A023890 -id:A023890 - cp's OEIS frontend

A354432 a(n) is the numerator of the sum of the reciprocals of the nonprime divisors of n.

1, 1, 1, 5, 1, 7, 1, 11, 10, 11, 1, 3, 1, 15, 16, 23, 1, 4, 1, 7, 22, 23, 1, 5, 26, 27, 31, 19, 1, 41, 1, 47, 34, 35, 36, 61, 1, 39, 40, 31, 1, 55, 1, 29, 6, 47, 1, 7, 50, 29, 52, 17, 1, 25, 56, 3, 58, 59, 1, 53, 1, 63, 74, 95, 66, 83, 1, 22, 70, 17, 1, 15, 1, 75, 28

Offset: 1

Ilya Gutkovskiy, May 28 2022

			1, 1, 1, 5/4, 1, 7/6, 1, 11/8, 10/9, 11/10, 1, 3/2, 1, 15/14, 16/15, 23/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A017665, A018252, A023890, A028235, A354433 (denominators).

Table[DivisorSum[n, 1/# &, !PrimeQ[#] &], {n, 75}] // Numerator

a(n) = numerator(sumdiv(n, d, if(!isprime(d), 1/d))) \\ Michael S. Branicky, May 28 2022

from fractions import Fraction
from sympy import divisors, isprime
def a(n): return sum(Fraction(1, d) for d in divisors(n, generator=True) if not isprime(d)).numerator
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, May 28 2022

from math import prod
from fractions import Fraction
from sympy import factorint
def A354432(n):
    f = factorint(n)
    return (Fraction(prod(p**(e+1)-1 for p, e in f.items()),prod(p-1 for p in f)*n) - sum(Fraction(1,p) for p in f)).numerator # Chai Wah Wu, May 28 2022

a(p) = 1 for prime p. - Michael S. Branicky, May 28 2022

A354433 a(n) is the denominator of the sum of the reciprocals of the nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 2, 1, 14, 15, 16, 1, 3, 1, 5, 21, 22, 1, 3, 25, 26, 27, 14, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 20, 1, 42, 1, 22, 5, 46, 1, 4, 49, 25, 51, 13, 1, 18, 55, 2, 57, 58, 1, 30, 1, 62, 63, 64, 65, 66, 1, 17, 69, 14, 1, 8, 1, 74, 25

Offset: 1

Ilya Gutkovskiy, May 28 2022

			1, 1, 1, 5/4, 1, 7/6, 1, 11/8, 10/9, 11/10, 1, 3/2, 1, 15/14, 16/15, 23/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007947, A017666, A018252, A023890, A354432 (numerators).

Table[DivisorSum[n, 1/# &, !PrimeQ[#] &], {n, 75}] // Denominator

a(n) = denominator(sumdiv(n, d, if(!isprime(d), 1/d))) \\ Michael S. Branicky, May 28 2022

from fractions import Fraction
from sympy import divisors, isprime
def a(n): return sum(Fraction(1, d) for d in divisors(n, generator=True) if not isprime(d)).denominator
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, May 28 2022

from math import prod
from fractions import Fraction
from sympy import factorint
def A354433(n):
    f = factorint(n)
    return (Fraction(prod(p**(e+1)-1 for p, e in f.items()),prod(p-1 for p in f)*n) - sum(Fraction(1,p) for p in f)).denominator # Chai Wah Wu, May 28 2022

a(p) = 1 for prime p. - Michael S. Branicky, May 28 2022

A194580 Nonprime numbers with a sum of nonprime divisors which is a perfect square.

Original entry on oeis.org

1, 15, 35, 143, 243, 323, 465, 899, 1183, 1386, 1763, 2065, 2352, 3060, 3599, 3612, 3696, 3887, 5183, 5358, 5590, 9889, 10403, 11663, 12337, 12740, 12879, 14329, 14455, 14645, 16401, 19043, 19097, 20835, 22477, 22499, 22678, 23427, 25553

Offset: 1

Michel Lagneau, Aug 29 2011

nonn

If n is prime, the sum is equal to 1.

			The divisors of 465 are {1, 3, 5, 15, 31, 93, 155, 465} and the sum of the nonprime divisors 1 + 15 + 93 + 155 + 465 = 729 = 27^2, hence 465 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A018252, A023890.

