A325311 -id:A325311 - cp's OEIS frontend

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)).

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)).

A156942 Odd abundant numbers whose abundance is odd.

Original entry on oeis.org

11025, 99225, 245025, 275625, 342225, 540225, 893025, 1334025, 1863225, 2205225, 2480625, 3080025, 3186225, 3980025, 4601025, 4862025, 5832225, 6125625, 6890625, 7868025, 8037225, 8555625, 9272025, 9828225, 10595025, 10989225

Offset: 1

Robert G. Wilson v, Feb 18 2009

nonn

Number of terms <10^n: 0, 0, 0, 0, 2, 7, 24, 83, 250, 792, 2484, 7988, 25383, 80082, ..., . Not all are a multiple of 25, i.e.; 81162081 = 9009^2 = (9*7*11*13)^2. See A156943.
Any term must be an odd square. Square roots are in A174830.
Indeed, the sum of divisors of any number isn't odd unless it's a square or twice a square (A028982), and to get the abundance, twice the number is subtracted, so the parity remains the same. - M. F. Hasler, Jan 26 2020
Question: Is this a subsequence of A379503? (Is A379504(a(n)) > 0 for all n? See A379951). The first 15000 terms are all included there. - Amiram Eldar and Antti Karttunen, Jan 06 2025
Question 2: Is A379505(a(n)) > 1 for all n, especially if there are no quasiperfect numbers (numbers k such that sigma(k) = 2k+1)? - Antti Karttunen, Jan 06 2025
From Amiram Eldar, Jan 16 2025: (Start)
The least term that is not divisible by 5 is a(75) = 81162081.
The least term that is not divisible by 3 is a(296889) = 1382511906801025.
The least term that is coprime to 15 is 15285071557677427358507559514565648611799881. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A005101, A005231, A028982, A033880, A156903, A174830, A379503, A379504, A379505, A379951 [= A379504(a(n))].

Subsequences: A156943, A325311 (thus also A379490), A347890, A379949 (terms that are primitive abundant).

fQ[n_] := Block[{ds = DivisorSigma[1, n] - 2 n}, ds > 0 && OddQ@ ds]; Select[ Range[1, 12006223, 2], fQ @# &]

is(n)=my(s=sigma(n)); n%2 && s>2*n && (s-2*n)%2 \\ Charles R Greathouse IV, Feb 21 2017

a(n) = A174830(n)^2. - M. F. Hasler, Jan 26 2020

Edited by Robert G. Wilson v at the suggestion of T. D. Noe, Mar 30 2010

A324647 Odd numbers k such that 2k is equal to bitwise-AND of 2k and sigma(k).

Original entry on oeis.org

1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

If this sequence has no terms common with A324649 (A324897, A324898), or no terms common with A324727, then there are no odd perfect numbers.
First 22 terms factored:
     1116225 = 3^2 * 5^2 * 11^2 * 41
     1245825 = 3^2 * 5^2 *  7^2 * 113
     1380825 = 3^2 * 5^2 * 19^2 * 17    [Here the unitary prime is not the largest]
     2127825 = 3^2 * 5^2 *  7^2 * 193
    10046025 = 3^4 * 5^2 * 11^2 * 41
    16813125 = 3^2 * 5^4 *  7^2 * 61
   203753025 = 3^2 * 5^2 *  7^2 * 18481
   252880425 = 3^2 * 5^2 *  7^2 * 22937
   408553425 = 3^2 * 5^2 *  7^2 * 37057
   415433025 = 3^2 * 5^2 *  7^4 * 769
   740361825 = 3^2 * 5^2 *  7^2 * 67153
   969523425 = 3^4 * 5^2 * 13^2 * 2833
  1369580625 = 3^2 * 5^4 *  7^2 * 4969
  1612924425 = 3^2 * 5^2 *  7^2 * 146297
  1763305425 = 3^2 * 5^2 *  7^2 * 159937
  2018027025 = 3^2 * 5^2 *  7^2 * 183041
  2048985225 = 3^2 * 5^2 *  7^2 * 185849
  2286684225 = 3^2 * 5^2 *  7^2 * 207409
  3341556225 = 3^2 * 5^2 *  7^2 * 303089
  3915517725 = 3^4 * 5^2 *  7^2 * 39461
  3985769025 = 3^4 * 5^2 *  7^2 * 40169
  4051698525 = 3^2 * 5^2 *  7^2 * 367501.
Compare the above factorizations to the various constraints listed for odd perfect numbers in the Wikipedia article. However, this is NOT a subsequence of A191218 (A228058), see below.
The first terms that do not belong to A191218 are 399736269009 = (3 * 7^2 * 11 * 17 * 23)^2 and 1013616036225 = (3^2 * 5 * 13 * 1721)^2, that both occur instead in A325311. The first terms with omega(n) <> 4 are 9315603297, 60452246925, 68923392525, and 112206463425. They factor as 3^2 * 7^2 * 11^2 * 13^2 * 1033, 3^2 * 5^2 * 7^2 * 17^2 * 18973, 3^2 * 5^2 * 13^2 * 19^2 * 5021, 3^2 * 5^2 * 7^2 * 199^2 * 257. - Giovanni Resta, Apr 21 2019
From Antti Karttunen, Jan 13 2025: (Start)
Because of the "monotonic property" of bitwise-and, this is a subsequence of nondeficient numbers (A023196).
Both odd perfect numbers, and quasiperfect numbers, if such numbers exist at all, would satisfy the condition for being included in this sequence. Furthermore, any term must be either an odd square with an odd abundancy (in A156942), which subset is given in A379490 (where quasiperfect numbers must thus reside, if they exist), or be included in A228058, i.e., satisfy the Euler's criteria for odd perfect numbers.
(End)

