A048272 -id:A048272 - cp's OEIS frontend

A101563 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+10*x^k).

1, 9, 101, 1009, 10001, 99909, 1000001, 10001009, 100000101, 999990009, 10000000001, 100000100909, 1000000000001, 9999999000009, 100000000010101, 1000000010001009, 10000000000000001, 99999999900099909

Offset: 0

Paul Barry, Dec 07 2004

G. C. Greubel, Table of n, a(n) for n = 0..990

Cf. A051731, A081295, A048272, A101561, A101562.

A101563:= func< n | (&+[(-1)^(n-k)*(10)^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101563(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024

A101563[n_]:= (-1)^n*DivisorSum[n+1, (-10)^(#-1) &];
Table[A101563[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

def A101563(n): return sum((-1)^(n+k)*(10)^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101563(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * (10)^k * A051731(n+1, k+1).

a(n) = (-1)^n * Sum_{d|n+1} (-10)^(d-1). - G. C. Greubel, Jun 25 2024

A158441 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1+x^n) /n ).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 7, 7, 9, 14, 18, 20, 31, 34, 42, 61, 69, 83, 109, 127, 156, 203, 228, 276, 347, 404, 477, 591, 683, 801, 990, 1132, 1323, 1598, 1837, 2148, 2560, 2929, 3405, 4018, 4608, 5319, 6244, 7124, 8184, 9569, 10877, 12465, 14457, 16412, 18761, 21633

Offset: 0

Paul D. Hanna, Mar 28 2009

nonn

Also the number of partitions of n in which each part occurs a triangle number (>=0) times. - Seiichi Manyama, May 11 2018

			From _Seiichi Manyama_, Mar 11 2018: (Start)
n | Partitions of n in which each part occurs a triangle number (>=0) times.
--+-------------------------------------------------------------------------
1 | 1;
2 | 2;
3 | 3 = 2+1 = 1+1+1;
4 | 4 = 3+1;
5 | 5 = 4+1 = 3+2 = 2+1+1+1;
6 | 6 = 5+1 = 4+2 = 3+2+1 = 3+1+1+1 = 2+2+2 = 1+1+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 4+1+1+1 = 2+2+2+1; (End)

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

Cf. A006171, A000203, A295794, A320250.

with(numtheory):
b:= proc(n) option remember; -add((-1)^d, d=divisors(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018

nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2018 *)

{a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1+x^m+x*O(x^n))/m)), n))}

N=99; x='x+O('x^N);
gf=1/prod(n=1,N,eta(x^n)^((-1)^(n-1)));
Vec(gf) /* Joerg Arndt, Jun 24 2011 */

Euler transform of A048272. [Vladeta Jovovic, Mar 28 2009]
G.f.: 1/prod(n>=1, P(x^n)^((-1)^(n-1)) ) where P(x) = prod(k>=1, 1-x^k ), see Pari code. [Joerg Arndt, Jun 24 2011]
G.f.: Product_{k>0} (Sum_{m>=0} x^(k*m*(m+1)/2)) = (1+x+x^3+x^6+...)*(1+x^2+x^6+x^12+...)*(1+x^3+x^9+x^18+...)*... . - Seiichi Manyama, May 11 2018
a(n) ~ (log(2))^(3/8) * exp(Pi*sqrt(2*log(2)*n/3)) / (2^(11/8) * 3^(3/8) * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 08 2018

A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).

Original entry on oeis.org

-2, 1, 2, 3, 4, 6, 6, 10, 8, 15, 10, 25, 12, 28, 10, 60, 16, 25, 18, 125, 0, 66, 22, 218, 24, 91, -30, 420, 28, -387, 30, 2011, -88, 153, 28, -1894, 36, 190, -182, 8902, 40, -3234, 42, 2398, -132, 276, 46, 2340, 48, -2678, -510, 4641, 52, -1754, -198, 108400

Offset: 1

Gus Wiseman, Aug 16 2017

sign

			2x/(1-x) = (1-x)^(-2) - 1 + (1-x^2)^1 - 1 + (1-x^3)^2 - 1 + (1-x^4)^3 - 1 + ...

