A348741 -id:A348741 - 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

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Claim: If there is an odd term y of A336702 larger than one, and it is the least one of such terms, then it should satisfy condition that for all nontrivial unitary divisor pairs d and x/d of x = A064989(y) [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348739. Proof: Applying A064989 to the odd terms of A336702 gives the fixed points of A326042. Suppose there are other odd terms in A336702 in addition to its initial 1, and let y be the least of these odd terms > 1 and x = A064989(y). Because A326042 (from here on indicated with f) is multiplicative, it follows that if we take any two nontrivial unitary divisors a and b of x, with x = a*b, gcd(a,b) = 1, 1 < a,b < x, then f(a)*f(b) = f(x) = x. Because f(x)/x = 1, we must have f(a)/a * f(b)/b = 1, as also the ratio f(n)/n is multiplicative. But f(a)/a and f(b)/b cannot be equal to 1, because then a and b would also be fixed by f, which contradicts our assumption that x were the least such fixed point larger than one. Therefore f(a) < a and f(b) > b, or vice versa. See also the comments in A348930, A348933.
Moreover, all odd perfect numbers (a subsequence of A336702), if such numbers exist, should also satisfy the same condition, regardless of whether they are the least of such numbers or not, because having a non-deficient proper divisor will push the abundancy index (ratio sigma(n)/n) of any number over 2. That is, for any such pair of nontrivial unitary divisors d and x/d, both A003961(d) and A003961(x/d) should be deficient, i.e., neither one should be in A337386. See also the condition given in A347383.
Terms that occur also in A337386 are: 120, 240, 360, 420, 480, 504, 540, 600, 630, ...

Positions of positive terms in A348736, positions of 1's in A348737 (characteristic function).
Almost complement of A348739.
Cf. A000203, A003961, A003973, A006039, A027687, A064989, A326042, A336702, A337386, A347383, A348741, A348748 (corresponding odd numbers).
Subsequences: A000040, A374464 (after its initial 1).
Cf. also A348930, A348933.

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[100], 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)));
isA348738(n) = (A326042(n)

A348748 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

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Sequence obtained when A003961 is applied to A348738 and the terms are sorted into ascending order.

The first squares in this sequence are: 169, 361, 961, 1369, 1849, 2209, 2809, 3721, 4489, 5329, 6241, 6889, ...

Cf. A000203, A003961, A064989, A326042, A348738, A348749, A348938 (terms of A228058 that occur here).

Cf. also A348741, A348753.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 200, 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)};
isA348748(n) = ((n%2)&&(A064989(sigma(n)) < A064989(n)));

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

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

A348931 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) < k.

Original entry on oeis.org

5, 11, 17, 23, 29, 35, 41, 47, 49, 53, 55, 59, 65, 71, 77, 83, 85, 89, 95, 101, 107, 113, 115, 119, 125, 131, 137, 143, 145, 149, 155, 161, 167, 169, 173, 179, 185, 187, 191, 197, 203, 205, 209, 215, 221, 227, 233, 235, 239, 245, 251, 253, 257, 263, 265, 269, 275, 281, 287, 293, 295, 299, 305, 311, 317, 319, 323

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

See comments in A348930.

Index entries for sequences related to sigma(n)

Cf. A007310, A348930, A348932, A348933.

Cf. also A348741, A348753.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[350], 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));
isA348931(n) = ((n%2)&&(n%3)&&(A348930(n)

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A161942 Odd part of sum of divisors of n.

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A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

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A348748 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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A348742 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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A348931 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) < k.

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