A247837 -id:A247837 - cp's OEIS frontend

A000593 Sum of odd divisors of n.

1, 1, 4, 1, 6, 4, 8, 1, 13, 6, 12, 4, 14, 8, 24, 1, 18, 13, 20, 6, 32, 12, 24, 4, 31, 14, 40, 8, 30, 24, 32, 1, 48, 18, 48, 13, 38, 20, 56, 6, 42, 32, 44, 12, 78, 24, 48, 4, 57, 31, 72, 14, 54, 40, 72, 8, 80, 30, 60, 24, 62, 32, 104, 1, 84, 48, 68, 18, 96, 48, 72, 13, 74, 38, 124

Offset: 1

N. J. A. Sloane

Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - Michael Somos, May 17 2013
A069289(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A001227 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
For the g.f.s given below by Somos Oct 29 2005, Jovovic, Oct 11 2002 and Arndt, Nov 09 2010, see the Hardy-Wright reference, proof of Theorem 382, p. 312, with x^2 replaced by x. - Wolfdieter Lang, Dec 11 2016
a(n) is also the total number of parts in all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017
It seems that a(n) divides A000203(n) for every n. - Ivan N. Ianakiev, Nov 25 2017 [Yes, see the formula dated Dec 14 2017].
Also, alternating row sums of A126988. - Omar E. Pol, Feb 11 2018
Where a(n) shows the number of equivalence classes of Hurwitz quaternions with norm n (equivalence defined by right multiplication with one of the 24 Hurwitz units as in A055672), A046897(n) seems to give the number of equivalence classes of Lipschitz quaternions with norm n (equivalence defined by right multiplication with one of the 8 Lipschitz units). - R. J. Mathar, Aug 03 2025

			G.f. = x + x^2 + 4*x^3 + x^4 + 6*x^5 + 4*x^6 + 8*x^7 + x^8 + 13*x^9 + 6*x^10 + 12*x^11 + ...

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 496, pp. 69-246, Ellipses, Paris, 2004.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003, p. 312.
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg, 1994, p. 133.
John Riordan, Combinatorial Identities, Wiley, 1968, p. 187.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Francesca Aicardi, Matricial formulas for partitions, arXiv:0806.1273 [math.NT], 2008.
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999; Canad. J. Math. 51 (1999), 1258-1276.
John A. Ewell, On the sum-of-divisors function, Fib. Q., 45 (2007), 205-207.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Kaya Lakein and Anne Larsen, A Proof of Merca's Conjectures on Sums of Odd Divisor Functions, arXiv:2107.07637 [math.NT], 2021.
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017, also here
Mircea Merca, Congruence identities involving sums of odd divisors function, Proceedings of the Romanian Academy, Series A, Volume 22, Number 2/2021, pp. 119-125.
Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
N. J. A. Sloane, Transforms.
H. J. Stephen Smith, Report on the Theory of Numbers. — Part VI., Report of the 35 Meeting of the British Association for the Advancement of Science (1866). See p. 336.
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Eric Weisstein's World of Mathematics, Partition Function Q.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000203, A000265, A001227, A006128, A050999, A051000, A051001, A051002, A065442, A078471 (partial sums), A069289, A247837 (subset of the primes).

Cf. A301799, A301800.

a000593 = sum . a182469_row  -- Reinhard Zumkeller, May 01 2012, Jul 25 2011

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[j*x^j/(1+x^j): j in [1..2*m]])  )); // G. C. Greubel, Nov 07 2018

[&+[d:d in Divisors(n)|IsOdd(d)]:n in [1..75]]; // Marius A. Burtea, Aug 12 2019

A000593 := proc(n) local d,s; s := 0; for d from 1 by 2 to n do if n mod d = 0 then s := s+d; fi; od; RETURN(s); end;

