A348743 -id:A348743 - 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

A348742 Odd numbers k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2205, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 9801, 10201, 10609, 11025, 11449, 11881, 12321

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

All odd squares (A016754) are present, but not all terms are squares. A348743 gives the nonsquare terms.

Odd terms of A336702 form a subsequence. Also all odd terms of A005820 would be present here, as well as any hypothetical quasi-perfect numbers (see comments and references in A332223, A336700), both in A016754. - Antti Karttunen, Nov 28 2024

Union of A016754 and A348743.
Cf. A161942, A162284 (subsequence), A336702, A348741 (complement among the odd numbers).
Cf. also A005820, A332223, A336700, A348739.

q:= n-> (t-> is(t/2^padic[ordp](t,2)>=n))(numtheory[sigma](n)):
select(q, [2*i-1$i=1..10000])[];  # Alois P. Heinz, Nov 28 2024

odd[n_] := n/2^IntegerExponent[n, 2]; Select[Range[1, 10^4, 2], odd[DivisorSigma[1, #]] >= # &] (* Amiram Eldar, Nov 02 2021, edited (because of the changed definition) by Antti Karttunen, Nov 28 2024 *)

A000265(n) = (n >> valuation(n, 2));
isA348742(n) = ((n%2)&&A000265(sigma(n))>=n); \\ revised by Antti Karttunen, Nov 28 2024

a(1) = 1 inserted as the initial term, because of the changed definition (from > to >=) - Antti Karttunen, Nov 28 2024

A065235 Odd numbers which can be written in precisely one way as sum of a subset of their proper divisors.

Original entry on oeis.org

8925, 32445, 351351, 442365, 159427275, 159587925, 159677175, 159784275, 159837825, 159855675, 159944925, 159962775, 160016325, 160105575, 160266225, 160284075, 160391175, 160444725, 160480425, 160533975, 160551825, 160766025, 161015925, 161033775, 161069475

Offset: 1

Jud McCranie, Oct 23 2001

nonn

From Antti Karttunen, Nov 28 2024: (Start)
Characteristic function of this sequence is c(n) = A000035(n)*A378448(n).
The only non-multiples of 25 among the first 10000 terms are a(2)..(4): 32445 = 3^2 * 5 * 7 * 103, 351351 = 3^3 * 7 * 11 * 13^2 and 442365 = 3 * 5 * 7 * 11 * 383, while the other 9997 terms are all of the form 25 * some squarefree number. No terms of A228058 occur among the initial 10000 terms. Compare also to A348743.
(End)

			See A064771 for an example when the number is even.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences where any odd perfect numbers must occur

Odd terms in A064771 (a unique subset of proper divisors sums to the number).

Cf. A000035, A033630, A065205, A136446, A228058, A348743, A378448.

{k such that k is odd and A065205(k) = 1}. - Antti Karttunen, Nov 28 2024

Definition clarified by M. F. Hasler, Apr 08 2008

More terms from Giovanni Resta, Oct 04 2019

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))

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

A386427 Odd nondeficient numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

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

Offset: 1

Antti Karttunen, Aug 18 2025

Nondeficient numbers (A023196) that satisfy Euler's condition for odd perfect numbers (A228058).

This is not equal to A348743, as that sequence contains also terms like 1279741205456530915782536871495922949062895982530933679752838870798129159675 and 15388519572341080054329140040512468358441210638435506649120749687401476705908239675, that are lacking from this sequence.

Intersection of A023196 and A228058.
Also the intersection of A083207 and A228058, and probably also of A005835 and A228058. - Antti Karttunen, Aug 21 2025
Subsequence of A348743, from which this eventually differs at some very large n.
Cf. A386426 (conjectured subsequence).

isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA386427(n) = ((sigma(n)>=(2*n)) && isA228058(n));

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A161942 Odd part of sum of divisors of n.

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A348742 Odd numbers k for which A161942(k) >= k, where A161942 is the odd part of sigma.

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A065235 Odd numbers which can be written in precisely one way as sum of a subset of their proper divisors.

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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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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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A386427 Odd nondeficient numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

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