A318468 -id:A318468 - cp's OEIS frontend

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

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,", ")))

A324639 Numbers k such that bitand(2k,sigma(k)) = 2*bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 16, 17, 20, 26, 28, 32, 36, 37, 38, 41, 44, 50, 64, 73, 74, 88, 98, 100, 104, 128, 130, 134, 136, 137, 149, 152, 153, 164, 172, 184, 256, 257, 261, 262, 264, 272, 277, 284, 293, 294, 304, 328, 337, 368, 392, 405, 410, 424, 442, 464, 496, 512, 520, 521, 522, 528, 529, 538, 548, 549, 550, 556, 560, 577

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

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

Cf. A004198, A318458, A318468.

Subsequences: A324643, A324718 (odd terms).

Select[Range[1000], Block[{s = DivisorSigma[1, #]}, BitAnd[2*#, s] == 2* BitAnd[#, s-#]] &] (* Paolo Xausa, Mar 11 2024 *)

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

A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).

Original entry on oeis.org

399736269009, 1013616036225, 1393148751631700625, 2998748839068013955625, 3547850289210724050225

Offset: 1

Antti Karttunen, Jan 13 2025

If there are any quasiperfect numbers, i.e., numbers x for which sigma(x) = 2*x+1, then they should occur also in this sequence.
Square roots of these terms are: 632247, 1006785, 1180317225, 54760833075, 59563833735.
Question: Are there any solutions to similar equations "Odd squares s such that 2*s is equal to bitwise-AND of 2*s and A001065(s)" and "Odd squares s such that 3*s is equal to bitwise-AND of 3*s and sigma(s)"? Such sequences would contain odd triperfect numbers, if they exist (cf. A005820, A347391, A347884). - Antti Karttunen, Aug 19 2025
a(6) > 4*10^21. - Giovanni Resta, Aug 19 2025

Paolo Cattaneo, Sui numeri quasiperfetti, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62.
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.

Odd squares in A324647.
Intersection of A016754 and A324647.
Subsequence of A325311, which is a subsequence of A005231.
Cf. A000203, A001065, A004198, A318468.
Cf. also A336700, A336701, A337339, A337342, A348742, A379474, A379503, A379505, A379949 for other conditions that quasiperfect numbers should satisfy.

k=0; forstep(n=1,oo,2, if(!((n-1)%(2^27)),print1("("n")")); if(!isprime(n) && omega(n)>=3, f = factor(n); sq=n^2; sig=prod(i=1,#f~,((f[i,1]^(1+(2*f[i,2])))-1) / (f[i,1]-1)); if(((2*sq)==bitand(2*sq, sig)), k++; print1(sq,", "))));

a(4) and a(5) from Giovanni Resta, Aug 19 2025

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

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A324639 Numbers k such that bitand(2k,sigma(k)) = 2*bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

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A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).

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cp's OEIS Frontend

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

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A324639 Numbers k such that bitand(2k,sigma(k)) = 2*bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

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A379490 Odd squares s such that 2*s is equal to bitwise-AND of 2*s and sigma(s).

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A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).