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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).
2
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
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
KEYWORD
nonn,mult
AUTHOR
Colin Defant, Jan 06 2015
STATUS
approved