A227133 A227134

Keyword

nonn,mult

Author

Michael Somos, Jul 02 2013

Extensions

More terms from Antti Karttunen, Nov 23 2017

Status

approved