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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).

%I A297365 #13 Nov 09 2023 16:57:45
%S A297365 5,11,19,71,247,271,991,2232,6200,8271,10295,16744,18496,18576,25704,
%T A297365 26656,102175,122607,166624,225939,301103,747967,7237384,7302592,
%U A297365 15760224,21770800,28121184,72967087,98617024,104577848,173859007,253496176,335610184,371191600
%N 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).
%C A297365 Equivalently, numbers k such that A191414(k) = A191414(k+1). - _Amiram Eldar_, Nov 09 2023
%H A297365 Amiram Eldar, <a href="/A297365/b297365.txt">Table of n, a(n) for n = 1..47</a> (terms below 10^11)
%e A297365 11 is in the sequence since uphi(11) * usigma(11) = 10 * 12 = uphi(12) * usigma(12) = 6 * 20 = 120.
%t A297365 usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
%t A297365 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]
%o A297365 (PARI) A191414(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(2*f[i, 2])-1); }
%o A297365 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
%Y A297365 The unitary version of A244439.
%Y A297365 Cf. A034448, A047994, A191414.
%K A297365 nonn
%O A297365 1,1
%A A297365 _Amiram Eldar_, Dec 29 2017