A297365 Numbers k such that uphi(k)*usigma(k) = uphi(k+1)*usigma(k+1), where uphi is the unitary totient function (A047994) and usigma the sum of unitary divisors (A034448).

5, 11, 19, 71, 247, 271, 991, 2232, 6200, 8271, 10295, 16744, 18496, 18576, 25704, 26656, 102175, 122607, 166624, 225939, 301103, 747967, 7237384, 7302592, 15760224, 21770800, 28121184, 72967087, 98617024, 104577848, 173859007, 253496176, 335610184, 371191600

Offset: 1

Author

Amiram Eldar, Dec 29 2017

Keywords

• nonn

Comments

Equivalently, numbers k such that A191414(k) = A191414(k+1). - Amiram Eldar, Nov 09 2023

Examples

			11 is in the sequence since uphi(11) * usigma(11) = 10 * 12 = uphi(12) * usigma(12) = 6 * 20 = 120.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..47 (terms below 10^11)

Crossrefs

The unitary version of A244439.

Cf. A034448, A047994, A191414.

Programs

Mathematica

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
uphi[n_] := (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]; u[n_] := uphi[n]*usigma[n]; aQ[n_] := u[n] == u[n + 1]; Select[Range[10^6], aQ]

PARI

A191414(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
lista(kmax) = {my(a1 = 1, a2); for(k = 2, kmax, a2 = A191414(k); if(a1 == a2, print1(k-1, ", ")); a1 = a2); } \\ Amiram Eldar, Nov 09 2023