Table[a := Select[Divisors[n], OddQ[ # ]&]; Sum[a[[i]], {i, 1, Length[a]}], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
f[n_] := Plus @@ Select[ Divisors@ n, OddQ]; Array[f, 75] (* Robert G. Wilson v, Jun 19 2011 *)
a[ n_] := If[ n < 1, 0, Sum[ -(-1)^d n / d, {d, Divisors[ n]}]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# n / # &]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d, {d, Divisors[ n]}]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # < 3, 1, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 15 2015 *)
Array[Total[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] &, {75}] (* Michael De Vlieger, Apr 07 2016 *)
Table[SeriesCoefficient[n Log[QPochhammer[-1, x]], {x, 0, n}], {n, 1, 75}] (* Vladimir Reshetnikov, Nov 21 2016 *)
Table[DivisorSum[n,#&,OddQ[#]&],{n,80}] (* Harvey P. Dale, Jun 19 2021 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d+1) * n/d))}; /* Michael Somos, May 29 2005 */

N=66; x='x+O('x^N); Vec( serconvol( log(prod(j=1,N,1+x^j)), sum(j=1,N,j*x^j)))  /* Joerg Arndt, May 03 2008, edited by M. F. Hasler, Jun 19 2011 */

s=vector(100);for(n=1,100,s[n]=sumdiv(n,d,d*(d%2)));s /* Zak Seidov, Sep 24 2011*/

a(n)=sigma(n>>valuation(n,2)) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import factorint
def A000593(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) # Chai Wah Wu, Sep 09 2021

[sum(k for k in divisors(n) if k % 2) for n in (1..75)] # Giuseppe Coppoletta, Nov 02 2016

Inverse Moebius Transform of [0, 1, 0, 3, 0, 5, ...].
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)).
a(2*n) = A000203(2*n)-2*A000203(n), a(2*n+1) = A000203(2*n+1). - Henry Bottomley, May 16 2000
a(2*n) = A054785(2*n) - A000203(2*n). - Reinhard Zumkeller, Apr 23 2008
Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d divides n} (-1)^(d+1)*n/d, Dirichlet convolution of A062157 with A000027. - Vladeta Jovovic, Sep 06 2002
Sum_{k=1..n} a(k) is asymptotic to c*n^2 where c=Pi^2/24. - Benoit Cloitre, Dec 29 2002
G.f.: Sum_{n>0} n*x^n/(1+x^n). - Vladeta Jovovic, Oct 11 2002
G.f.: (theta_3(q)^4 + theta_2(q)^4 -1)/24.
G.f.: Sum_{k>0} -(-x)^k / (1 - x^k)^2. - Michael Somos, Oct 29 2005
a(n) = A050449(n)+A050452(n); a(A000079(n))=1; a(A005408(n))=A000203(A005408(n)). - Reinhard Zumkeller, Apr 18 2006
From Joerg Arndt, Nov 09 2010: (Start)
G.f.: Sum_{n>=1} (2*n-1) * q^(2*n-1) / (1-q^(2*n-1)).
G.f.: deriv(log(P)) = deriv(P)/P where P = Product_{n>=1} (1 + q^n). (End)
Dirichlet convolution of A000203 with [1,-2,0,0,0,...]. - R. J. Mathar, Jun 28 2011
a(n) = Sum_{k = 1..A001227(n)} A182469(n,k). - Reinhard Zumkeller, May 01 2012
G.f.: -1/Q(0), where Q(k) = (x-1)*(1-x^(2*k+1)) + x*(-1 +x^(k+1))^4/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) = Sum_{k=1..n} k*A000009(k)*A081362(n-k). - Mircea Merca, Feb 26 2014
a(n) = A000203(n) - A146076(n). - Omar E. Pol, Apr 05 2016
a(2*n) = a(n). - Giuseppe Coppoletta, Nov 02 2016
a(n) = n * [x^n] log((-1; x)inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 21 2016
From Wolfdieter Lang, Dec 11 2016: (Start)
G.f.: Sum_{n>=1} x^n*(1+x^(2*n))/(1-x^(2*n))^2, from the second to last equation of the proof to Theorem 382 (with x^2 -> x) of the Hardy-Wright reference, p. 312.
a(n) = Sum_{d|n} (-d)*(-1)^(n/d), commutating factors of the D.g.f. given above by Jovovic, Oct 11 2002. See also the a(n) version given by Jovovic, Sep 06 2002. (End)
a(n) = A000203(n)/A038712(n). - Omar E. Pol, Dec 14 2017
a(n) = A000203(n)/(2^(1 + (A183063(n)/A001227(n))) - 1). - Omar E. Pol, Nov 06 2018
a(n) = A000203(2n) - 2*A000203(n). - Ridouane Oudra, Aug 28 2019
From Peter Bala, Jan 04 2021: (Start)
a(n) = (2/3)*A002131(n) + (1/3)*A002129(n) = (2/3)*A002131(n) + (-1)^(n+1)*(1/3)*A113184(n).
a(n) = A002131(n) - (1/2)*A146076; a(n) = 2*A002131(n) - A000203(n). (End)
a(n) = A000203(A000265(n)) - John Keith, Aug 30 2021
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = A065442 - 1 = 0.60669... . - Amiram Eldar, Dec 14 2024