Cf. A048272, A220418, A260685, A281145, A289078, A289501, A290261, A290971.

a:= n-> add(binomial(n/d-1-a(d), n/d), d=
        numtheory[divisors](n) minus {n})-2:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 27 2017

nn=60;
rus=SolveAlways[Normal[Series[2x/(1-x)==Sum[(1-x^n)^a[n]-1,{n,nn}],{x,0,nn}]],x];
Array[a,nn]/.First[rus]

For all n > 0 we have: 2 = Sum_{d|n} binomial(-a(d) + n/d - 1, n/d).

A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, -1, 1, 7, 10, 1, 2, 1, 15, 28, 11, 6, 0, 1, 31, 82, 55, 26, 4, 2, 1, 63, 244, 239, 126, 30, 8, -2, 1, 127, 730, 991, 626, 196, 50, 1, 3, 1, 255, 2188, 4031, 3126, 1230, 344, 43, 13, 0, 1, 511, 6562, 16255, 15626, 7564, 2402, 439, 91, 6, 2, 1, 1023, 19684, 65279, 78126, 45990, 16808, 3823, 757, 78, 12, -2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
   0,  1,   3,    7,    15,    31,  ...
   2,  4,  10,   28,    82,   244,  ...
  -1,  1,  11,   55,   239,   991,  ...
   2,  6,  26,  126,   626,  3126,  ...
   0,  4,  30,  196,  1230,  7564,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557.

Cf. A109974, A279394, A279396, A285425, A321438 (diagonal), A322082, A322083, A322084.

Table[Function[k, Sum[(-1)^(n/d + 1) d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, (-1)^(n/d+1)*d^k)}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^j).

A325939 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k).

Original entry on oeis.org

0, 1, -1, 2, -1, 1, -1, 3, -2, 1, -1, 3, -1, 1, -3, 4, -1, 1, -1, 3, -3, 1, -1, 5, -2, 1, -3, 3, -1, 1, -1, 5, -3, 1, -3, 4, -1, 1, -3, 5, -1, 1, -1, 3, -5, 1, -1, 7, -2, 1, -3, 3, -1, 1, -3, 5, -3, 1, -1, 5, -1, 1, -5, 6, -3, 1, -1, 3, -3, 1, -1, 7, -1, 1, -5, 3, -3, 1, -1, 7

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

Number of even divisors of n minus number of odd strong divisors of n (i.e. odd divisors > 1).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A032741, A048272, A075997 (partial sums), A325937.

nmax = 80; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^# &, # > 1 &], {n, 1, 80}]

A325939(n) = sumdiv(n, d, if(1==d,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

G.f.: Sum_{k>=2} (-1)^k * x^k / (1 - x^k).
a(n) = Sum_{d|n, d>1} (-1)^d.
a(n) = 1 - A048272(n).

A333809 G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, -1, 2, 0, 1, 0, 1, 0, 2, -1, 1, 2, 1, -1, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, -1, 3, 0, 1, -1, 1, 1, 2, -1, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, -1, 2, 0, 1, -1, 2, 2, 1, -2, 1, 0, 3, -1, 2, 0, 1, -1, 2, 0, 1, 0, 2, 0, 2, -2, 1, 2

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

Number of odd divisors of n that are < sqrt(n) minus number of even divisors of n that are < sqrt(n).

Cf. A048272, A056924, A333781, A333805, A333810, A258998, A348515.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

From Ridouane Oudra, Sep 06 2025: (Start)
a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(d+1).
a(n) = A333781(n) - A258998(n).
a(n) = A048272(n) - A348515(n). (End)

A348951 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A048272, A056924, A113652, A228441, A333809, A348515, A348952, A348953, A348954, A348955, A348956, A258998, A333781.

Table[DivisorSum[n, (-1)^(n/#) &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348951(n) = sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k*(k + 1)) / (1 + x^k).
a(n) = A258998(n) - A348515(n). - Ridouane Oudra, Aug 21 2025
a(n) = A333781(n) - A048272(n). - Ridouane Oudra, Sep 01 2025

A356389 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.

