A253629 - OEIS

%I #30 Nov 30 2022 10:01:51

%S 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,

%T 16,30,24,32,16,48,18,48,24,38,20,56,24,42,32,44,24,72,24,48,32,56,30,

%U 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

%N 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).

%C 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.

%H G. C. Greubel, <a href="/A253629/b253629.txt">Table of n, a(n) for n = 1..1000</a>

%H Colin Defant, <a href="http://arxiv.org/abs/1501.00971"> An arithmetic function arising from the Dedekind psi function</a>, arXiv:1501.00971 [math.NT], 2015.

%F a(1) is put to 1.

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

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

%p seq(x * mul(`if`(p>2,p+1,1)/p, p=numtheory:-factorset(x)), x = 1..100);

%p # _Robert Israel_, Jan 08 2015

%t 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}]

%o (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

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

%K nonn,mult

%O 1,3

%A _Colin Defant_, Jan 06 2015