A023890 := proc(n) a := 0 ; for d in numtheory[divisors](n) do if not isprime(d) then a := a+d; end if; end do; a; end proc:
for n from 1 do if issqr(A023890(A018252(n))) then  print(A018252(n)) ;  end if;
end do: # R. J. Mathar, Sep 06 2011

f[n_] := IntegerQ[Sqrt[Total[Select[Divisors[n], ! PrimeQ[#] &]]]]; Select[Range[25553], ! PrimeQ[#] && f[#] &] (* T. D. Noe, Sep 06 2011 *)

isok(n) = !isprime(n) && issquare(sumdiv(n, d, d*(1-isprime(d)))); \\ Michel Marcus, Aug 25 2019

{A018252(j): A023890(A018252(j)) in A000290}. - R. J. Mathar, Sep 06 2011

A225028 Numbers m such that A206773(m) = k and A206773(k) = m.

Original entry on oeis.org

36, 42, 50, 1316, 12540240, 29559057, 131080256

Offset: 1

Michel Lagneau, Apr 24 2013

Members of a pair (m,k) such that m = sum of aliquot parts of k minus the sum of the prime distinct divisors of k and k = sum of aliquot parts of m minus the sum of the prime distinct divisors of m.
We introduce in a single sequence the “pseudo-amicable numbers” and the “pseudo-perfect numbers” in the case m = k. In this sequence 42 and 1316 are pseudo-perfect numbers:  for example, the sum of the aliquot parts of 42 is 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 and 54 - (2 + 3 + 7) = 54 - 12 = 42.
The pseudo-amicable numbers are (36, 50), (12540240, 29559057). (see the example).
Is the sequence finite?
a(8) > 5*10^9. - Donovan Johnson, Apr 25 2013
a(8) <= 37778715690312487141376. It is easy to verify that if p = (2^k-1) and q = 4^k-2*2^k-1 are two primes, then n = 2^(k-1)*p*q is in the sequence, because A206773(n) = n. This happens for k = 2, 3, 7, and 19, which give the terms 42, 1316, 131080256, and 3777871569031248714137, respectively. Up to 3*10^12 there are no other n for which A206773(n) = n. - Giovanni Resta, May 03 2016
a(8) <= 72872313094554244192. - Giovanni Resta, Jan 28 2020

			The divisors of m = 50 are {1, 2, 5, 10, 25, 50} and (1 + 2 + 5 + 10 + 25) - (2 + 5) = 43 - 7 = 36;
The divisors of k = 36 are {1, 2, 3, 4, 6, 9, 12, 18, 36} and (1+2+3+4+6+9+12+18) - (2 + 3) = 55 - 5 = 50.

Cf. A023890, A001065, A000203, A000396, A063990, A206773.

with(numtheory):for n from 1 to 10^7 do:x:=factorset(n):n1:=nops(x):s:=sum('x[i] ', 'i'=1..n1):s1:=sigma(n)-n-s: y:=factorset(s1):n2:=nops(y): ss:=sum('y[i] ', 'i'=1..n2):s2:=sigma(s1)-s1-ss:if s2=n then printf(`%d, `,n):else fi:od:

a(7) from Donovan Johnson, Apr 25 2013

A279051 Sum of odd nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 16, 1, 1, 10, 1, 1, 22, 1, 1, 1, 26, 1, 37, 1, 1, 16, 1, 1, 34, 1, 36, 10, 1, 1, 40, 1, 1, 22, 1, 1, 70, 1, 1, 1, 50, 26, 52, 1, 1, 37, 56, 1, 58, 1, 1, 16, 1, 1, 94, 1, 66, 34, 1, 1, 70, 36, 1, 10, 1, 1, 116, 1, 78, 40, 1, 1, 118, 1, 1, 22, 86, 1, 88, 1, 1, 70, 92, 1, 94, 1, 96, 1, 1

Offset: 1

Ilya Gutkovskiy, Jan 17 2017

nonn

			a(9) = 10 because 9 has 3 divisors {1, 3, 9} among which 2 are odd nonprime {1, 9} therefore 1 + 9 = 10.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000593, A014076, A023890, A005069, A033272, A093641.

with(numtheory):
a:= n-> add(`if`(d::even or d::prime, 0, d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 18 2017

Table[DivisorSum[n, #1 &, Mod[#1, 2] == 1 && ! PrimeQ[#1] &], {n, 97}]
nmax = 97; Rest[CoefficientList[Series[Sum[k x^k/(1 + x^k), {k, 1, nmax}] - Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]]

a(n) = sumdiv(n, d, !isprime(d)*(d%2)*d); \\ Michel Marcus, Sep 18 2017

G.f.: A(x) = B(x) - C(x), where B(x) = Sum_{k>=1} k*x^k/(1 + x^k), C(x) = Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)).
a(n) = Sum_{d|n, d odd nonprime} d.
a(A093641(n)) = 1.

