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

399736269009, 1013616036225, 1393148751631700625, 2998748839068013955625, 3547850289210724050225

Offset: 1

Author

Antti Karttunen, Jan 13 2025

Keywords

• nonn
• more

Comments

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

Links

• 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.

Crossrefs

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.

Programs

PARI

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,", "))));

Extensions

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