A247820 Numbers k such that sigma(2*k-1) is a prime p.

Original entry on oeis.org

5, 13, 145, 365, 841, 1201, 1741, 2521, 3961, 5101, 7813, 8581, 13945, 14281, 14965, 41761, 42925, 73345, 139921, 229165, 245701, 265721, 276025, 289561, 298765, 341965, 351961, 353641, 367225, 414961, 595141, 601705, 676285, 697381, 711625, 740545, 942565

Offset: 1

Jaroslav Krizek, Sep 24 2014

nonn

Supersequence of A247789.

Corresponding values of primes p for a(n) are A008438(a(n)-1) or A247837.

			Number 13 is in sequence because sigma(2*13-1) = sigma(25) = 31 (prime).

Martin Møller Skarbiniks Pedersen, Table of n, a(n) for n = 1..1000

Cf. A000203, A008438, A247789, A247837.

[n: n in [1..10000000] | IsPrime(SumOfDivisors(2*n-1))];

for(n=1,10^7,if(isprime(sigma(2*n-1)),print1(n,", "))) \\ Derek Orr, Sep 25 2014

A278911 Odd numbers with prime sum of divisors.

Original entry on oeis.org

9, 25, 289, 729, 1681, 2401, 3481, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 83521, 85849, 146689, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089

Offset: 1

Jaroslav Krizek, Nov 30 2016

nonn

Also odd numbers with prime number and sum of divisors; if the sum of divisors is prime, then the number of divisors is prime.
Values of prime sums are sorted in A247837.
Subsequence of A050150 (odd numbers with prime number of divisors).
Odd terms of A023194.
All terms are squares of the form p^e such that p is odd prime and e+1 is a prime.

			sigma(9) = 13 (prime).

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

Cf. A000203, A005408, A023194, A050150, A193070, A247837, A278914.

[n: n in[2..10^7] | IsOdd(n) and IsPrime(SumOfDivisors(n)) and IsPrime(NumberOfDivisors(n))];

N:= 10^7: # to get all terms <= N
Ps:= select(isprime, [seq(i,i=3..floor(N^(1/2)),2)]):
es:= map(`-`,select(isprime, [seq(i,i=3..floor(log[3](N))+1,2)]),1):
Pes:= [seq(seq([p,e],p=Ps),e=es)]:
filter:= proc(pe) local v; v:= (pe[1]^(pe[2]+1)-1)/(pe[1]-1); pe[1]^pe[2] <= N and isprime(v) end proc:
sort(map(pe -> pe[1]^pe[2], select(filter, Pes))); # Robert Israel, Jan 22 2019

Select[Range[1, 2*10^6, 2], PrimeQ@DivisorSigma[1, #] &] (* Michael De Vlieger, Dec 01 2016 *)

isok(n) = (n % 2) && isprime(sigma(n)); \\ Michel Marcus, Dec 01 2016

a(n) = A193070(n)^2. - Michel Marcus, Dec 01 2016

A247836 Primes p of the form sigma(2q-1) where q is a prime.