A279290 Sum of cubes of nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 65, 1, 217, 1, 577, 730, 1001, 1, 2009, 1, 2745, 3376, 4673, 1, 6778, 1, 9065, 9262, 10649, 1, 16345, 15626, 17577, 20413, 24761, 1, 31592, 1, 37441, 35938, 39305, 42876, 55226, 1, 54873, 59320, 73577, 1, 86310, 1, 95897, 95230, 97337, 1, 131033, 117650, 141626, 132652, 158249, 1, 183925, 166376

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

			a(4) = 65 because 4 has 2 nonprime divisors {1,4} and 1^3 + 4^3 = 65.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of divisors.

Cf. A001158, A005064, A023890, A189120.

Table[DivisorSum[n, #1^3 & ,  !PrimeQ[#1] & ], {n, 55}]
Table[DivisorSigma[3, n] - DivisorSum[n, #1^3 & , PrimeQ[#1] & ], {n, 55}]
Table[Total[Select[Divisors[n],!PrimeQ[#]&]^3],{n,60}] (* Harvey P. Dale, Aug 02 2024 *)

a(n) = {my(f = factor(n)); sigma(f, 3) - sum(i=1, #f~, f[i, 1]^3);} \\ Amiram Eldar, Jan 11 2025

a(n) = A001158(n) - A005064(n).
a(n) = 1 when n = 1 or n is prime.
a(p^k) = (p^(3*k+3) - 1)/(p^3 - 1) - p^3 when p is prime.

A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

Original entry on oeis.org

42, 1316, 131080256

Offset: 1

G. L. Honaker, Jr., Jan 26 2020

The number 37778715690312487141376 is also in the sequence. - Daniel Suteu, Jan 27 2020
The first 3 terms have the form (2^p-1)*(2^(p-1))*((2^p-1)^2-2), i.e., a Perfect number times a Carol prime. - G. L. Honaker, Jr., Jan 27 2020
In other words, the values of p are given by the intersection of A091515 and A000043. Currently, only four such values of p are known: {2, 3, 7, 19}. - Daniel Suteu, Jan 27 2020
From Bernard Schott, Jan 28 2020: (Start)
Proposition: If a number N_p is of the form Q_p * C_p where Q_p = (2^(p-1)) * (2^p - 1) is a perfect number and C_p = (2^p-1)^2-2 is a Carol prime then, the sum of the nonprime proper divisors of N_p called S_p(N_p) is equal to N_p.
Proof:
The sum of the nonprime proper divisors of N_p is:
S_p(N_p) = (2* Q_p - 2 - (2^p-1)) + ((Q_p - 1) * C_p).
In the first parenthesis, there is the sum of the nonprime proper divisors of N_p coming only from the perfect number Q_p, then in the second parenthesis, there is the sum of the nonprime proper divisors of N_p coming from C_p.
Then, this sum of the nonprime proper divisors of N_p, S_p(N_p) is indeed equal to N_p = (2^(p-1)) * (2^p-1) * ((2^p-1)^2-2).
Hence, (2^19-1)*(2^(19-1))*((2^19-1)^2-2) = 37778715690312487141376 is a term. (End)
10^13 < a(4) <= 72872313094554244192 = 2^5 * 109 * 151 * 65837 * 2101546957. - Giovanni Resta, Jan 28 2020

			42 is a term because 42 = 1 + 6 + 14 + 21.
1316 is a term because 1316 = 1 + 4 + 14 + 28 + 94 + 188 + 329 + 658.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 42
Wikipedia, 42 (number)

Cf. A001065, A018252, A023890, A331858.

Cf. A000043, A091515, A091516 (Carol primes).

fun[p_, e_] := (p^(e+1) - 1)/(p - 1); npsigma[n_] := Times @@ fun @@@ (f = FactorInteger[n]) - Plus @@ First /@ f;; Select[Range[2, 1500], npsigma[#] == 2# &] (* Amiram Eldar, Jan 26 2020 *)

isok(n) = sigma(n) - n - vecsum(factor(n)[,1]) == n; \\ Daniel Suteu, Jan 27 2020

a(2) from Chuck Gaydos

a(3) from Amiram Eldar, Jan 26 2020

A376154 Numbers that are not the sum of the nonprime divisors of k for any k.

