A007609 -id:A007609 - cp's OEIS frontend

A111865 Expansion of g.f. Product_{k>=1} 1/(1-x^sigma(k)).

1, 1, 1, 2, 3, 3, 5, 7, 9, 11, 14, 17, 24, 29, 36, 46, 57, 66, 85, 103, 125, 151, 182, 213, 264, 310, 368, 440, 524, 604, 724, 849, 998, 1164, 1363, 1573, 1854, 2136, 2481, 2879, 3336, 3807, 4427, 5079, 5844, 6698, 7695, 8754, 10072, 11451, 13075, 14898, 16988

Offset: 0

Jon Perry, Nov 23 2005

nonn

Number of partitions of n into parts of size p = sigma(k) for some k, when there are A054973(p) kinds of part p.

			a(6) = 5 : We have sigma(1)=1, sigma(2)=3, sigma(3)=4, sigma(5)=6 so 111111, 1113, 114, 6 and 33.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Cf. A000203, A002191, A007609, A054973, A305320.

with(numtheory):
seq(coeff(series(mul(1/(1-x^sigma(k)),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, May 31 2018

CoefficientList[ Series[Product[1/(1 - x^DivisorSigma[1, k]), {k, 47}], {x, 0, 52}], x] (* Robert G. Wilson v, Nov 25 2005 *)

lista(nn) = Vec(prod(k=1, nn, 1/(1-x^sigma(k))+ O(x^nn))) \\ Michel Marcus, May 30 2018

G.f.: Product_{k>=1} 1/(1-x^sigma(k)).

More terms from Robert G. Wilson v, Nov 25 2005

a(0)=1 prepended by Seiichi Manyama, May 30 2018

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).

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 *)

A206030 Numbers m with at least two divisors d with the same sigma(d).

Original entry on oeis.org

66, 132, 170, 198, 210, 260, 264, 322, 330, 340, 345, 396, 400, 420, 456, 462, 510, 520, 528, 594, 630, 644, 651, 660, 680, 690, 726, 780, 792, 800, 820, 840, 850, 858, 912, 924, 966, 990, 1020, 1035, 1040, 1050, 1056, 1066, 1092, 1122, 1155, 1160, 1188

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Complement of sequence contains numbers whose divisors d have distinct values of sigma(d).

			66 is in sequence because two divisors d (6 and 11) of 66 have the same sigma(d) = 12.

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

Cf. A000203, A007609, A085790.

Select[Range[1200], Length[DivisorSigma[1, Divisors[#]]] != Length[Union[DivisorSigma[1, Divisors[#]]]] &] (* T. D. Noe, Feb 10 2012 *)
[Range[1200],Max[Tally[DivisorSigma[1,Divisors[#]]][[;;,2]]]>1&] (* Harvey P. Dale, Sep 27 2024 *)

ok(n)={my(v=apply(sigma, divisors(n))); #Set(v) < #v} \\ Andrew Howroyd, Aug 01 2018

A159953 Values in A054973 larger than 1.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 3, 3, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 5, 2, 2, 6, 4, 2, 2, 5, 2, 5, 3, 3, 3, 7, 3, 6, 2, 3, 2, 2, 6, 3, 2, 4, 2, 3, 8, 2, 9, 4, 2, 6, 2, 2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 2, 3, 4, 3, 9, 2, 10, 2, 3, 2, 4, 4, 3, 4, 2, 2, 11, 5, 2, 5, 2, 3, 4, 2, 2, 3, 5, 3, 8, 7, 4, 15, 2, 4, 7, 8

Offset: 1

Jaroslav Krizek, Apr 27 2009

nonn

This is a survey of how many solutions the equation sigma(x)=k has for k in A159886, or about the lengths of the plateaus in A007609.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1095 from Jean-François Alcover)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A007609, A054973, A159886.

read("transforms3") ; a054973 := BFILETOLIST("b054973.txt") ;
for i from 1 to 1000 do if op(i,a054973) > 1 then printf("%d,", op(i,a054973)) ; fi; od: # R. J. Mathar, May 22 2009

b[n_] := Sum[Boole[DivisorSigma[1, k] == n], {k, 1, n}];
Select[Array[b, 1000], # > 1&] (* Jean-François Alcover, Apr 06 2020 *)

list(lim) = {my(s); for(k = 1, lim, s = invsigmaNum(k); if(s > 1, print1(s, ", ")));} \\ Amiram Eldar, Dec 25 2024, using Max Alekseyev's invphi.gp

Edited and extended by R. J. Mathar, May 22 2009

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.

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.

A162967 Values taken by the sigma(sigma(n)) function A051027, with repetition, sorted into ascending order.

Original entry on oeis.org

1, 4, 7, 8, 12, 14, 15, 24, 24, 28, 28, 32, 32, 39, 39, 42, 56, 56, 60, 60, 60, 60, 63, 63, 72, 80, 84, 90, 91, 96, 96, 96, 96, 104, 112, 114, 120, 120, 120, 120, 124, 124, 124, 126, 128, 128, 133, 160, 168, 168, 168, 168, 171, 171, 186, 186, 195, 195, 195, 195, 195

Offset: 1

Jaroslav Krizek, Jul 19 2009

nonn

Removal of duplicates generates A070286. - R. J. Mathar, Jul 21 2009

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

Cf. A007609, A002191. - R. J. Mathar, Jul 21 2009

Cf. A051027, A070286.

f(k) = {my(v = invsigma(k), c = 0); for(i = 1, #v, c += invsigmaNum(v[i])); c;} \\ using Max Alekseyev's invphi.gp
list(lim) = {my(m); for(k = 1, lim, m = f(k); for(i = 1, m, print1(k, ", ")));} \\ Amiram Eldar, Dec 26 2024

A253248 Number of k <= n with A000203(k) <= A000203(n).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 8, 10, 8, 12, 10, 13, 14, 16, 13, 18, 14, 20, 19, 20, 17, 24, 20, 25, 24, 27, 19, 30, 23, 31, 29, 30, 30, 36, 25, 35, 34, 39, 30, 42, 31, 41, 41, 41, 34, 48, 38, 48, 44, 51, 36, 53, 46, 55, 48, 51, 42, 60, 43, 57, 59, 63, 52

Offset: 1

Robert Israel, Jun 04 2015

			A000203(7) = 8 >= A000203(k) for k = 1,2,3,4,5,7, so a(7) = 6.

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

Cf. A000203, A002093, A007609.

N:= 1000:
B:= map(numtheory:-sigma,[$1..N]):
M:= max(B):
X:= Vector(M):
for n from 1 to N do
  b:= B[n];
  X[b..-1]:= X[b..-1] + <(1$(M-b+1))>;
  A[n]:= X[b];
od:
seq(A[n],n=1..N);

f[v_] := Count[v, ?(# <= v[[-1]] &)]; seq[lim] := Module[{v = DivisorSigma[1, Range[lim]]}, f[v[[1 ;; #]]] & /@ Range[Length[v]]]; seq[65] (* Amiram Eldar, Dec 19 2024 *)

a(n) <= n, with equality if and only if n is in A002093.

Empirically it appears that lim inf_(n -> infinity) a(n)/n = 2/3, with minimum value a(29)/29 = 19/29.

cp's OEIS Frontend

A111865 Expansion of g.f. Product_{k>=1} 1/(1-x^sigma(k)).

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

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A206030 Numbers m with at least two divisors d with the same sigma(d).

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A159953 Values in A054973 larger than 1.

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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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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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A162967 Values taken by the sigma(sigma(n)) function A051027, with repetition, sorted into ascending order.

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