Original entry on oeis.org

13, 31, 2801, 3541, 5113, 10303, 17293, 30941, 88741, 292561, 732541, 1191373, 1395943, 2922391, 3281533, 7450171, 7781311, 12190573, 15916111, 22292563, 28792661, 67757593, 93886411, 103052953, 112137511, 124802413, 135059263, 181427431

Offset: 1

Jaroslav Krizek, Sep 24 2014

nonn

Subsequence of A247837.

			Prime 13 is in sequence because there is prime 5 such that sigma(2*5-1) = sigma(9) = 13.

Cf. A000203, A247837, A247820.

Sort([a: p in PrimesUpTo(5000000) | IsPrime(a) where a is SumOfDivisors(2*p-1)]); // corrected by Jaroslav Krizek, Nov 16 2014

Select[DivisorSigma[1,2Prime[Range[3*10^6]]-1],PrimeQ] (* The program generates the first 23 terms of the sequence. *) (* Harvey P. Dale, Jul 17 2025 *)

forprime(p=1,10^9,if(isprime(sigma(2*p-1)),print1(sigma(2*p-1),", "))) \\ Derek Orr, Sep 25 2014

A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

Original entry on oeis.org

18, 50, 578, 1458, 3362, 4802, 6962, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 167042, 171698, 293378, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562, 1468898, 1659842, 2380562, 2406818, 2705138, 2789522

Offset: 1

Michel Lagneau, May 30 2016

nonn

a(n) is of the form 2q^2 where q is an odd numbers for which sigma(q^2) is prime (A193070).
The corresponding primes p are 13, 31, 307, 1093, 1723, 2801, 3541, 5113, 8011, 10303, 19531, 17293, 28057, 30941, 30103, 88741, 86143, 147073, 292561, 459007, 492103, 797161, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 830833, 1191373, 1204507, ...
We observe an interesting property: each prime p is element of A053183 (primes of the form m^2 + m + 1 when m is prime) or element of A247837 (primes of the form sigma(2m-1) for a number m) or element of both A053183 and A247837.
Examples:
The numbers 13, 31, 307, 1723, 3541, 5113,... are in A053183;
The numbers 13, 31, 307, 1093, 1723, 2801, 3541,...are in A247837;
The numbers 13, 31, 307, 1723, 3541,... are in A053183 and A247837.

			18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18}. The sum of the odd divisors is 1 + 3 + 9 = 13 and the sum of the even divisors is 2 + 6 + 18 = 26 = 2*13.

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

Cf. A002384, A053183, A193070, A247837, A278911.

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        if isprime(s1) and s0=2*s1
        then
        printf(`%d, `, n):
         else fi:
     od:

Select[Range[2, 3000000, 2], And[PrimeQ[Total@ Select[#, EvenQ]/2], PrimeQ@ Total@ Select[#, OddQ]] &@ Divisors@ # &] (* Michael De Vlieger, May 30 2016 *)
sodpQ[n_]:=Module[{d=Divisors[n],s},s=Total[Select[d,OddQ]];PrimeQ[ s] && Total[ Select[d,EvenQ]]==2s]; Select[Range[2,279*10^4,2],sodpQ] (* Harvey P. Dale, Dec 01 2020 *)
2 * Select[Range[1, 1200, 2]^2, PrimeQ@DivisorSigma[1, #] &] (* Amiram Eldar, Jul 19 2022 *)

is(n)=my(t); n%4==2 && issquare(n/2,&t) && isprime(n/2+t+1) \\ Charles R Greathouse IV, Jun 08 2016

a(n) >> n^2. - Charles R Greathouse IV, Jun 08 2016

a(n) = 2 * A278911(n) = 2 * A193070(n)^2. - Amiram Eldar, Jul 19 2022

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A000593 Sum of odd divisors of n.

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A247820 Numbers k such that sigma(2*k-1) is a prime p.

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A278911 Odd numbers with prime sum of divisors.

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A247836 Primes p of the form sigma(2q-1) where q is a prime.

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A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

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A273938 Sum of the divisors of the n-th odd prime power.

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