A347391 -id:A347391 - cp's OEIS frontend

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

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

A386423 Odd numbers k such that k/(1+A347381(k)) obtains record values, where A347381 gives the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree.

Original entry on oeis.org

1, 3, 5, 9, 13, 15, 35, 63, 77, 81, 99, 105, 135, 175, 189, 455, 765, 775, 819, 945, 2125, 6375, 9261, 21275, 43011, 43125, 43475, 44469, 45441, 45617, 45619, 46189, 46305, 155363, 161257, 203203, 318835, 401625, 1016015, 1128799, 1773827, 3048045, 3255075, 3386397, 4044555

Offset: 1

Antti Karttunen, Jul 21 2025

Odd terms of A347391 probably form a subsequence, especially if there are no odd perfect numbers or other odd terms larger than one in A336702.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A005940, A336702, A347381, A347391.

Cf. also A119239, A171929, A228059.

m=-1; n=-1; k=0; while(m!=0, n+=2; if(!((n-1)%(2^25)),print1("("n")")); my(r=n/(1+A347381(n))); if(r>m, m=r; k++; write("b386423.txt", k, " ", n); print1(n, ", ")));

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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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A386423 Odd numbers k such that k/(1+A347381(k)) obtains record values, where A347381 gives the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree.

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cp's OEIS Frontend

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

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Extensions

A386423 Odd numbers k such that k/(1+A347381(k)) obtains record values, where A347381 gives the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree.

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