Original entry on oeis.org

1, 2, 10, 34, 218, 1308, 10596, 74688, 793152, 7931520, 94504320, 1054218240, 14662840320, 205279764480, 3427909632000, 50923531008000, 907545606912000, 16335820924416000, 323185344975360000, 6220416698689536000, 140360358705186816000, 3087927891514109952000

Offset: 1

Seiichi Manyama, Aug 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A356390, A356391.

Cf. A048272, A356297, A356392.

Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)

a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k)/(1-x)))

a(n) = n! * Sum_{k=1..n} A048272(k)/k.
E.g.f.: (1/(1-x)) * Sum_{k>0} log(1 + x^k)/k.
a(n) ~ n! * log(2) * (log(n) + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 07 2022

A060184 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 0, -1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 1, -2, 0, 1, 3, 1, 5, 6, 0, 0, -1, -1, 2, 1, 5, 5, -2, 0, -2, -3, 2, 2, 9, 10, 0, 1, 4, 3, 0, 4, 0, 2, 9, 9, -3, 1, 3, -2, -7, 2, 0, 3, 14, 16, 0, 2, 6, -1, -9, 2, 0, 3, 15, 17, -2, 1, 8, 19, 10, -6, 4, 0, -1, 0, 15, 22, 0, 1, 9, 21, 7, -13, 2, 0, -2, -4, 11, 20, -4, 2, 15, 33, 14, -15, 3, 0, -4, -10, 10, 28, 0, 3

Offset: 1

N. J. A. Sloane, Mar 20 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
  1  0  2 -1  2  0  2 -2  3  0  2 ...
        1  0  1  2  1  1  1  6 -1 ...
              1  0  1  0  5 -1  5 ...

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A048272, A060185, A060186.

Cf. A060043, A060044, A060047, A060177.

max = 27(*rows*); t[n_, k_] := Module[{m, mm, q, s}, mm = Array[m, k]; s = Sum[q^Total[mm]/Times @@ (1+q^mm), Evaluate[Sequence @@ Transpose[{mm, Join[{1}, Most[mm]+1], max-Range[k-1, 0, -1]}]]]; SeriesCoefficient[s, {q, 0, n}]]; Table[Print[an = Table[t[n, k], {k, Floor[(Sqrt[8*n+1]-1)/2], 1, -1}]]; an, {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 21 2014 *)

G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1+q^m_1)*(1+q^m_2)*...*(1+q^m_k)) = Sum_n T(n, k)*q^n.

More terms from Vladeta Jovovic, Sep 20 2007

A069262 a(n) = 4*prime(n)^2.

Original entry on oeis.org

16, 36, 100, 196, 484, 676, 1156, 1444, 2116, 3364, 3844, 5476, 6724, 7396, 8836, 11236, 13924, 14884, 17956, 20164, 21316, 24964, 27556, 31684, 37636, 40804, 42436, 45796, 47524, 51076, 64516, 68644, 75076, 77284, 88804, 91204, 98596

Offset: 1

Benoit Cloitre, Apr 14 2002

Previous name was: Numbers n such that sum(d|n,(-1)^d) = 3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001248, A048272.

[4*p^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

4 Prime[Range[40]]^2 (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = 4*prime(n)^2; \\ Michel Marcus, Mar 23 2016

lista(nn) = {forprime(p=2, nn, print1(4*p^2, ", "));} \\ Altug Alkan, Mar 23 2016

a(n) = 4*prime(n)^2 = 4*A001248(n).
Numbers k such that A048272(k) = -3.
Sum_{n>=1} 1/a(n) = P(2)/4, where P is the prime zeta function. - Amiram Eldar, Dec 19 2020

New name from existing formula by Michel Marcus, Mar 23 2016

cp's OEIS Frontend

A101563 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+10*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A158441 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1+x^n) /n ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A322081 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+1)*d^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A325939 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A333809 G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k*(k + 1)) / (1 - x^k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A348951 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A356389 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.

Views

Author