A048050 -id:A048050 - cp's OEIS frontend

A255369 a(n) = (sigma(n)-n-1)*(2-mu(n)), where sigma(n) is the sum of the divisors of n and mu(n) is the Möbius function.

-1, 0, 0, 4, 0, 5, 0, 12, 6, 7, 0, 30, 0, 9, 8, 28, 0, 40, 0, 42, 10, 13, 0, 70, 10, 15, 24, 54, 0, 123, 0, 60, 14, 19, 12, 108, 0, 21, 16, 98, 0, 159, 0, 78, 64, 25, 0, 150, 14, 84, 20, 90, 0, 130, 16, 126, 22, 31, 0, 214, 0, 33, 80, 124, 18, 231, 0, 114

Offset: 1

Wesley Ivan Hurt, May 04 2015

sign

a(n) = 0 if and only if n is prime. If n is semiprime, then a(n) = sopfr(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Antti Karttunen, Sequence plotted up to n=10000, showing the details better

Cf. A000203 (sigma), A008683 (Möbius function), A001414 (sopfr).

Cf. A048050 (Chowla's function), A228483 (2-mu(n)).

[(SumOfDivisors(n)-n-1)*(2-MoebiusMu(n)): n in [1..80]]; // Vincenzo Librandi, May 05 2015

with(numtheory): a:=n->(sigma(n)-n-1)*(2-mobius(n)): seq(a(n), n=1..100);

Table[(DivisorSigma[1, n] - n - 1) (2 - MoebiusMu[n]), {n, 100}]

a(n)=(sigma(n)-n-1)*(2-moebius(n)) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; say +(divisor_sum($)-$-1)*(2-moebius($)) for 1..80;  # _Dana Jacobsen, May 13 2015

a(n) = A048050(n) * A228483(n) for n > 1, a(1) = -1.

Formula corrected for case n=1 by Antti Karttunen, Feb 25 2018

A257533 Sum of the proper divisors of the n-th semiprime.

Original entry on oeis.org

2, 5, 3, 7, 9, 8, 10, 13, 5, 15, 14, 19, 12, 21, 16, 25, 7, 20, 16, 22, 31, 33, 18, 26, 39, 18, 43, 22, 45, 32, 20, 34, 49, 24, 55, 40, 28, 61, 24, 11, 63, 44, 46, 26, 69, 50, 73, 24, 34, 75, 36, 81, 56, 30, 85, 13, 62, 91, 64, 42, 28, 99, 70

Offset: 1

R. J. Mathar, Apr 28 2015

For purposes of this sequence, the proper divisors of a number include all divisors other than 1 and the number itself. - Harvey P. Dale, Mar 15 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A083681.

Maple
```
seq(A048050(A001358(n)),n=1..80) ;
```

Total[Rest[Most[Divisors[#]]]]&/@Select[Range[250],PrimeOmega[#]==2&] (* Harvey P. Dale, Mar 15 2022 *)

go(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), listput(v,[p*q,if(qu[2],v) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A048050(A001358(n)).

A083681(n)-a(n) = A088707(n).

A273460 Numbers n such that sum of the divisors of n (except 1 and n) is equal to the product of the digits of n.

Original entry on oeis.org

98, 101, 103, 107, 109, 307, 329, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901

Offset: 1

Michel Lagneau, May 23 2016

Or numbers n such that A048050(n) = A007954(n).

Most of the terms are primes which have at least one 0 among their digits (A056709). The composite numbers of the sequence are 98, 329, 3383, 4343, 5561, 6623, 12773, 17267, 21479, 57721, 129383, 136259, 142943, 172793, 246959, 256631, 292571,...

			sigma(98) - 98 - 1 = 171 - 98 - 1 = 72 and 8*9 = 72 so 98 is in the sequence.

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

Cf. A007954, A048050, A056709, A069675.

with(numtheory):
for n from 1 to 3000 do:
  q:=convert(n,base,10):n0:=nops(q):
  pr:=product('q[i]', 'i'=1..n0):p:=sigma(n)-n-1:
   if p=pr
    then
    printf(`%d, `,n):
    else
   fi:
od:

Do[If[DivisorSigma[1, n]-n-1==Apply[Times, IntegerDigits[n]], Print[n]], {n, 2000}]
Select[Range[2,2000],Total[Most[Rest[Divisors[#]]]]==Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Jul 20 2019 *)

A281006 a(n) = A000203(n) - A052928(n-1).

Original entry on oeis.org

1, 3, 2, 5, 2, 8, 2, 9, 5, 10, 2, 18, 2, 12, 10, 17, 2, 23, 2, 24, 12, 16, 2, 38, 7, 18, 14, 30, 2, 44, 2, 33, 16, 22, 14, 57, 2, 24, 18, 52, 2, 56, 2, 42, 34, 28, 2, 78, 9, 45, 22, 48, 2, 68, 18, 66, 24, 34, 2, 110, 2, 36, 42, 65, 20, 80, 2, 60, 28, 76, 2, 125, 2, 42, 50, 66, 20, 92, 2, 108, 41, 46, 2, 142, 24, 48, 34, 94, 2

Offset: 1

Omar E. Pol, Jan 23 2017

nonn

a(n) = 2 iff n is an odd prime (A065091).

