A113998 -id:A113998 - cp's OEIS frontend

A113999 a(n) = Sum_{ k, k|n } 10^(k-1).

1, 11, 101, 1011, 10001, 100111, 1000001, 10001011, 100000101, 1000010011, 10000000001, 100000101111, 1000000000001, 10000001000011, 100000000010101, 1000000010001011, 10000000000000001, 100000000100100111

Offset: 1

Paul Barry, Nov 12 2005

A034729 to base 2. Stacking elements of the sequence gives A113998.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), this sequence (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A113999:= func< n | (&+[10^(d-1): d in Divisors(n)]) >;
[A113999(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

A113999[n_]:= DivisorSum[n, 10^(#-1) &];
Table[A113999[n], {n, 40}] (* G. C. Greubel, Jun 26 2024 *)

PARI
```
a(n)=if(n<1,0,sumdiv(n,k,10^(k-1)));
```

def A113999(n): return sum(10^(k-1) for k in (1..n) if (k).divides(n))
[A113999(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

G.f.: Sum_{n>0} x^n/(1-10*x^n).

a(n) ~ 10^(n-1). - Vaclav Kotesovec, Jun 05 2021

A081307 a(n) = (n+1)*tau(n) - sigma(n).

Original entry on oeis.org

1, 3, 4, 8, 6, 16, 8, 21, 17, 26, 12, 50, 14, 36, 40, 54, 18, 75, 20, 84, 56, 56, 24, 140, 47, 66, 72, 118, 30, 176, 32, 135, 88, 86, 96, 242, 38, 96, 104, 238, 42, 248, 44, 186, 198, 116, 48, 366, 93, 213, 136, 220

Offset: 1

Benoit Cloitre, Apr 20 2003

nonn

Old name was: Sum_{k=1..n} Sum_{m=1..k} 1/(1-x^m).

Number of positive integer pairs (s,t) with s <= t <= n, such that s|n. For example, when n = 6, the 16 pairs are (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (6,6). - Wesley Ivan Hurt, Nov 15 2021

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A000203 (sigma), A094471, A113998.

Table[(n + 1) DivisorSigma[0, n] - DivisorSigma[1, n], {n, 100}] (* Wesley Ivan Hurt, Nov 15 2021 *)

a(n)=if(n<1,0,polcoeff(sum(k=1,n,sum(l=1,k,1/(1-x^l)),x*O(x^n)),n))

a(n)=sum(j=1, n, sum(k=1, j, n%k==0)) \\ Hugo Pfoertner, Jul 09 2025

Sum_{k=1..n} Sum_{m=1..k} 1/(1-x^m).

a(n) = Sum_{k=1..n} k*A113998(n,k). - Philippe Deléham, Feb 03 2007

Name changed by Wesley Ivan Hurt, Nov 16 2021 using formula from Vladeta Jovovic, Jan 22 2005

A286236 Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

Original entry on oeis.org

1, 1, 2, 3, 0, 4, 3, 0, 2, 7, 10, 0, 0, 0, 11, 3, 0, 0, 5, 4, 16, 21, 0, 0, 0, 0, 0, 22, 10, 0, 0, 0, 5, 0, 7, 29, 21, 0, 0, 0, 0, 0, 8, 0, 37, 10, 0, 0, 0, 0, 14, 0, 0, 11, 46, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 10, 0, 0, 0, 0, 0, 5, 0, 8, 12, 16, 67, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 22, 92, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 17, 0, 106

Offset: 1

Antti Karttunen, May 05 2017

This is transpose of A286237, see comments there.

			The top left 12 X 12 corner of the array:
   1,  1,  3,  3, 10, 3, 21, 10, 21, 10, 55, 10
   2,  0,  0,  0,  0, 0,  0,  0,  0,  0,  0,  0
   4,  2,  0,  0,  0, 0,  0,  0,  0,  0,  0,  0
   7,  0,  5,  0,  0, 0,  0,  0,  0,  0,  0,  0
  11,  4,  0,  5,  0, 0,  0,  0,  0,  0,  0,  0
  16,  0,  0,  0, 14, 0,  0,  0,  0,  0,  0,  0
  22,  7,  8,  0,  0, 5,  0,  0,  0,  0,  0,  0
  29,  0,  0,  0,  0, 0, 27,  0,  0,  0,  0,  0
  37, 11,  0,  8,  0, 0,  0, 14,  0,  0,  0,  0
  46,  0, 12,  0,  0, 0,  0,  0, 27,  0,  0,  0
  56, 16,  0,  0, 19, 0,  0,  0,  0, 14,  0,  0
  67,  0,  0,  0,  0, 0,  0,  0,  0,  0, 65,  0
The first 15 rows when viewed as a triangle:
   1,
   1, 2,
   3, 0, 4,
   3, 0, 2, 7,
  10, 0, 0, 0, 11,
   3, 0, 0, 5,  4, 16,
  21, 0, 0, 0,  0,  0, 22,
  10, 0, 0, 0,  5,  0,  7, 29,
  21, 0, 0, 0,  0,  0,  8,  0, 37,
  10, 0, 0, 0,  0, 14,  0,  0, 11, 46,
  55, 0, 0, 0,  0,  0,  0,  0,  0,  0, 56,
  10, 0, 0, 0,  0,  0,  5,  0,  8, 12, 16, 67,
  78, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 0, 0, 0,  0,  0,  0, 27,  0,  0,  0,  0, 22, 92,
  36, 0, 0, 0,  0,  0,  0,  0,  0,  0, 19,  0, 17,  0, 106

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array

Transpose: A286237.

Cf. A000010, A000027, A113998, A286156, A286234, A286246.

