A348742 -id:A348742 - cp's OEIS frontend

A161942 Odd part of sum of divisors of n.

1, 3, 1, 7, 3, 3, 1, 15, 13, 9, 3, 7, 7, 3, 3, 31, 9, 39, 5, 21, 1, 9, 3, 15, 31, 21, 5, 7, 15, 9, 1, 63, 3, 27, 3, 91, 19, 15, 7, 45, 21, 3, 11, 21, 39, 9, 3, 31, 57, 93, 9, 49, 27, 15, 9, 15, 5, 45, 15, 21, 31, 3, 13, 127, 21, 9, 17, 63, 3, 9, 9, 195, 37, 57, 31, 35, 3, 21, 5, 93, 121, 63

Offset: 1

Franklin T. Adams-Watters, Jun 22 2009

It is conjectured that iteration of this function will always reach 1. This implies the nonexistence of odd perfect numbers. This is equivalent to the same question for A000593, which can be expressed as the sum of the divisors of the odd part of n.
Up to 20000000, there are only two odd numbers with a(n) and a(a(n)) both >= n: 81 and 18966025. See A162284.
For the nonexistence proof of odd perfect numbers, it is enough to show that this sequence has no fixed points beyond the initial one. This is equivalent to a similar condition given for A326042. - Antti Karttunen, Jun 17 2019

Cf. A000265, A000203, A000593, A162284, A326042, A336698. A348992, A351544.
Cf. A348741, A348742, A348743.
One less than A337194.

oddPart[n_] := n/2^IntegerExponent[n, 2]; a[n_] := oddPart[ DivisorSigma[1, n]]; Table[a[n], {n, 1, 82}] (* Jean-François Alcover, Sep 03 2012 *)

oddpart(n)=n/2^valuation(n,2);
a(n)=oddpart(sigma(n));

from sympy import divisor_sigma
def A161942(n): return (m:=int(divisor_sigma(n)))>>(~m&m-1).bit_length() # Chai Wah Wu, Mar 17 2023

(define (A161942 n) (A000265 (A000203 n))) ;; [For the implementations of A000203 and A000265, see under the respective entries]. - Antti Karttunen, Nov 18 2017

Multiplicative with a(p^e) = oddpart((p^{e+1}-1)/(p-1)), where oddpart(n) = A000265(n) is the largest odd divisor of n.
a(n) = A000265(A000203(n)).
a(n) = A337194(n)-1. - Antti Karttunen, Nov 30 2024

A348749 Odd numbers k for which A064989(sigma(k)) > A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

9, 25, 45, 49, 75, 81, 117, 121, 225, 243, 289, 325, 333, 405, 441, 529, 549, 605, 625, 657, 675, 729, 841, 925, 1053, 1089, 1125, 1215, 1225, 1413, 1445, 1521, 1525, 1575, 1665, 1681, 1737, 1825, 1875, 2025, 2205, 2401, 2475, 2493, 2601, 2817, 2825, 2925, 2997, 3025, 3033, 3125, 3249, 3481, 3573, 3645, 3675, 3789

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Sequence obtained when A003961 is applied to A348739 and the terms are sorted into ascending order.
From Robert Israel, Nov 12 2024: (Start)
If a and b are terms and are coprime, then a * b is a term.
If p > 2 is in A053182, Legendre's conjecture implies p^2 is in this sequence. (End)

Cf. A000203, A003961, A053182, A064989, A326042, A348739, A348748, A348939 (terms of A228058 that occur here).

Cf. also A348742, A348754.

g:= prevprime: g(2):= 1:
f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  mul(g(t[1])^t[2],t=F)
end proc:
select(t -> f(numtheory:-sigma(t)) > f(t), [seq(i,i=1..4000,2)]); # Robert Israel, Nov 12 2024

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 4000, 2], s[DivisorSigma[1, #]] > s[#] &] (* Amiram Eldar, Nov 04 2021 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA348749(n) = ((n%2)&&(A064989(sigma(n)) > A064989(n)));

A348739 Numbers k for which A326042(k) > k.

Original entry on oeis.org

4, 9, 12, 16, 18, 25, 32, 36, 44, 48, 49, 64, 72, 81, 96, 99, 100, 108, 124, 144, 147, 162, 169, 176, 180, 192, 196, 225, 236, 243, 252, 256, 279, 284, 288, 300, 320, 324, 361, 372, 396, 400, 405, 432, 441, 448, 450, 468, 484, 486, 496, 507, 512, 529, 531, 567, 576, 588, 604, 612, 625, 639, 648, 675, 676, 700, 704

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Terms that occur also in A337386 are: 180, 300, 720, 900, 960, 1008, 1200, 1440, 1620, 1800, 2016, 2400, ...

