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%I #18 Jul 28 2022 09:18:06
%S 1,3,7,15,21,31,33,69,91,93,105,127,135,141,217,231,273,285,381,465,
%T 483,573,651,775,819,861,889,945,987,1023,1149,1185,1365,1419,1485,
%U 1561,1743,1891,1905,1995,2139,2295,2325,2667,2821,3003,3105,3255,3507,3937,4011,4095,4185,4191,4371,4683,5425,5673,6279,6345
%N Odd numbers k for which A348942(A064989(k)) = 1.
%C Applying A064989 to these numbers and sorting the results to ascending order gives A355942, the positions of 1's in A348942.
%C From _Antti Karttunen_, Jul 23 2022: (Start)
%C Odd numbers k such that A064989(k) is a multiple of A350073(k) = A064989(sigma(k)).
%C If both x and y are terms and gcd(x, y) = 1, then x*y is also present.
%C (End)
%H <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348942(A064989(a(n))) = 1 for all n >= 1.
%t f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; s3[n_] := (sn = s2[n])/GCD[n, sn]; Select[Range[1, 6500, 2], s3[s[#]] == 1 &] (* _Amiram Eldar_, Nov 05 2021 *)
%o (PARI)
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A326042(n) = A064989(sigma(A003961(n)));
%o A348942(n) = { my(u=A326042(n)); (u / gcd(n, u)); };
%o isA348943(n) = ((n%2)&&(1==A348942(A064989(n))));
%o (PARI)
%o \\ Alternatively, as:
%o A355943(n) = ((n%2)&&!(A064989(n)%A064989(sigma(n))));
%o isA348946(n) = A348943(n);
%Y Cf. A003961, A064989, A326042, A348942, A355942, A355943 (characteristic function).
%Y Cf. A000668, A046528 (subsequences).
%Y Cf. also A349756.
%K nonn
%O 1,2
%A _Antti Karttunen_, Nov 04 2021
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