Has a symmetric representation as a narrow pyramid with holes, in the same way as A249351.

			A000203    A052928     a(n)
.   1    -    0    =    1
.   3    -    0    =    3
.   4    -    2    =    2
.   7    -    2    =    5
.   6    -    4    =    2
.  12    -    4    =    8
...

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A004526, A048050, A052928, A065091, A176059, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A262626, A279387, A281010.

A281006(n) = (sigma(n) - 2*((n-1)>>1)); \\ Antti Karttunen, Sep 25 2018

a(n) = sigma(n) - 2*floor((n - 1)/2) = A000203(n) - 2*A004526(n-1).

a(n) = A048050(n) + A176059(n), n >= 2.

A281626 a(n) = (sum of trivial divisors of n) - (sum of nontrivial divisors of n).

Original entry on oeis.org

1, 3, 4, 3, 6, 2, 8, 3, 7, 4, 12, -2, 14, 6, 8, 3, 18, -1, 20, 0, 12, 10, 24, -10, 21, 12, 16, 2, 30, -10, 32, 3, 20, 16, 24, -17, 38, 18, 24, -8, 42, -10, 44, 6, 14, 22, 48, -26, 43, 9, 32, 8, 54, -10, 40, -6, 36, 28, 60, -46, 62, 30, 24, 3, 48, -10, 68, 12

Offset: 1

Jaroslav Krizek, Feb 11 2017

sign

Trivial divisors of n are numbers 1 and n.
a(n) = 0 for numbers in A088831 (numbers n whose abundance is 2).
a(n) <= 0 for numbers in A005101 (abundant numbers).
a(n) > 0 for numbers in A263837 (non-abundant numbers).

			a(20) = (20+1) - (2+4+5+10) = 0.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000079, A000396, A005101, A006881, A048050, A088831, A263837.

[1] cat [2*(n+1) - SumOfDivisors(n): n in [2..100]];

Table[If[n == 1, 1, 2 (n + 1) - DivisorSigma[1, n]], {n, 68}] (* Michael De Vlieger, Feb 11 2017 *)

a(1) = 1; for n>1, a(n) = (n+1) - (sigma(n) - n - 1) = 2*(n+1) - sigma(n) = n + 1 - A048050(n).
a(A000396(n)) = 2 for n >= 1.
a(A000079(n)) = 3 for n >= 1.
a(A006881(n)) = phi(n).
a(p) = p + 1 for p prime.

A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).

Original entry on oeis.org

0, 0, 0, 4, 0, 13, 0, 20, 9, 29, 0, 65, 0, 53, 34, 84, 0, 130, 0, 145, 58, 125, 0, 273, 25, 173, 90, 265, 0, 399, 0, 340, 130, 293, 74, 614, 0, 365, 178, 609, 0, 735, 0, 625, 340, 533, 0, 1105, 49, 754, 298, 865, 0, 1183, 146, 1113, 370, 845, 0, 1859, 0, 965, 580, 1364, 194, 1743, 0, 1465, 538, 1599, 0, 2550, 0, 1373, 884

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Sum of squares of divisors of n except 1 and n^2 (sum of squares of nontrivial divisors of n).

			a(6) = 13 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial {2, 3} and 2^2 + 3^2 = 13.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A000290, A001157, A001248, A001358, A005063, A008578, A027751, A048050, A053807, A067558, A070824.

nmax = 75; Rest[CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
Join[{0}, Table[DivisorSigma[2, n] - n^2 - 1, {n, 2, 75}]]

A291108(n) = sumdiv(n,d,if((1==d)||(n==d),0,d^2)); \\ Antti Karttunen, Jan 22 2025

G.f.: Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).
a(n) = A001157(n) -  A000290(n) - 1 for n > 1.
a(n) = A067558(n) - 1 for n > 1.
a(n) = A005063(n) if n is a semiprime (A001358).
a(n) = 0 if n is a prime or 1 (A008578).
a(n) = n if n is a square of prime (A001248).
a(p^k) = (p^(2*k) - p^2)/(p^2 - 1) for p is a prime and k > 0.

A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

Original entry on oeis.org

15, 20, 35, 95, 104, 119, 143, 207, 209, 287, 319, 323, 377, 464, 527, 559, 650, 779, 899, 923, 989, 1007, 1023, 1189, 1199, 1343, 1349, 1519, 1763, 1919, 1952, 2015, 2159, 2507, 2759, 2911, 2915, 2975, 3239, 3599, 3827, 4031, 4199, 4607, 5183, 5207, 5249

Offset: 1

Zdenek Cervenka, Oct 23 2017

nonn

Numbers k such that sigma(k)/(sigma(k)-k-1) is a positive integer.

