A167344 Totally multiplicative sequence with a(p) = (p-1)*(p+1) = p^2-1 for prime p.

1, 3, 8, 9, 24, 24, 48, 27, 64, 72, 120, 72, 168, 144, 192, 81, 288, 192, 360, 216, 384, 360, 528, 216, 576, 504, 512, 432, 840, 576, 960, 243, 960, 864, 1152, 576, 1368, 1080, 1344, 648, 1680, 1152, 1848, 1080, 1536, 1584, 2208, 648, 2304, 1728

Offset: 1

Author

Jaroslav Krizek, Nov 01 2009

Keywords

• nonn
• mult

Links

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

Crossrefs

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

Cf. also A335915.

Programs

Mathematica

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 *)

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

Formula

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).

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

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

a(n) = A340323(n) * A340368(n). - Antti Karttunen, Jan 31 2021

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