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

1,2

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

Table of n, a(n) for n=1..60.

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)

Formula

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

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 *)

Prog

(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);

Crossrefs

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

Cf. A000668, A046528 (subsequences).

Cf. also A349756.

Sequence in context: A153829 A153830 A080550 * A043725 A182247 A121712

Adjacent sequences: A348940 A348941 A348942 * A348944 A348945 A348946

Keyword

nonn

Author

Antti Karttunen, Nov 04 2021

Status

approved