A324647 -id:A324647 - cp's OEIS frontend

A324880 Even numbers k such that A324863(k) = A324874(k).

16, 64, 90, 256, 484, 490, 750, 756, 810, 988, 1000, 1024, 1296, 1440, 2116, 2622, 2662, 3630, 3710, 4004, 4116, 4624, 4896, 4900, 5880, 6426, 6724, 6760, 7290, 7744, 7840, 7920, 8100, 8924, 9604, 10000

Offset: 1

Antti Karttunen, Mar 27 2019

nonn

Of 36 such numbers found in the range 1..10000, the following ten: 90, 490, 750, 810, 2622, 2662, 3630, 3710, 6426, 7290, are such that a positive integer of the form 4m+1 is produced when A156552 is applied to them: 45, 105, 117, 189, 225, 405, 765, 2205, 2565, 262185 (when sorted into ascending order). See also A324647.

Cf. A156552, A323243, A324398, A324647, A324863, A324866, A324874.

Subsequence of A324879.

for(n=1,10000,if(!(n%2)&&A324863(n)==A324874(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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A324880 Even numbers k such that A324863(k) = A324874(k).

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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

A324880 Even numbers k such that A324863(k) = A324874(k).

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Author

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PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

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