Giovanni Resta, Table of n, a(n) for n = 1..500
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Odd terms of A324652.

Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311, A379490 (square terms).

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

{Odd k such that 2k = A318468(k)}.

a(23)-a(24) from Giovanni Resta, Apr 21 2019

A115414 Odd abundant numbers not divisible by 3.

Original entry on oeis.org

5391411025, 26957055125, 28816162375, 33426748355, 34393484125, 37739877175, 40342627325, 48150877775, 50866790975, 53356378075, 55959128225, 59305521275, 60711976325, 61164628525, 63395557225, 64899009175, 67275433225, 68972878975, 70088343325, 74922022175, 75665665075

Offset: 1

Sergio Pimentel, Mar 08 2006

nonn

An odd abundant number (A005231) not divisible by 3 must have at least 7 distinct prime factors (e.g., 5^4*7^2*11^2*13*17*19*23) and be >= 5*29#/3# = 5^2*7*11*13*17*19*23*29 = 5391411025 = A047802(2) = a(1). This is most easily seen by writing the relative abundancy A(N) = sigma(N)/2N = sigma[-1](N) as A(Product p_i^e_i) = (1/2)*Product (p_i-1/p_i^e_i)/(p_i-1) < (1/2)*Product p_i/(p_i-1). See A064001 for odd abundant numbers not divisible by 5. - M. F. Hasler, Jul 27 2016

This is not a subsequence of A248150. For example, 81324229811825 and 37182145^2 = 1382511906801025 are terms, with sigma(.) == 2 (mod 4) and sigma(.) == 3 (mod 4) respectively. - Amiram Eldar, Aug 24 2020

			a(1)=5391411025 because it is the smallest abundant number (sigma(n)/n =~ 2.003) that is not divisible by 2 or 3.

David A. Corneth, Table of n, a(n) for n = 1..42320 (terms < 10^14, terms 1..394 from Donovan Johnson, terms 395..4343 from Giovanni Resta)
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A005101, A005231, A064001, A112640, A112644, A248150, A325311.

is(n)=gcd(n,6)==1 && sigma(n,-1)>2 \\ Charles R Greathouse IV, Jul 28 2016

Added missing term 55959128225 and a(14)-a(16) from Donovan Johnson, Dec 29 2008
a(17)-a(20) from Donovan Johnson, Dec 01 2011
More terms from M. F. Hasler, Jul 28 2016

A324899 Odd numbers k for which sigma(k) == 3 (mod 4).

Original entry on oeis.org

25, 169, 225, 289, 841, 1225, 1369, 1521, 1681, 2025, 2601, 2809, 3025, 3721, 5329, 7569, 7921, 8281, 9025, 9409, 10201, 11025, 11881, 12321, 12769, 13225, 13689, 14161, 15129, 15625, 18225, 18769, 20449, 22201, 23409, 24025, 24649, 25281, 27225, 29929, 32761, 33489, 34969, 37249, 38809, 41209, 46225, 47961, 52441, 54289, 55225

Offset: 1

Antti Karttunen, Apr 20 2019

nonn

These are all squares. Square roots are in A324909.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Intersection of A016754 and A072462.
Cf. A000203, A324909 (square roots).
Cf. A325311 (a subsequence).

Select[Range[1, 235, 2]^2, Mod[DivisorSigma[1, #], 4] == 3 &] (* Amiram Eldar, Apr 05 2024 *)

isA324899(n) = ((n%2)&&3==(sigma(n)%4));

a(n) = A324909(n)^2. - Amiram Eldar, Apr 05 2024

A337342 Numbers k such that A048673(k) divides 1+A003973(k).