			15 is in the sequence since sigma(15)/(sigma(15)-15-1) = 24/8 = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002808 (composite numbers).
Cf. A000203, A048050.
Cf. A088831 (k=2), A063906 (k=3).

Quiet@ Select[Range[2, 5300], And[IntegerQ[#], # > 1] &[#2/(#2 - #1 - 1)] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Oct 24 2017 *)

lista(nn) = forcomposite(n=1, nn, if (denominator(sigma(n)/(sigma(n)-n-1)) == 1, print1(n, ", "))); \\ Michel Marcus, Oct 24 2017

list(lim)=my(v=List(),s,t); forfactored(n=9,lim\1, s=sigma(n); t=s-n[1]-1; if(t && s%t==0, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 11 2017

This sequence gives all numbers a(n) in increasing order which satisfy A000203(a(n))/A048050(a(n)) = A000203(a(n))/(A000203(a(n)) - (a(n)+1)) = k(n), with a positive integer k(n) for n >= 1. - Wolfdieter Lang, Nov 10 2017

Edited by Wolfdieter Lang, Nov 10 2017

Name corrected by Michel Marcus, Nov 12 2017

A309400 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 5, 5, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11

Offset: 1

Omar E. Pol, Nov 30 2019

The number of parts in row n equals sigma(n) = A000203(n), the sum of the divisors of n. More generally, the number of parts congruent to 0 (mod m) in row m*n equals sigma(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of the aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The sum of row n equals n*A000005(n) = A038040(n).
Records in row n give the n-th row of A027750.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A056538(n,k) parts.
Column 1 gives A000012.
Right border gives A000027.

			Triangle begins:
[1];
[1,1], [2];
[1,1,1], [3];
[1,1,1,1], [2,2], [4];
[1,1,1,1,1], [5];
[1,1,1,1,1,1], [2,2,2], [3,3], [6];
[1,1,1,1,1,1,1], [7];
[1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8];
[1,1,1,1,1,1,1,1,1], [3,3,3], [9];
[1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10];
[1,1,1,1,1,1,1,1,1,1,1], [11];
[1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [12];
[1,1,1,1,1,1,1,1,1,1,1,1,1], [13];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2], [7,7], [14];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [3,3,3,3,3], [5,5,5], [15];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2,2], [4,4,4,4], [8,8], [16];
...

Mirror of A244051.

Cf. A000005, A000012, A000027, A000203, A000217, A001065, A001227, A002131, A027750, A038040, A048050, A056538, A126988, A237593, A245579, A299765, A328365.

A372836 a(n) is the numerator of Sum_{d|n, 1 < d < n} 1/d.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 0, 3, 1, 7, 0, 5, 0, 9, 8, 7, 0, 10, 0, 21, 10, 13, 0, 35, 1, 15, 4, 27, 0, 41, 0, 15, 14, 19, 12, 3, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 25, 1, 21, 20, 45, 0, 65, 16, 9, 22, 31, 0, 107, 0, 33, 40, 31, 18, 7, 0, 57, 26, 73, 0, 61, 0, 39, 16, 63, 18, 89, 0, 21

Offset: 1

Ilya Gutkovskiy, May 14 2024

			0, 0, 0, 1/2, 0, 5/6, 0, 3/4, 1/3, 7/10, 0, 5/4, 0, 9/14, 8/15, 7/8, 0, 10/9, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A017665, A048050, A277790, A277791.

nmax = 80; CoefficientList[Series[Sum[x^(2 k)/(k (1 - x^k)), {k, 2, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = numerator(sumdiv(n, d, if ((d>1) && (dMichel Marcus, May 14 2024

Numerators of coefficients in expansion of Sum_{k>=2} x^(2*k) / (k * (1 - x^k)).

A062973 Chowla function of n is not divisible by phi(n).

Original entry on oeis.org

6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98

Offset: 1

Jason Earls, Jul 24 2001

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

okcfQ[n_]:=!Divisible[DivisorSigma[1,n]-n-1,EulerPhi[n]]
Select[Range[100],okcfQ]  (* Harvey P. Dale, Feb 25 2011 *)

j=[]; for(n=1,400, if(Mod(sigma(n)-n-1,eulerphi(n))!=0,j=concat(j,n))); j

cp's OEIS Frontend

A255369 a(n) = (sigma(n)-n-1)*(2-mu(n)), where sigma(n) is the sum of the divisors of n and mu(n) is the Möbius function.

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A257533 Sum of the proper divisors of the n-th semiprime.

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A273460 Numbers n such that sum of the divisors of n (except 1 and n) is equal to the product of the digits of n.

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A281006 a(n) = A000203(n) - A052928(n-1).

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A281626 a(n) = (sum of trivial divisors of n) - (sum of nontrivial divisors of n).

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A291108 Expansion of Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).

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A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

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A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).