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

Author

Antti Karttunen, Jun 09 2019

Keywords

• nonn
• more

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

Examples

			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.
		

Links

• 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)

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.

Programs

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 *)

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); };

Extensions

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