A322078 a(n) = n^2 * Sum_{p|n} 1/p^2, where p are primes dividing n.

0, 1, 1, 4, 1, 13, 1, 16, 9, 29, 1, 52, 1, 53, 34, 64, 1, 117, 1, 116, 58, 125, 1, 208, 25, 173, 81, 212, 1, 361, 1, 256, 130, 293, 74, 468, 1, 365, 178, 464, 1, 673, 1, 500, 306, 533, 1, 832, 49, 725, 298, 692, 1, 1053, 146, 848, 370, 845, 1, 1444, 1, 965, 522

Offset

1,4

Comments

Generalized formula is f(n,m) = n^m * Sum_{p|n} 1/p^m, where f(n,0) = A001221(n) and f(n,1) = A069359(n).

Links

Daniel Suteu, Table of n, a(n) for n = 1..10000

Formula

Sum_{k=1..n} a(k) ~ A085541 * A000330(n).

G.f.: Sum_{k>=1} x^prime(k) * (1 + x^prime(k)) / (1 - x^prime(k))^3. - Ilya Gutkovskiy, Oct 10 2019

a(n) = 1 <=> n is prime. _ Alois P. Heinz, Oct 11 2019

Dirichlet g.f.: zeta(s-2)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^2/n^s) Sum_{p|n} 1/p^2. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^2*(p*j)^(s-2)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-2) = zeta(s-2)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 02 2023

Example

a(40) = 464 because the prime factors of 40 are 2 and 5, so we have 40^2 * (1/2^2 + 1/5^2) = 464.

Maple

a:= n-> n^2*add(1/i[1]^2, i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 11 2019

Mathematica

f[p_, e_] := 1/p^2; a[n_] := If[n==1, 0, n^2*Plus@@f@@@FactorInteger[n]]; Array[a, 60] (* Amiram Eldar, Nov 26 2018 *)

Prog

(PARI) a(n) = my(f=factor(n)[, 1]~); sum(k=1, #f, n^2\f[k]^2);
(Magma) [0] cat [n^2*&+[1/p^2:p in PrimeDivisors(n)]:n in [2..70]]; // Marius A. Burtea, Oct 10 2019

Crossrefs

Cf. A000040, A085541.

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), this sequence (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Sequence in context: A373394 A002564 A287640 * A019428 A303547 A184753

Adjacent sequences: A322075 A322076 A322077 * A322079 A322080 A322081

Keyword

nonn

Author

Daniel Suteu, Nov 25 2018

Status

approved