Original entry on oeis.org

0, 2, 3, 4, 6, 8, 9, 12, 14, 17, 18, 19, 20, 21, 24, 25, 28, 30, 31, 32, 33, 38, 41, 42, 43, 44, 45, 46, 48, 49, 51, 53, 54, 57, 60, 64, 65, 67, 68, 69, 72, 73, 74, 76, 77, 79, 80, 81, 82, 85, 89, 90, 91, 93, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 113, 114, 117, 121, 126

Offset: 1

Robert Israel, Sep 12 2024

nonn

Nonnegative numbers that do not appear in A023890.

If k > 1 and k = A023890(m), then m must be composite with m < k. Thus k is a term of this sequence if k does not appear in A023890(1..k-1).

			a(6) = 8 is a term because 8 is not A023890(1) = 1 or A023890(4) = 5 or A023890(6) = 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A023890.

N:= 100: # to get terms <= N
f:= proc(n) convert(remove(isprime, numtheory:-divisors(n)),`+`) end proc:
S:= {$0..N} minus map(f, {$1..N-1}):
sort(convert(S,list));

A380441 Sum of the nonprimes dividing n and the number of distinct primes dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 9, 2, 14, 11, 13, 2, 25, 2, 17, 18, 30, 2, 36, 2, 37, 24, 25, 2, 57, 27, 29, 38, 49, 2, 65, 2, 62, 36, 37, 38, 88, 2, 41, 42, 85, 2, 87, 2, 73, 72, 49, 2, 121, 51, 88, 54, 85, 2, 117, 58, 113, 60, 61, 2, 161, 2, 65, 96, 126, 68, 131, 2, 109, 72, 133, 2, 192, 2, 77, 118, 121, 80, 153, 2, 181, 119, 85, 2, 215, 88, 89, 90, 169, 2, 227, 94, 145, 96

Offset: 1

Wesley Ivan Hurt, Jun 21 2025

Inverse Möbius transform of A005451(n).

For each divisor d of n, add 1 if d is prime, else add d.

Cf. A000203 (sigma), A001221 (omega), A005171 (char nonprimes), A005451, A008472 (sopf), A023890.

Table[DivisorSigma[1, n] - Sum[p, {p, Select[Divisors[n], PrimeQ]}] + PrimeNu[n], {n, 100}]

a(n) = sigma(n) - sopf(n) + omega(n).
a(n) = Sum_{d|n} d^c(d), where c = A005171.
a(n) = Sum_{d|n} A005451(d).
a(p^k) = 1 - p + (p^(k+1)-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025
a(n) = A023890(n) + A001221(n). - Wesley Ivan Hurt, Aug 31 2025

A380444 Sum of the nonprimes dividing n and the squares of the primes dividing n.

Original entry on oeis.org

1, 5, 10, 9, 26, 20, 50, 17, 19, 40, 122, 36, 170, 68, 50, 33, 290, 47, 362, 64, 80, 148, 530, 68, 51, 200, 46, 100, 842, 100, 962, 65, 164, 328, 110, 99, 1370, 404, 218, 112, 1682, 146, 1850, 196, 104, 580, 2210, 132, 99, 115, 350, 256, 2810, 128, 202, 164, 428, 904, 3482, 196, 3722, 1028, 152, 129, 260, 262, 4490, 400, 608, 208, 5042, 203, 5330, 1448, 150, 484

Offset: 1

Wesley Ivan Hurt, Jun 21 2025

Inverse Möbius transform of A103164(n).

			a(12) = 1 + 2^2 + 3^2 + 4 + 6 + 12 = 36.

Cf. A000005 (tau), A000203 (sigma), A005063, A008472 (sopf), A010051, A023890, A103164.

Table[DivisorSigma[1, n] + Sum[p (p - 1), {p, Select[Divisors[n], PrimeQ]}], {n, 100}]

a(n) = sigma(n) - sopf(n) + sopf_2(n), where sopf_2(n) = Sum_{p|n, p prime} p^2.
a(n) = Sum_{d|n} d^tau(d^c(d)), where c = A010051.
a(n) = A023890(n) + A005063(n).
a(p^k) = (p^(k+1)+p^3-2*p^2+p-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025

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A354432 a(n) is the numerator of the sum of the reciprocals of the nonprime divisors of n.

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A354433 a(n) is the denominator of the sum of the reciprocals of the nonprime divisors of n.

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A194580 Nonprime numbers with a sum of nonprime divisors which is a perfect square.

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A225028 Numbers m such that A206773(m) = k and A206773(k) = m.

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A279051 Sum of odd nonprime divisors of n.

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A279290 Sum of cubes of nonprime divisors of n.

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A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

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