A308039 -id:A308039 - cp's OEIS frontend

A054973 Number of numbers whose divisors sum to n.

1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 3, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 4, 0

Offset: 1

Henry Bottomley, May 16 2000

nonn

a(n) = frequency of values n in A000203(m), where A000203(m) = sum of divisors of m. a(n) >= 1 for such n that A175192(n) = 1, a(n) >= 1 if A000203(m) = n for any m. a(n) = 0 for such n that A175192(n) = 0, a(n) = 0 if A000203(m) = n has no solution. - Jaroslav Krizek, Mar 01 2010
First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368. - Robert G. Wilson v, May 14 2014
a(n) is also the number of positive terms in the n-th row of triangle A299762. - Omar E. Pol, Mar 14 2018
Also the number of integer partitions of n whose parts form the set of divisors of some number (necessarily the greatest part). The Heinz numbers of these partitions are given by A371283. For example, the a(24) = 3 partitions are: (23,1), (15,5,3,1), (14,7,2,1). - Gus Wiseman, Mar 22 2024

			a(12) = 2 since 11 has factors 1 and 11 with 1 + 11 = 12 and 6 has factors 1, 2, 3 and 6 with 1 + 2 + 3 + 6 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sum-of-divisors function).
For partial sums see A074753.
Cf. A002191, A007609, A308039.
The non-strict version is A371284, ranks A371288.
These partitions have ranks A371283, unsorted version A275700.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
Cf. A001221, A002865, A008289, A371286, A371285.

nn = 105; t = Table[0, {nn}]; k = 1; While[k < 6 nn^(3/2)/Pi^2, d = DivisorSigma[1, k]; If[d < nn + 1, t[[d]]++]; k++]; t (* Robert G. Wilson v, May 14 2014 *)
Table[Length[Select[IntegerPartitions[n],#==Reverse[Divisors[Max@@#]]&]],{n,30}] (* Gus Wiseman, Mar 22 2024 *)

a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i));); #v; \\ Michel Marcus, Oct 22 2013

a(n)=sum(k=1,n,sigma(k)==n) \\ Charles R Greathouse IV, Nov 12 2013

first(n)=my(v=vector(n),t); for(k=1,n, t=sigma(n); if(t<=n, v[t]++)); v \\ Charles R Greathouse IV, Mar 08 2017

A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308039. - Amiram Eldar, Dec 23 2024

Incorrect comment deleted by M. F. Hasler, Nov 21 2019

A074753 Number of integers k such that sigma(k) < n.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 6, 6, 8, 9, 10, 11, 11, 11, 13, 13, 14, 14, 14, 14, 17, 17, 17, 17, 18, 18, 19, 21, 23, 23, 23, 23, 24, 24, 25, 26, 27, 27, 30, 30, 31, 31, 31, 31, 34, 34, 34, 34, 34, 34, 36, 36, 38, 39, 39, 39, 42, 42, 43, 44, 44, 44, 44, 44, 45, 45, 45, 45, 50, 50

Offset: 1

Benoit Cloitre, Sep 28 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert E. Dressler, An elementary proof of a theorem of Erdős on the sum of divisors function, Journal of Number Theory, Vol. 4, No. 6 (1972), pp. 532-536.

Partial sums of A054973.

Cf. A000203, A308039.

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
for n from 1 to N-2 do
  s:= numtheory:-sigma(n)+1;
  if s <= N then V[s]:= V[s]+1 fi;
od:
ListTools:-PartialSums(V); # Robert Israel, Jan 08 2018

Table[Length[Select[Range[n], DivisorSigma[1,#] < n&]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *)

a(n)=sum(i=1,n,if(1+sign(sigma(i)-n),0,1))

list(nmax) = my(s = 0); for(n = 1, nmax, s += invsigmaNum(n); print1(s, ", ")); \\ Amiram Eldar, Dec 23 2024, using Max Alekseyev's invphi.gp

a(n) = card( k : sigma(k) < n ).
a(n) is asymptotic to c*n with c = 0.67...
a(n) = c * n + o(n), where c = 0.6727383... = A308039 (Dressler, 1972). - Amiram Eldar, Dec 23 2024

A212717 Numerator of Sum_{k=1..n} 1/sigma(k).

Original entry on oeis.org

1, 4, 19, 145, 53, 83, 353, 607, 8171, 75359, 78089, 79259, 11657, 2963, 12047, 378137, 386197, 389917, 397171, 2804377, 11344453, 11457293, 11626553, 11694257, 11825297, 11922017, 12023573, 12096113, 12231521, 12287941, 6207443, 6239683, 3140999, 9479417

Offset: 1

Michel Lagneau, May 25 2012

			1, 4/3, 19/12, 145/84, 53/28, 83/42, 353/168, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000203, A212718 (denominators), A308039, A345327.

with(numtheory): a:=n->numer(sum(1/sigma(k), k=1..n)): seq(a(n), n=1..50);

