A349867 Generalized multiplicative Euler phi function of order 2 (see El Hindi et al. link).

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 4, 2, 2, 8, 8, 2, 6, 2, 2, 4, 10, 1, 8, 4, 6, 2, 12, 2, 8, 16, 4, 8, 4, 2, 12, 6, 4, 2, 16, 2, 12, 4, 4, 10, 22, 8, 12, 8, 8, 4, 24, 6, 8, 2, 6, 12, 28, 2, 16, 8, 4, 32, 8, 4, 20, 8, 10, 4, 24, 2, 24, 12, 8, 6, 8, 4, 24, 16

Offset

1,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Mohammad El-Hindi and Therrar Kadri, On The Generalized Multiplicative Euler Phi Function, arXiv:2111.13474 [math.NT], 2021. Gives a formula.

A. N. El-Kassar and H. Y. Chehade, Generalized Group of Units, Math. Balkanica, Vol. 20 (2006), Fasc. 3-4.

Judy L. Smith and Joseph A. Gallian, Factoring Finite Factor Rings, Mathematics Magazine, 58, (1985), 93-95.

Mathematica

phig[n_Integer, k_Integer]:=phig[n, k]=Module[{f=FactorInteger[n], p, a}, If[k==1, Return[EulerPhi[n]]]; If[n==1, Return[1]]; Times@@Table[{p, a}=f[[j]]; Power@@If[OddQ[p], If[a<k, {phig[p-1, k-1]*Product[phig[p, i], {i, k-a+1, k-1}], 1}, {phig[p-1, k-1]*p^(a-k)*Product[phig[p, i], {i, 1, k-1}], 1}], If[a<2 k, {1, 1}, {2^(a-1), 1}]], {j, Length[f]}]]; a[n_]:=phig[n, 2]; Table[a[n], {n, 1, 80}] (* Vincenzo Librandi, Nov 06 2025 *)

Prog

(PARI) phig(n, k) = {my(f=factor(n)); if (k==1, return(eulerphi(f))); for (j=1, #f~, my(p=f[j, 1], alfa=f[j, 2]); if (p % 2, f[j, 1] = if (alfa < k, phig(p-1, k-1)*prod(i=k-alfa+1, k-1, phig(p, i)), phig(p-1, k-1)*p^(alfa-k)*prod(i=1, k-1, phig(p, i))), f[j, 1] = if (alfa < 2*k, 1, 2^(alfa-1)); ); f[j, 2] = 1; ); factorback(f); }
a(n) = phig(n, 2);

Crossrefs

Cf. A000010, A010554.

Sequence in context: A206941 A327790 A143179 * A089610 A102566 A328771

Adjacent sequences: A349864 A349865 A349866 * A349868 A349869 A349870

Keyword

nonn,mult

Author

Michel Marcus, Dec 03 2021

Status

approved