A297366 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).

6, 10, 12, 15, 18, 22, 24, 26, 28, 36, 40, 46, 48, 52, 58, 63, 72, 80, 82, 88, 96, 100, 106, 108, 112, 124, 136, 148, 162, 166, 172, 178, 192, 196, 226, 232, 242, 250, 262, 268, 285, 288, 292, 316, 346, 352, 358, 382, 388, 400, 432, 448, 466, 478, 486, 502

Offset

1,1

Comments

The unitary version of A145749.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

6 is in the sequence since uphi(6) + usigma(6) = 2 + 12 = uphi(7) + usigma(7) = 6 + 8 = 14.

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[10^3], aQ]

Prog

(PARI) u(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1) + prod(i = 1, #f~, f[i, 1]^f[i, 2]+1); }
list(kmax) = {my(u1 = u(1), u2); for(k = 2, kmax, u2 = u(k); if(u1 == u2, print1(k-1, ", ")); u1 = u2); } \\ Amiram Eldar, Jun 30 2025

Crossrefs

Cf. A034448, A047994, A145749.

Sequence in context: A143958 A294278 A373677 * A378616 A325472 A305188

Adjacent sequences: A297363 A297364 A297365 * A297367 A297368 A297369

Keyword

nonn

Author

Amiram Eldar, Dec 29 2017

Status

approved