Positions of negative terms in A348736.
Cf. A326182 (subsequence after its initial 1), A348738.
Cf. A000203, A003961, A064989, A161942, A191218, A326042, A337386, A348742, A348749 (corresponding odd numbers), A348942.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[700], s1[DivisorSigma[1, s2[#]]] > # &] (* Amiram Eldar, Nov 04 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
isA348739(n) = (A326042(n)>n);

A348741 Odd numbers k for which A161942(k) < k, where A161942 is the odd part of sigma.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Differs from A088828 for the first time at n=1080, where a(1080) = 2207, while A088828(1080) = 2205 = A348743(1), the value which is missing from this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..16291
Index entries for sequences related to sigma(n)

Setwise difference A088828 \ A348743.
Cf. A161942, A348742 (complement among the odd numbers).
Cf. also A348738, A348753, A348931.

odd[n_] := n/2^IntegerExponent[n, 2]; Select[Range[1, 150, 2], odd[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Nov 02 2021 *)

A000265(n) = (n >> valuation(n, 2));
isA348741(n) = ((n%2)&&A000265(sigma(n))

A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

2205, 19845, 108045, 143325, 178605, 187425, 236925, 266805, 319725, 353925, 372645, 407925, 452025, 462825, 584325, 637245, 646425, 658125, 672525, 789525, 796005, 804825, 845325, 920205, 972405, 981225, 1007325, 1055925, 1069425, 1102725, 1113525, 1116225, 1166445, 1201725, 1245825, 1289925, 1378125, 1380825, 1442925

Offset: 1

Antti Karttunen, Nov 02 2021

The first non-multiples of 5 are a(103) = 6243237 and a(125) = 8164233.
From Antti Karttunen, Nov 28 2024: (Start)
This is not a subsequence of A228058. At least k = A000040(28)*(A002110(27)/2)^2 = 15388519572341080054329140040512468358441210638435506649120749687401476705908239675 is a number of the form 4m+3 such that A161942(k) >= k.
Another such number is A000040(28)*81*(A002110(25)/6)^2 = 1279741205456530915782536871495922949062895982530933679752838870798129159675.
Question: What is the smallest term of this sequence that is of the form 4m+3, and thus not in A386427 (in A191218 and in A228058)?
(End)

Intersection of A088828 and A348742.
Cf. A386427 (a subsequence, which agrees for a very long time).
Cf. A033630, A065205, A083206, A161942, A191218, A228058, A336702, A348741.
Cf. also A065235, A162284.

A000265(n) = (n >> valuation(n, 2));
isA348743(n) = ((n%2)&&!issquare(n)&&A000265(sigma(n))>=n); \\ Edited Nov 28 2024

Definition changed (from > to >=) to formally include also any hypothetical odd perfect numbers - Antti Karttunen, Nov 28 2024

Comment removed, because it was more related to sequence A386427. - Antti Karttunen, Aug 21 2025

A162284 Odd numbers n such that b(n) >= n and b(b(n)) >= n, where b(n) = A161942(n) = oddpart(sigma(n)).

Original entry on oeis.org

1, 81, 18966025, 135187129, 164275489, 350561925, 445421025, 598047025, 649587169, 748733769, 850830561, 960362325, 1055925025, 1341097641, 1406175001, 1476326929, 1520766009, 1536248025, 1591004025, 1735566525

Offset: 1

Franklin T. Adams-Watters, Jun 29 2009

nonn

Applying b a third time produces a smaller value for every number in this sequence up to 10^10, except 1.
Of the first 43 terms, only the following nine are not squares: 350561925, 960362325, 1591004025, 1735566525, 1753206525, 1831175325, 4558583925, 6745097205, 8766517725, and incidentally, all of them are terms of A228058. - Antti Karttunen, Jun 16 2019
But see also comments in A348743. - Antti Karttunen, Nov 30 2024

Cf. A161942, A228058, A326042.
Subsequence of A348742.
Cf. also A348743.

Values for n > 3 from Robert Gerbicz

A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499, 511, 523, 529, 541

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

See comments in A348930.

Index entries for sequences related to sigma(n)

Cf. A007310, A348930, A348931.

Cf. also A348742, A348754.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[550], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #]] > # &] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348932(n) = ((n%2)&&(n%3)&&(A348930(n)>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

A161942 Odd part of sum of divisors of n.

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A348749 Odd numbers k for which A064989(sigma(k)) > A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

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A348739 Numbers k for which A326042(k) > k.

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A348741 Odd numbers k for which A161942(k) < k, where A161942 is the odd part of sigma.

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A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

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A162284 Odd numbers n such that b(n) >= n and b(b(n)) >= n, where b(n) = A161942(n) = oddpart(sigma(n)).

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A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

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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).