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

1,1

Comments

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

Links

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

Example

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

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]

Prog

(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

Crossrefs

The unitary version of A244439.

Cf. A034448, A047994, A191414.

Sequence in context: A267603 A323007 A370682 * A128927 A145934 A033913

Adjacent sequences: A297362 A297363 A297364 * A297366 A297367 A297368

Keyword

nonn

Author

Amiram Eldar, Dec 29 2017

Status

approved