A167344 - OEIS

%I #22 Nov 12 2022 05:25:17

%S 1,3,8,9,24,24,48,27,64,72,120,72,168,144,192,81,288,192,360,216,384,

%T 360,528,216,576,504,512,432,840,576,960,243,960,864,1152,576,1368,

%U 1080,1344,648,1680,1152,1848,1080,1536,1584,2208,648,2304,1728

%N Totally multiplicative sequence with a(p) = (p-1)*(p+1) = p^2-1 for prime p.

%H G. C. Greubel, <a href="/A167344/b167344.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = ((p-1)*(p+1))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+1))^e(k).

%F a(n) = A003958(n) * A003959(n).

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 - 2)) = 1.884261780923861906728291280746835210118330549695678826316037127832097567... - _Vaclav Kotesovec_, Sep 20 2020

%F a(n) = A340323(n) * A340368(n). - _Antti Karttunen_, Jan 31 2021

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - 1/(p^3 - p^2 + 1)) = 0.2487962948... . - _Amiram Eldar_, Nov 12 2022

%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* _G. C. Greubel_, Jun 10 2016 *)

%o (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,1]^2-1); factorback(f); \\ _Michel Marcus_, Jan 31 2021

%Y Cf. A003958, A003959, A306709, A340323, A340368.

%Y Cf. also A335915.

%K nonn,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009