A253629 Multiplicative function defined for prime powers by a(p^e) = p^(e-1)(p+1) if p > 2 and a(2^e) = 2^(e-1).

1, 1, 4, 2, 6, 4, 8, 4, 12, 6, 12, 8, 14, 8, 24, 8, 18, 12, 20, 12, 32, 12, 24, 16, 30, 14, 36, 16, 30, 24, 32, 16, 48, 18, 48, 24, 38, 20, 56, 24, 42, 32, 44, 24, 72, 24, 48, 32, 56, 30, 72, 28, 54, 36, 72, 32, 80, 30, 60, 48, 62, 32, 96, 32, 84, 48, 68, 36, 96, 48, 72, 48, 74, 38, 120, 40, 96, 56, 80, 48, 108, 42, 84, 64, 108, 44, 120, 48, 90, 72, 112, 48

Offset

1,3

Comments

This arithmetic function is a sort of modification of the Dedekind psi function (A001615). This modification is made in order to construct the additive arithmetic function A253630.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Colin Defant, An arithmetic function arising from the Dedekind psi function, arXiv:1501.00971 [math.NT], 2015.

Formula

a(1) is put to 1.

a(n) = A001615(n) if n is odd and a(n) = A001615(n)/3 if n is even.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 9/(2*Pi^2) = 0.4559453... (A088245). - Amiram Eldar, Nov 30 2022

Maple

seq(x * mul(`if`(p>2, p+1, 1)/p, p=numtheory:-factorset(x)), x = 1..100);
# Robert Israel, Jan 08 2015

Mathematica

Table[If[EvenQ[n], (1/3) If[n > 1, n Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1], If[n > 1, n Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1]], {n, 260}]

Prog

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]-1)*if(f[i, 1]>2, f[i, 1]+1, 1)) \\ Charles R Greathouse IV, Jan 08 2015

Crossrefs

Cf. A001615 (Dedekind psi function), A088245, A253630.

Sequence in context: A161912 A162339 A200697 * A327095 A176836 A329287

Adjacent sequences: A253626 A253627 A253628 * A253630 A253631 A253632

Keyword

nonn,mult

Author

Colin Defant, Jan 06 2015

Status

approved