A306709 -id:A306709 - cp's OEIS frontend

A373369 a(n) = gcd(A001414(n), A059975(n)), where A001414 and A059975 are fully additive with a(p) = p and a(p) = p-1, respectively.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 6, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3

Offset: 1

Antti Karttunen, Jun 05 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A059975, A345452 (positions of even terms).

Cf. also A082299, A306709, A373362, A373363, A373364, A373365.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };
A373369(n) = gcd(A001414(n), A059975(n));

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

Original entry on oeis.org

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

Jaroslav Krizek, Nov 01 2009

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

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

Cf. also A335915.

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 *)

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

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

cp's OEIS Frontend

A373369 a(n) = gcd(A001414(n), A059975(n)), where A001414 and A059975 are fully additive with a(p) = p and a(p) = p-1, respectively.

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