A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 15370304, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056

Offset

1,1

Comments

No further terms below 2^31.

See also comments in A326133.

The quotients A000120(k)/(sigma(k)-A005187(k)) for these terms are: 1, 1, 5, 1, 3, 1, 5, 9, 2, 2, 1, 2, 2. Ones occur at the positions of perfect numbers.

a(18) > 10^11. - Amiram Eldar, Jan 03 2021

137438691328, 546409576448, 219521427324928, 747406020889133775, 2305843008139952128, 2596609131644715008, 2606591654705281630208, and 2465798898636590010500907008 are all terms of this sequence. - Alexander Violette, Sep 26 2025

Links

Table of n, a(n) for n=1..17.

Index entries for sequences related to binary expansion of n

Index entries for sequences where any odd perfect numbers must occur

Index entries for sequences related to sigma(n)

Example

110 is "1101110" in binary, thus A000120(110) = 5. Sigma(110) = 216, while A005187(110) = 215, thus as 5 = 5*(216-215), 110 is included in this sequence.

Mathematica

q[n_] := Module[{bw = DigitCount[n, 2, 1], ab = DivisorSigma[1, n] - 2*n, sum}, (sum = ab + bw) > 0 && Divisible[bw, sum]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 03 2021 *)

Prog

(PARI)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

Crossrefs

Intersection of A326132 and A326133, also of A326132 and A326138.

Cf. A000120, A000396 (subsequence), A005187, A294898, A295296, A295297, A295298, A295299, A326130.

Cf. also A325981, A326141.

Sequence in context: A332751 A263942 A326138 * A098470 A253068 A055220

Adjacent sequences: A326128 A326129 A326130 * A326132 A326133 A326134

Keyword

nonn,more

Author

Antti Karttunen, Jun 09 2019

Extensions

a(14)-a(17) from Amiram Eldar, Jan 03 2021

Status

approved