T[n_, m_] := ((n + m)^2 - n - 3*m + 2)/2
t[n_, k_] := If[Mod[n, k] != 0, 0, T[EulerPhi[k], n/k]]
Table[Reverse[t[n, #] & /@ Range[n]], {n, 1, 20}] (* David Radcliffe, Jun 12 2025 *)

from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return 0 if n%k!=0 else T(totient(k), n//k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)][::-1]) # Indranil Ghosh, May 10 2017

(define (A286236 n) (A286236bi (A002260 n) (A004736 n)))
(define (A286236bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A000010 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286236 n) (A286236tr (A002024 n) (A002260 n)))
(define (A286236tr n k) (A286236bi k (+ 1 (- n k))))

T(n,k) = A113998(n,k) * A286234(n,k).

A067951 a(0) = 1; a(n) = Sum_{1 <= k <= n and k|n} a(n-k).

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 17, 18, 42, 60, 110, 111, 341, 342, 702, 1154, 2240, 2241, 6037, 6038, 15580, 22320, 38012, 38013, 122544, 138125, 261012, 389594, 796173, 796174, 2259345, 2259346, 5439649, 7737007, 13178898, 16234417, 45367492, 45367493

Offset: 0

Naohiro Nomoto, Mar 07 2002

With offset 1, eigensequence of triangle A113998. - Gary W. Adamson, Sep 12 2016

a(n) = a(n-1)+1 iff n is prime. - Robert Israel, Sep 13 2016

Robert Israel, Table of n, a(n) for n = 0..4680

Cf. A113998.

f:=proc(n) option remember;
   add(procname(n-k), k=numtheory:-divisors(n))
end proc:
f(0):= 1:
seq(f(n),n=0..50); # Robert Israel, Sep 13 2016

a[0] = 1; a[n_] := a[n] = Sum[a[n - k], {k, Divisors@ n}]; Table[a@ n, {n, 0, 37}] (* Michael De Vlieger, Sep 13 2016 *)

a(n)=if (n==0, return(1)); my(an = 0); fordiv(n, k, an += a(n-k)); an; \\ Michel Marcus, Jul 14 2013

A286246 Square array A(n,k) = P(A046523(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

Original entry on oeis.org

1, 3, 2, 3, 0, 4, 10, 0, 5, 7, 3, 0, 0, 0, 11, 21, 0, 0, 5, 8, 16, 3, 0, 0, 0, 0, 0, 22, 36, 0, 0, 0, 14, 0, 12, 29, 10, 0, 0, 0, 0, 0, 8, 0, 37, 21, 0, 0, 0, 0, 5, 0, 0, 17, 46, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 78, 0, 0, 0, 0, 0, 27, 0, 19, 12, 23, 67, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 30, 92, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 17, 0, 106

Offset: 1

Antti Karttunen, May 06 2017

			The top left 12 X 12 corner of the array:
   1,  3,  3, 10,  3, 21,  3, 36, 10, 21,  3, 78
   2,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   4,  5,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   7,  0,  5,  0,  0,  0,  0,  0,  0,  0,  0,  0
  11,  8,  0, 14,  0,  0,  0,  0,  0,  0,  0,  0
  16,  0,  0,  0,  5,  0,  0,  0,  0,  0,  0,  0
  22, 12,  8,  0,  0, 27,  0,  0,  0,  0,  0,  0
  29,  0,  0,  0,  0,  0,  5,  0,  0,  0,  0,  0
  37, 17,  0, 19,  0,  0,  0, 44,  0,  0,  0,  0
  46,  0, 12,  0,  0,  0,  0,  0, 14,  0,  0,  0
  56, 23,  0,  0,  8,  0,  0,  0,  0, 27,  0,  0
  67,  0,  0,  0,  0,  0,  0,  0,  0,  0,  5,  0
The first fifteen rows of triangle:
   1,
   3, 2,
   3, 0, 4,
  10, 0, 5, 7,
   3, 0, 0, 0, 11,
  21, 0, 0, 5,  8, 16,
   3, 0, 0, 0,  0,  0, 22,
  36, 0, 0, 0, 14,  0, 12, 29,
  10, 0, 0, 0,  0,  0,  8,  0, 37,
  21, 0, 0, 0,  0,  5,  0,  0, 17, 46,
   3, 0, 0, 0,  0,  0,  0,  0,  0,  0, 56,
  78, 0, 0, 0,  0,  0, 27,  0, 19, 12, 23, 67,
   3, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 0, 0, 0,  0,  0,  0,  5,  0,  0,  0,  0, 30, 92,
  21, 0, 0, 0,  0,  0,  0,  0,  0,  0,  8,  0, 17,  0, 106

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Transpose: A286247.

Cf. A000027, A046523, A113998, A286156, A286244, A286236.

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k): return 0 if (n + k - 1)%k!=0 else T(a046523(k), (n + k - 1)//k)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 09 2017

(define (A286246 n) (A286246bi (A002260 n) (A004736 n)))
(define (A286246bi row col) (if (not (zero? (modulo (+ row col -1) col))) 0 (let ((a (A046523 col)) (b (/ (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286246 n) (A286246tr (A002024 n) (A002260 n)))
(define (A286246tr n k) (A286246bi k (+ 1 (- n k))))

T(n,k) = A113998(n,k) * A286244(n,k).

cp's OEIS Frontend

A113999 a(n) = Sum_{ k, k|n } 10^(k-1).

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A081307 a(n) = (n+1)*tau(n) - sigma(n).

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A286236 Square array A(n,k) = P(A000010(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

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A067951 a(0) = 1; a(n) = Sum_{1 <= k <= n and k|n} a(n-k).

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Maple

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A286246 Square array A(n,k) = P(A046523(k), (n+k-1)/k) if k divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

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