A074753 -id:A074753 - cp's OEIS frontend

A308039 Decimal expansion of lim_{i->oo} c(i)/i, where c(i) is the number of integers k such that sigma(k) < i (A074753).

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

Offset: 0

Amiram Eldar, May 10 2019

Erdös proved the existence of this constant. Dressler found its explicit form.

			0.6727383092174097953276872030889868689708768294102327312357145188219...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Robert E. Dressler, An elementary proof of a theorem of Erdös on the sum of divisors function, Journal of Number Theory, Vol. 4, No. 6 (1972), pp. 532-536.
Paul Erdős, Some remarks on Euler's phi function and some related problems, Bulletin of the American Mathematical Society, Vol. 51, No. 8 (1945), pp. 540-544.
Leonid G. Fel, Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization, arXiv:1108.0957 [math.NT], 2011, p. 6.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y1).
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, equation (4.2)-(4.3) on p. 162.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv:1608.00795 [math.NT], 2016.
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, A074753, A345327.

$MaxExtraPrecision = 1000; m = 1000; f[p_] := (1 - 1/p)*(p - 1)*Sum[1/(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]]

Equals Product_{p prime} (1 - 1/p) * Sum_{k>=0} 1/sigma(p^k) = Product_{p prime} ((p - 1)^2/p) * Sum_{k>=1} 1/(p^k - 1) = Product_{p prime} 1 - ((p - 1)^2/p) * Sum_{k>=1} 1/((p^k - 1)*(p^(k+1) - 1)).
Equals lim_{n->oo} (1/log(n))*Sum_{k=1..n} 1/sigma(k).
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} k/sigma(k) (the asymptotic mean of k/sigma(k)). - Amiram Eldar, Dec 23 2020
Sum_{k=1..n} 1/A000203(k) ~ Y1*log(n) + Y1*(gamma + Y2), where gamma = A001620, Y1 = A308039, Y2 = A345327. - Vaclav Kotesovec, Jun 14 2021

More digits from Vaclav Kotesovec, Jun 13 2021

A202275 Differences between A074753 (number of integers k such that sigma(k) <= n) and A202276 (number of integers k <= n such that sigma(x) = k has no solution); sigma = A000203.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 3, 4, 3, 2, 1, 3, 4, 5, 6, 5, 4, 6, 5, 6, 5, 4, 3, 6, 5, 4, 3, 4, 3, 4, 6, 8, 7, 6, 5, 6, 5, 6, 7, 8, 7, 10, 9, 10, 9, 8, 7, 10, 9, 8, 7, 6, 5, 7, 6, 8, 9, 8, 7, 10, 9, 10, 11, 10, 9, 8, 7, 8, 7, 6, 5, 10, 9, 10, 9, 8, 7, 8, 7, 9, 8, 7, 6, 9

Offset: 1

Jaroslav Krizek, Dec 25 2011

sign

Conjectures: Max a(n) = 15 for n = 195, 403, 434. For n >= 687, a(n) < 0.

First term < 0: a(538) = -1.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202277, A202276, A175523, A007369, A007368, A007609.

a(n) = A074753(n) - A202276(n).

A202277 Numbers m such that number of integers k such sigma(k) <= m (A074753) is equal to number of integers k <= m such that sigma(x) = k has no solution (A202276).

Original entry on oeis.org

2, 537, 639, 647, 653, 655, 657, 661, 663, 672, 674, 684, 686

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Numbers m such that A074753(m) - A202276(m) = 0.
Numbers m such that A202275(m) = 0 (see graph of A202275).
Conjecture: sequence is finite with 13 terms.

Cf. A074753, A202275, A202276, A175523, A007369, A007368, A007609.

A175091 Numbers k such that A074753(k) == (2,4) (mod 6).

Original entry on oeis.org

4, 7, 13, 15, 21, 22, 23, 24, 40, 49, 50, 51, 52, 53, 54, 57, 64, 65, 66, 67, 68, 73, 74, 79, 80, 94, 95, 96, 99, 100, 101, 102, 111, 112, 122, 123, 124, 127, 134, 135, 136, 137, 138, 145, 146, 147, 148, 149, 150, 159, 160, 172, 173, 174, 177, 178, 179, 180, 181, 182

Offset: 1

Giovanni Teofilatto, Jan 29 2010

nonn

Cf. A074753.

A074753 := proc(n) option remember ; local a,k ; a := 0 ; for k from 1 to n do if numtheory[sigma](k) < n then a := a+1 ; end if; end do ; a ; end proc: isA175091 := proc(n) return ( (A074753(n) mod 6 )in {2,4}) ; end proc ; for n from 1 to 300 do if isA175091(n) then printf("%d,",n) ; end if; end do ; # R. J. Mathar, Feb 08 2010

Corrected by R. J. Mathar, Feb 08 2010

A054973 Number of numbers whose divisors sum to n.

Original entry on oeis.org

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

A202276 Number of integers k <= n such that sigma(x) = k has no solution, sigma = A000203.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 15, 15, 15, 16, 17, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 30, 30, 31, 32, 32, 33, 33, 33, 34, 35, 36, 37, 37, 38, 39

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Partial sums of A175253.

			a(9) = 3 because sigma(x) = k has no solution for k = 2, 5 and 9.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202277, A202275, A175523, A007369, A007368, A007609.

seq[max_] := Module[{t = Table[0, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 1; Accumulate@ t]; seq[100] (* Amiram Eldar, Dec 20 2024 *)

A175253 a(n) = characteristic function of numbers k such that A000203(m) = k has no solution for any m, where A000203(m) = sum of divisors of m.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1

Offset: 1

Jaroslav Krizek, Mar 14 2010

nonn

a(n) = characteristic function of numbers from A007369(n). a(n) = 1 if A000203(m) not equal to n for any m, else 0. a(n) = 1 for such n that A054973(n) = 0. a(n) = 0 for such n that A054973(n) >= 1. a(n) + A175192(n) = A000012(n).

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202275, A202276, A175523, A007369, A007368, A007609.

seq[max_] := Module[{t = Table[0, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 1; t]; seq[100] (* Amiram Eldar, Mar 22 2024 *)

More terms from Jaroslav Krizek, Dec 25 2011.

cp's OEIS Frontend

A308039 Decimal expansion of lim_{i->oo} c(i)/i, where c(i) is the number of integers k such that sigma(k) < i (A074753).

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A202275 Differences between A074753 (number of integers k such that sigma(k) <= n) and A202276 (number of integers k <= n such that sigma(x) = k has no solution); sigma = A000203.

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A202277 Numbers m such that number of integers k such sigma(k) <= m (A074753) is equal to number of integers k <= m such that sigma(x) = k has no solution (A202276).

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A175091 Numbers k such that A074753(k) == (2,4) (mod 6).

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A054973 Number of numbers whose divisors sum to n.

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A202276 Number of integers k <= n such that sigma(x) = k has no solution, sigma = A000203.

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A175253 a(n) = characteristic function of numbers k such that A000203(m) = k has no solution for any m, where A000203(m) = sum of divisors of m.

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