A184387 -id:A184387 - cp's OEIS frontend

A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

1, 3, 3, 6, 3, 10, 3, 10, 6, 10, 3, 21, 3, 10, 10, 15, 3, 21, 3, 21, 10, 10, 3, 36, 6, 10, 10, 21, 3, 36, 3, 21, 10, 10, 10, 45, 3, 10, 10, 36, 3, 36, 3, 21, 21, 10, 3, 55, 6, 21, 10, 21, 3, 36, 10, 36, 10, 10, 3, 78, 3, 10, 21, 28, 10, 36, 3, 21, 10, 36, 3, 78

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Length of row n in triangle A187207. - Omar E. Pol, Aug 07 2011

Number of pairs of even divisors of 2n, (d1,d2), such that d1<=d2. - Wesley Ivan Hurt, Aug 24 2020

			For n = 4; tau(4) = 3; a(4) = 1+2+3 = 6.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A000005 (tau), A000217 (triangular numbers).
Cf. A184387, A184388, A184390, A184391, A130674.
Cf. A035116, A066446.
Cf. A187207, A135539, A337362, A129308.

a184389 = a000217 . a000005'  -- Reinhard Zumkeller, Sep 08 2015

A184389 := proc(n)
    A000217(numtheory[tau](n)) ;
end proc: # R. J. Mathar, Oct 04 2014

((#+1)#)/2&/@DivisorSigma[0,Range[80]] (* Harvey P. Dale, Feb 27 2013 *)

a(n) = my(nd=numdiv(n)); nd*(nd+1)/2; \\ Michel Marcus, Jun 25 2016

a(n) = A000217(A000005(n)) = (1/2)*A000005(n)*(A000005(n)+1).
a(n) = A066446(n) + A000005(n) = A035116(n) - A066446(n). - Reinhard Zumkeller, Sep 08 2015
Dirichlet g.f.: zeta(s)^2*(zeta(s)^2 + zeta(2*s))/(2*zeta(2*s)). - Ilya Gutkovskiy, Jun 25 2016
a(n) = Sum_{d1|(2*n), d2|(2*n), d1 and d2 even, d1<=d2} 1. - Wesley Ivan Hurt, Aug 24 2020
a(n) = Sum_{d|n} A018892(d). - Daniel Suteu, Jan 08 2021
a(n) = Sum_{d|n} A135539(n,d). - Ridouane Oudra, May 29 2025
a(n) = A337362(n) + A129308(n). - Ridouane Oudra, May 30 2025

A184388 a(n) = product of numbers from 1 to sigma(n), where sigma(n) = A000203(n).

Original entry on oeis.org

1, 6, 24, 5040, 720, 479001600, 40320, 1307674368000, 6227020800, 6402373705728000, 479001600, 304888344611713860501504000000, 87178291200, 620448401733239439360000, 620448401733239439360000, 8222838654177922817725562880000000, 6402373705728000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

			For n = 4; sigma(4) = 7; a(n) = 1*2*3*4*5*6*7 = 5040.

Cf. A184387, A184389, A184390, A184391, A130674.

DivisorSigma[1,Range[20]]! (* Harvey P. Dale, Apr 09 2019 *)

a(n)=sigma(n)! \\ Charles R Greathouse IV, Feb 13 2013

a(n) = sigma(n)! = Product_(i = 1,…,sigma(n)) i = A000142(A000203(n)) = (A000203(n))!.

Corrected and extended by Harvey P. Dale, Apr 09 2019

A184390 a(n) = sum of numbers from 1 to pi(n), where pi(n) = A007955(n).

Original entry on oeis.org

1, 3, 6, 36, 15, 666, 28, 2080, 378, 5050, 66, 1493856, 91, 19306, 25425, 524800, 153, 17009028, 190, 32004000, 97461, 117370, 276, 55037822976, 7875, 228826, 266085, 240956128, 435, 328050405000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

			For n = 6; pi(6) = 36; a(n) = (1/2)*36*37 = 666.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A184387, A184388, A184389, A184391, A130674.

# (#+1)/2&/@Array[Times@@Divisors[#]&,40] (* Harvey P. Dale, Oct 05 2012 *)

from math import isqrt
from sympy import divisor_count
def A184390(n): return (m:=((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)))*(m+1)//2 # Chai Wah Wu, Jun 25 2022

a(n) = Sum_{i = 1..pi(n)} i = A000217(A007955(n)) = (1/2)*A007955(n)*(A007955(n)+1).

A184391 a(n) = product of numbers from 1 to A007955(n).

Original entry on oeis.org

1, 2, 6, 40320, 120, 371993326789901217467999448150835200000000, 5040, 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000, 10888869450418352160768000000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Robert P. P. McKone, Table of n, a(n) for n = 1..11

Cf. A184387, A184388, A184389, A184390, A130674.

A184391[n_] := Product[i, {i, 1, n^(DivisorSigma[0, n]/2)}]; Table[A184391[n], {n, 1, 9}] (* Robert P. P. McKone, Feb 04 2021 *)

from math import isqrt, factorial
from sympy import divisor_count
def A184391(n): return factorial((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(n)! = A000142(A007955(n)).

a(9) from Robert P. P. McKone, Feb 04 2021

A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 78, 16, 120, 52, 144, 24, 406, 28, 192, 192, 496, 36, 780, 40, 903, 256, 288, 48, 1830, 124, 336, 320, 1596, 60, 2628, 64, 2016, 384, 432, 384, 4186, 76, 480, 448, 4095, 84, 4656, 88, 2688, 2184, 576, 96, 7750, 228, 2976, 576, 3136, 108, 7260

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 200 terms from Vincenzo Librandi)
Jon Maiga, Computer-generated formulas for A237290, Sequence Machine.

