A348943 Odd numbers k for which A348942(A064989(k)) = 1.

1, 3, 7, 15, 21, 31, 33, 69, 91, 93, 105, 127, 135, 141, 217, 231, 273, 285, 381, 465, 483, 573, 651, 775, 819, 861, 889, 945, 987, 1023, 1149, 1185, 1365, 1419, 1485, 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

Offset: 1

Author

Antti Karttunen, Nov 04 2021

Keywords

• nonn

Comments

Applying A064989 to these numbers and sorting the results to ascending order gives A355942, the positions of 1's in A348942.

From Antti Karttunen, Jul 23 2022: (Start)

Odd numbers k such that A064989(k) is a multiple of A350073(k) = A064989(sigma(k)).

If both x and y are terms and gcd(x, y) = 1, then x*y is also present.

(End)

Links

• Index entries for sequences where odd perfect numbers must occur, if they exist at all
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A003961, A064989, A326042, A348942, A355942, A355943 (characteristic function).

Cf. A000668, A046528 (subsequences).

Cf. also A349756.

Programs

Mathematica

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 *)

PARI

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A326042(n) = A064989(sigma(A003961(n)));
A348942(n) = { my(u=A326042(n)); (u / gcd(n, u)); };
isA348943(n) = ((n%2)&&(1==A348942(A064989(n))));

PARI

\\ Alternatively, as:
A355943(n) = ((n%2)&&!(A064989(n)%A064989(sigma(n))));
isA348946(n) = A348943(n);

Formula

A348942(A064989(a(n))) = 1 for all n >= 1.