A324643 Numbers k such that bitand(2k,sigma(k))/2 = k = bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

6, 20, 28, 36, 88, 100, 104, 264, 272, 304, 368, 392, 464, 496, 550, 784, 1032, 1040, 1044, 1056, 1068, 1104, 1120, 1184, 1232, 1312, 1376, 1504, 1696, 1888, 1952, 2140, 3222, 4100, 4128, 4160, 4288, 4512, 4544, 4624, 4640, 4672, 5056, 5312, 5696, 6208, 6328, 6464, 6592, 6808, 6848, 6976, 7232, 7304, 8128, 8288, 8968, 9256, 10184

Offset: 1

Author

Antti Karttunen, Mar 14 2019

Keywords

• nonn
• look

Comments

Numbers k for which k = A318458(k)/2 = A318468(k).

Intersection of A324649 and A324652.

It is conjectured that there are no odd terms in this sequence, which is equivalent to the conjecture that there are no odd perfect numbers.

Question: Where do the densest clusters of terms occur? See also the scatter plot. - Antti Karttunen, Mar 12 2024

As A324649 and A324652 are both subsequences of nondeficient numbers (A023196), also this sequence is, which stems from the "monotonic property" of bitwise-and. - Antti Karttunen, Jan 08 2025

Links

• Antti Karttunen, Table of n, a(n) for n = 1..17020
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..17020, labeling cusps in the plot.
• 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

Cf. A004198, A318458, A318468.

Intersection of A324649 and A324652.

Subsequence of A023196 and of A324639.

Programs

Mathematica

Select[Range[10^4], Block[{s = DivisorSigma[1, #]}, # == BitAnd[#, s-#] && 2*# == BitAnd[2*#, s]] &] (* Paolo Xausa, Mar 11 2024 *)

PARI

for(n=1,oo,if( (bitand(n, sigma(n)-n)==n) && (bitand(n+n, sigma(n))==2*n),print1(n,", ")))