A343570 If n = Product (p_j^k_j) then a(n) = Product (p_j^k_j + 3), with a(1) = 1.

1, 5, 6, 7, 8, 30, 10, 11, 12, 40, 14, 42, 16, 50, 48, 19, 20, 60, 22, 56, 60, 70, 26, 66, 28, 80, 30, 70, 32, 240, 34, 35, 84, 100, 80, 84, 40, 110, 96, 88, 44, 300, 46, 98, 96, 130, 50, 114, 52, 140, 120, 112, 56, 150, 112, 110, 132, 160, 62, 336, 64, 170, 120, 67, 128, 420, 70, 140, 156, 400, 74

Offset: 1

Author

Ilya Gutkovskiy, Apr 20 2021

Keywords

• nonn
• mult

Comments

The unitary analog of A007430.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A007430, A034444, A034448, A107759.

Programs

Mathematica

a[1] = 1; a[n_] := Times @@ ((#[[1]]^#[[2]] + 3) & /@ FactorInteger[n]); Table[a[n], {n, 71}]

PARI

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,1]^f[k,2] + 3; f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 20 2021

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d) * 2^omega(n/d).

a(n) = Sum_{d|n, gcd(d, n/d) = 1} A107759(d).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2 - 3/p^3) = 1.1848008127... . - Amiram Eldar, Nov 13 2022