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

1,2

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Cf. also A335915.

Sequence in context: A191487 A176205 A099256 * A025615 A297324 A223331

Adjacent sequences: A167341 A167342 A167343 * A167345 A167346 A167347

Keyword

nonn,mult

Author

Jaroslav Krizek, Nov 01 2009

Status

approved