A337342 -id:A337342 - cp's OEIS frontend

A007593 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.

21, 2133, 19521, 176661, 129127041, 328256967373616371221

Offset: 1

67585198634817522935331173030319681 and 443426488243037769923934299701036035201 are also in the sequence, but their positions are unknown. - Jud McCranie, Dec 16 1999; updated by Max Alekseyev, Jun 03 2025
For all k in A014224, 3^(k-1)*(3^k-2) is in this sequence. - M. F. Hasler, Apr 25 2012
The known examples are all of the form 3^(k-1)*(3^k-2), where 3^k-2 is prime (cf. A014224). Conversely, from sigma(3^(k-1)*p)=(3^k-1)/2*(p+1) it is immediate that 2*sigma(n)=3n+1 for such numbers, i.e., they are 2-hyperperfect. (This is "form 3" with p=3 in McCranie's paper.) - M. F. Hasler, Apr 25 2012
Numbers k for which sigma(k) = (3k+1)/2, thus numbers k such that A000203(k) = A014682(k). Sequence A064989(a(n)), n >= 1, forms a subsequence of A337342. - Antti Karttunen, Aug 26 2020

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 21, p. 7, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302.
Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, p. 561.
Daniel Minoli, Voice Over MPLS, McGraw-Hill, 2002, New York, NY, see pp. 112-134.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, PME Journal, Fall 1975, pp. 153-157.
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 144.

Antal Bege, Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.
J. S. McCranie, A study of hyperperfect numbers, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.
Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645.
Daniel Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 3.
Eric Weisstein's World of Mathematics, Hyperperfect Number

Cf. A000203, A000396, A014682, A028499, A028500, A028501, A028502, A034916, A220290, A337342.

Select[Range[2*10^5], #==2(DivisorSigma[1,#]-#-1)+1 &] (* Stefano Spezia, Sep 24 2024 *)

is(n)=n==2*(sigma(n)-n-1) + 1; \\ Charles R Greathouse IV, May 01 2013

a(6) from Jud McCranie confirmed and added by Max Alekseyev, Jun 03 2025

A336700 Numbers k such that the odd part of (1+k) divides (1 + odd part of sigma(k)).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 2431, 2943, 3775, 4095, 8191, 13311, 14335, 16383, 17407, 21951, 22527, 32767, 34335, 44031, 57855, 65535, 85375, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1562623, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 10059775, 16777215

Offset: 1

Antti Karttunen, Aug 02 2020

nonn

Numbers k for which A337194(k) = 1+A161942(k) is a multiple of A000265(1+k).

Conjecture: After 1, all terms are of the form 4u+3 (in A004767). If this could be proved, then it would refute at once the existence of both the odd perfect numbers and the quasiperfect numbers. Concentrating on the latter is probably easier, as it is known that all quasiperfect numbers must be odd squares, thus k is of the form 4u+1, in which case the condition given in A336701 that A000265(1+A000265(sigma(k))) must be equal to A000265(1+k) reduces to a simpler form, A000265(1+sigma(k)) = (1+k)/2, and as k = s^2, with s odd, so (s^2 + 1)/2 should divide 1+sigma(s^2). Does that condition allow any other solutions than s=1 ? See A337339.

Antti Karttunen, Table of n, a(n) for n = 1..77; all terms < 2^32
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.
Eric Weisstein's World of Mathematics, Quasiperfect Number
Index entries for sequences where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Cf. A000203, A000265, A004767, A016754, A161942, A228058, A336697, A336698, A337339, A337342.

Subsequences: A000225, A336701 (terms where the quotient is a power of 2).

Block[{f}, f[n_] := n/2^IntegerExponent[n, 2]; Select[Range[2^20], Mod[f[1 + f[DivisorSigma[1, #]]], f[1 + #]] == 0 &] ] (* Michael De Vlieger, Aug 22 2020 *)

A000265(n) = (n>>valuation(n,2));
isA336700(n) = !((1+A000265(sigma(n)))%A000265(1+n));

A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 109, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 377, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 8821, 6613, 865

Offset: 1

Antti Karttunen, Aug 24 2020

All terms are members of A007310, because all terms of A337336 and A337337 are.
No 1's after the initial one at a(1) => No quasiperfect numbers. See comments in A336700 & A337342.
If any quasiperfect numbers qp exist, they must occur also in A325311.
Question: Is there any reliable lower bound for this sequence? See A337340, A337341.
Duplicate values are rare, but at least two cases exist: a(21) = a(74) = 1513 and a(253) = a(554) = 71065. - Antti Karttunen, Jan 03 2024

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003961, A003973, A007310, A048673, A074627, A074630, A209922, A324899, A325311, A336697, A336700, A336844, A337194, A337336, A337337, A337340, A337341, A337342.
Cf. A337338 (numerators).
Cf. also A336848, A336849.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337339(n) = { my(s=(A003961(n)^2),u=(s+1)/2); (u/gcd(1+sigma(s), u)); };
\\ Or alternatively as:
A337339(n) = { my(s=A003961(n^2)); denominator((1+sigma(s))/((s+1)/2)); };

a(n) = A337336(n) / A337337(n) = A048673(n^2) / gcd(A048673(n^2), A336844(n^2)).

a(n) = A337336(n) / gcd(A337336(n), 1+A003973(n^2)).

A337335 a(n) = gcd(A048673(n), 1+A003973(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 19, 1, 13, 1, 1, 1, 13, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

Cf. A003961, A003973, A048673, A337337, A337342 (positions where equal to A048673).

Cf. also A336850.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337335(n) = { my(s=A003961(n)); gcd((1+s)/2, 1+sigma(s)); };

a(n) = gcd(A048673(n), 1+A003973(n)) = (n) = gcd((1+A003961(n))/2, 1+sigma(A003961(n))).

For all n>= 1, a(A000290(n)) = A337337(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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A007593 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1.

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A336700 Numbers k such that the odd part of (1+k) divides (1 + odd part of sigma(k)).

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A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

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A337335 a(n) = gcd(A048673(n), 1+A003973(n)).

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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).