Original entry on oeis.org

1, 10, 584, 3824, 23008, 5033216

Offset: 1

Antti Karttunen, Aug 24 2020

Numbers k such that A048673(k) = A337335(k). Equivalently, numbers k such that (A003961(k)+1)/2 divides 1+A003973(k).
No squares larger than one in this sequence => No quasiperfect numbers. See also A337339. For any x corresponding to a quasiperfect number qp = A003961(x), the quotient (1+A003973(x)) / A048673(x) should be 4. Thus that A003961(x) should also be a member of A325311.
At least for the terms x = a(2) .. a(6) here, the quotient (1+A003973(x)) / A048673(x) = 3. The terms for which the quotient is 3 are precisely those which by prime shifting become the terms of A007593 (that are all odd), thus the terms y = A064989(A007593(n)), for n >= 1, form a subsequence of this sequence.
a(7) > 2^28.
Terms 65810851904356352, 30943274395471606363637940224, 40102483616531202199118491418624 are also in the sequence, but their positions are unknown. (Adapted from Jud McCranie's Dec 16 1999 comment in A007593).

Cf. A003961, A003973, A007593, A048673, A064989, A209922, A324899, A325311, A336700, A337335, A337339.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337342(n) = { my(s=A003961(n)); !((1+sigma(s))%((1+s)/2)); };

A325312 Odd numbers k for which sigma(k^2) == 3 (mod 4) and sigma(k^2) > 2*k^2.

Original entry on oeis.org

105, 315, 495, 735, 945, 1155, 1485, 1995, 2205, 2415, 2625, 2835, 2925, 3135, 3255, 3315, 3465, 3795, 4455, 4515, 4935, 5115, 5145, 5445, 5655, 5985, 6195, 6615, 6825, 7035, 7095, 7245, 7455, 7605, 7755, 7875, 8085, 8295, 8505, 8715, 8775, 8925, 9009, 9405, 9735, 9765, 9945, 10395, 10725, 10815, 11235, 11385, 11781, 12375

Offset: 1

Antti Karttunen, Apr 20 2019

nonn

Square roots of A325311, odd abundant numbers k for which sigma(k) == 3 mod 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000196, A325311.

Intersection of A174830 and A324909.

Select[Range[1, 12375 , 2], Mod[(s = DivisorSigma[1, #^2]), 4] == 3 && s > 2*#^2 &] (* Amiram Eldar, Apr 05 2024 *)

isA325312(n) = { my(t=sigma(n*n)); (n%2 && 3==(t%4) && t>(2*n*n)); };

a(n) = sqrt(A325311(n)).

A347890 Odd numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.

Original entry on oeis.org

245025, 540225, 893025, 2205225, 3080025, 4862025, 6125625, 6890625, 7868025, 10989225, 13505625, 14402025, 19847025, 22896225, 23474025, 26471025, 27720225, 29648025, 43758225, 45765225, 55130625, 57836025, 60140025, 65367225, 70812225, 72335025, 76475025, 77000625, 94770225, 121550625, 153140625, 156125025

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

Odd numbers k such that A033880(k) is positive but A342926(k) is negative.
This is a subsequence of A156942, "odd abundant numbers whose abundance is odd". Proof: If sigma(k) > 2*k, and sigma(k) were even, then sigma(k)/2 would be an integer and a divisor of sigma(k), and we could compute A003415(sigma(k)) as A003415(2)*(sigma(k)/2) + 2*A003415(sigma(k)/2) by the definition of the arithmetic derivative. But that value is certainly larger than k, because sigma(k)/2 > k, therefore sigma(k) must be an odd number, with also its abundance sigma(k)-(2k) odd. This also entails that all terms are squares. See A347891 for the square roots.
The first term that is not a multiple of 25 is a(146) = 6800806089 = 82467^2.
This is not a subsequence of A325311. The first term that is not present there is a(5) = 3080025.

Index entries for sequences related to sigma(n)

Intersection of A005231 and A343216.
Subsequence of A016754, of A156942 and of A347889 (its odd terms).
Cf. A000203, A003415, A033880, A325311, A342926, A347891 (the square roots).

\\ Using the program given in A347891 would be much faster than this:
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347890(n) = ((n%2)&&(A003415(sigma(n))(2*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

cp's OEIS Frontend

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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A156942 Odd abundant numbers whose abundance is odd.

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A324647 Odd numbers k such that 2*k is equal to bitwise-AND of 2*k and sigma(k).

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A115414 Odd abundant numbers not divisible by 3.

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A324899 Odd numbers k for which sigma(k) == 3 (mod 4).

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A337342 Numbers k such that A048673(k) divides 1+A003973(k).

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A325312 Odd numbers k for which sigma(k^2) == 3 (mod 4) and sigma(k^2) > 2*k^2.

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A347890 Odd numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.

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A324647 Odd numbers k such that 2k is equal to bitwise-AND of 2k and sigma(k).

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