OFFSET
0,3
LINKS
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
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Jul 02 2013
EXTENSIONS
More terms from Antti Karttunen, Nov 23 2017
STATUS
approved