A297366 - OEIS

%I #13 Jul 01 2025 01:03:11

%S 6,10,12,15,18,22,24,26,28,36,40,46,48,52,58,63,72,80,82,88,96,100,

%T 106,108,112,124,136,148,162,166,172,178,192,196,226,232,242,250,262,

%U 268,285,288,292,316,346,352,358,382,388,400,432,448,466,478,486,502

%N 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).

%C The unitary version of A145749.

%H Amiram Eldar, <a href="/A297366/b297366.txt">Table of n, a(n) for n = 1..10000</a>

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

%t usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];

%t 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]

%o (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);}

%o 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

%Y Cf. A034448, A047994, A145749.

%K nonn

%O 1,1

%A _Amiram Eldar_, Dec 29 2017