A326131 - OEIS

%I #35 Jan 03 2021 02:54:58

%S 6,28,110,496,884,8128,18632,85936,116624,15370304,33550336,73995392,

%T 815634435,3915380170,5556840416,6800695312,8589869056

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

%C No further terms below 2^31.

%C See also comments in A326133.

%C 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.

%C a(18) > 10^11. - _Amiram Eldar_, Jan 03 2021

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e 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.

%t 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 *)

%o (PARI)

%o A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };

%o isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

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

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

%Y Cf. also A325981, A326141.

%K nonn,more

%O 1,1

%A _Antti Karttunen_, Jun 09 2019

%E a(14)-a(17) from _Amiram Eldar_, Jan 03 2021