Numerator[Table[Sum[1/DivisorSigma[1,k],{k,1,n}],{n,1,50}]]
Accumulate[1/DivisorSigma[1,Range[40]]]//Numerator (* Harvey P. Dale, Aug 13 2023 *)

a(n)/A212718(n) = c * (log(n) + gamma + Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p))) + O(log(n)^(2/3)*log(log(n))^(4/3)/n), where alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), and c = Product_{p prime} alpha(p) = A308039 (Sita Ramaiah and Suryanarayana, 1979). - Amiram Eldar, Oct 16 2022

A308041 Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

7, 6, 8, 7, 1, 8, 3, 6, 2, 4, 4, 6, 4, 8, 5, 1, 9, 8, 6, 7, 2, 7, 3, 4, 3, 3, 2, 4, 5, 5, 3, 5, 0, 5, 2, 5, 2, 3, 4, 2, 5, 5, 7, 4, 0, 4, 1, 1, 9, 0, 4, 1, 1, 0, 7, 0, 1, 5, 4, 1, 3, 5, 2, 9, 3, 4, 8, 6, 0, 7, 7, 6, 8, 3, 3, 7, 9, 0, 8, 0, 3, 9, 3, 3, 2, 8, 8, 0, 7, 6, 4, 8, 9, 6, 9, 1, 4, 7, 5, 9, 5, 3, 3, 7, 2, 4

Offset: 0

Amiram Eldar, May 10 2019

			0.76871836244648519867273433245535052523425574041190...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y3).
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 3.8-3.9, p. 1352-1353).
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.9, p. 29).

Cf. A034448, A063974, A308039 (corresponding limit with sigma).

$MaxExtraPrecision = 1000; m = 1000; f[p_] := 1 - (p - 1)/p*Sum[1/p^k/(p^k + 1), {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[0, m]];RealDigits[f[2]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

From Amiram Eldar, Dec 23 2024: (Start)
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k+1))).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A063974(k). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

A345327 Decimal expansion of a constant Y2 related to the asymptotics of A000203.

Original entry on oeis.org

5, 0, 7, 3, 3, 8, 8, 8, 2, 5, 8, 3, 0, 8, 4, 3, 7, 8, 1, 0, 0, 4, 9, 7, 8, 7, 6, 5, 1, 5, 9, 5, 2, 6, 7, 7, 3, 8, 9, 0, 1, 9, 6, 3, 4, 8, 2, 8, 1, 6, 4, 4, 8, 0, 8, 0, 4, 9, 7, 4, 5, 8, 7, 7, 2, 4, 5, 0, 6, 9, 4, 6, 1, 7, 3, 0, 2, 8, 6, 5, 1, 6, 3, 0, 0, 5, 6, 8, 8, 3, 9, 1, 7, 6, 3, 0, 2, 4, 6, 5, 9, 6, 0, 5, 8, 0

Offset: 0

Vaclav Kotesovec, Jun 14 2021

			0.5073388825830843781004978765159526773890196348281644808049...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y2).
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 161-162, formula (4.1)-(4.4).
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.3, p. 18).

Cf. A000203, A308039.

$MaxExtraPrecision = 1000; Do[ratfun = (p - 1)^2 * Sum[j/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}]/(p*(1 - (p - 1)^2/p * Sum[1/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}])); zetas = 0; ratab = Table[konfun = Together[ratfun + c/(p^power - 1)]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 40}]; Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 10, 250, 10}]

Equals Sum_{p primes} (p-1)^2 * g(p) * log(p) / (p*f(p)), where f(p) = 1 - (p-1)^2/p * Sum_{j>=1} 1/((p^j - 1)*(p^(j+1) - 1)) and g(p) = Sum_{j>=1} j/((p^j - 1)*(p^(j+1) - 1)).

Sum_{k=1..n} 1/A000203(k) ~ Y1*log(n) + Y1*(gamma + Y2), where gamma = A001620, Y1 = A308039, Y2 = A345327.

A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

Original entry on oeis.org

1, 7, 29, 129, 571, 2525, 11167, 49372, 218295, 965177, 4267457, 18868240, 83424514, 368855252, 1630865929, 7210751807, 31881800153

Offset: 1

Labos Elemer, Aug 29 2002

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 129, p. 44, Ellipses, Paris, 2008.

Cf. A000203 (sigma), A002387, A004080, A046024, A074631, A074633, A074467, A212717, A308039.

{s=0, s1=0}; Do[s=s+(1/DivisorSigma[1, n]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

Limit_{n->oo} a(n+1)/a(n) = exp(1/c) = 4.42142525588146107878... where c = A308039. - Amiram Eldar, May 05 2024

2 more terms from Lekraj Beedassy, Jul 14 2008
a(11)-a(15) from Donovan Johnson, Aug 22 2011
a(16)-a(17) from Amiram Eldar, May 05 2024

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A054973 Number of numbers whose divisors sum to n.

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A074753 Number of integers k such that sigma(k) < n.

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A212717 Numerator of Sum_{k=1..n} 1/sigma(k).

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A308041 Decimal expansion of lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/usigma(k), where usigma(k) is the sum of unitary divisors of k (A034448).

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A345327 Decimal expansion of a constant Y2 related to the asymptotics of A000203.

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A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

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