Cf. A000203, A119348, A005153, A119347 (count of the same numbers), A184387, A229335, A237287, A237289.

isSumDist := proc(n,k)
    local dvs,s ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237290 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
end proc:
seq(A237290(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* Giovanni Resta, Mar 13 2014 *)

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek));); sum(i=1, #vks, vks[i]);} \\ Michel Marcus, Mar 09 2014

A237290(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); vecsum(c); }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A237290(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237289(n).
a(p) = 2(p+2) for odd primes p.
a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).
a(n) = A000203(n) * (A119347(n)+1) / 2. [Found by Sequence Machine and easily seen to be true. Compare for example to the formulas of A229335.] - Antti Karttunen, Nov 29 2024

A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 9, 0, 20, 0, 39, 27, 54, 0, 77, 108, 108, 0, 135, 0, 170, 0, 272, 378, 252, 0, 372, 567, 500, 0, 405, 0, 464, 0, 792, 1053, 792, 0, 665, 1350, 1148, 0, 819, 0, 902, 882, 897, 2052, 1080, 0, 1425, 1395, 2052, 1715, 1377, 0, 2052, 0, 2600, 3375, 1710

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 2 + 3 + 4 = 9 (numbers 2, 3 and 4 are not a sum of any subset of distinct divisors of 5).
Numbers n = 14 and 15 are an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237290(14) = A237290(15).
a(14) = 4+5+6+11+12+13+18+19+20 = a(15) = 2+7+10+11+12+13+14+17+22 = 108.
a(6) = 0 as 6 is practical; the sums into distinct divisors from 1 through 12 are 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 2 + 3, 7 through 12 are (1 through 6) + 6. So none are not a sum distinct divisors of 6. - _David A. Corneth_, Jul 22 2025

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A119348, A005153, A184387, A237287, A237290.

isSumDist := proc(n,k)
    local dvs ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237289 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if not isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
    a ;
end proc:
seq(A237289(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Block[{d = Divisors@n, s}, s = Plus @@ d; s*(s + 1)/2 - Plus @@ Union[Plus @@@ Subsets@d]]; m = Array[a, 59] (* Giovanni Resta, Mar 13 2014 *)

from sympy import divisors
def A237289(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return (s*(s+1)>>1)-sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237290(n).
a(p) = p(p - 1) / 2 - 1 for p = prime > 2.
a(n) = 0 for practical numbers (A005153), a(n) > 0 for numbers that are not practical (A237287).
a(n) = A184387(n) - A229335(n) for numbers n such that A119347(n) = A100587(n).

a(55) and a(57)-a(59) corrected by Giovanni Resta, Mar 13 2014

A206028 a(n) is the sum of distinct values of sigma(d) where d runs over the divisors of n and sigma = A000203.

Original entry on oeis.org

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 248

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A007429: a(66) = 248, A007429(66) = 260. Number 66 is the smallest number with at least two divisors d with the same sigma(d); see A206030.
In A007429 all values of sigma(d) of the divisors d of n are included in the sum with repetitions allowed. In this sequence only the distinct values of sigma(d) of the divisors d of n are included in the sum.
If a term is a prime p when n = 2^j then p = 2^(j+2)-(j+3) is also a term of A099440 (primes of the form 2^n-n-1). Greater of twin primes are terms. - Metin Sariyar, Apr 03 2020

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Sum of k = 1+3+4+12 = 20.
For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Sum of k = 1+3+4+12+36+48+144 = 248 (note that only one twelve is added.).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000203, A000217, A007429, A184387, A206029, A206030.

Table[Total[Union[DivisorSigma[1, Divisors[n]]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)={vecsum(Set(apply(sigma, divisors(n))))} \\ Andrew Howroyd, Aug 01 2018

a(p) = p+2, a(pq) = (p+2)*(q+2) for p, q = distinct primes.
a(n) = A184387(n) - A206029(n) = A000217(A000203(n)) - A206029(n).
a(2^n) = 2^(n+2) - (n+3). - Metin Sariyar, Apr 09 2020

Name clarified by David A. Corneth, Aug 01 2018

a(62)-a(66) from Andrew Howroyd, Aug 01 2018

A206029 a(n) = sum of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

0, 2, 5, 17, 14, 58, 27, 94, 73, 143, 65, 351, 90, 264, 265, 439, 152, 708, 189, 826, 483, 614, 275, 1700, 458, 843, 762, 1497, 434, 2488, 495, 1896, 1111, 1409, 1113, 3988, 702, 1746, 1521, 3913, 860, 4476, 945, 3427, 2955, 2528, 1127, 7465, 1587, 4219

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

In sequence A007429 are added all values of sigma(d) of all divisors d of numbers n, in sequence A206028 are added only distinct values of sigma(d) of all divisors d of numbers n and in sequence a(n) are added numbers k (1<=k<=sigma(n)) such that sigma(d) = k has no solution for neither divisor d of number n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Sum of k = 2+5+6+7+8+9+10+11 = 58.

Cf. A000203, A206030, A007429, A206028.

Table[Total[Complement[Range[DivisorSigma[1, n]], DivisorSigma[1, Divisors[n]]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n) = A184387(n) - A206028 = A000217(A000203(n)) - A206028.

cp's OEIS Frontend

A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

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A184388 a(n) = product of numbers from 1 to sigma(n), where sigma(n) = A000203(n).

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A184390 a(n) = sum of numbers from 1 to pi(n), where pi(n) = A007955(n).

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A184391 a(n) = product of numbers from 1 to A007955(n).

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A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

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A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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A206028 a(n) is the sum of distinct values of sigma(d) where d runs over the divisors of n and sigma = A000203.

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