A324639 -id:A324639 - cp's OEIS frontend

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

1, 5, 9, 17, 37, 41, 73, 137, 149, 153, 257, 261, 277, 293, 337, 405, 521, 529, 549, 577, 593, 641, 661, 673, 677, 1025, 1033, 1061, 1093, 1097, 1109, 1153, 1193, 1289, 1297, 1301, 1321, 1361, 2053, 2069, 2081, 2089, 2097, 2113, 2129, 2209, 2213, 2225, 2309, 2341, 2377, 2389, 2593, 2633, 2689, 2693, 2729, 2825, 4129, 4133, 4177, 4229

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Odd numbers n for which 2*A318458(n) = A318468(n). If there are no common terms with A324719, then there are no odd perfect numbers.

This is not a subsequence of A191218, because terms 1, 9, 529, 2209, 10609, 77841, 83521, 263169, 279841, 330625, 528529, ... are not present in A191218.

Subsequence of A324639.

Cf. A004198, A191218, A318458, A318468, A324647, A324719, A324727.

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

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

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

Original entry on oeis.org

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

Antti Karttunen, Mar 14 2019

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

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)

Cf. A004198, A318458, A318468.
Intersection of A324649 and A324652.
Subsequence of A023196 and of A324639.

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

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

cp's OEIS Frontend

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

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