A038044 -id:A038044 - cp's OEIS frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

1, 4, 6, 12, 10, 24, 14, 32, 27, 40, 22, 72, 26, 56, 60, 80, 34, 108, 38, 120, 84, 88, 46, 192, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 324, 74, 152, 156, 320, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 448, 228, 232, 118

Offset: 1

Christian G. Bower

Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - Michael Somos, Jun 08 2000
Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008
Sum of all parts of all partitions of n into equal parts. - Omar E. Pol, Jan 18 2013

			For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The sum of all parts is 6 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 24 equalling 6 times the number of divisors of 6, so a(6) = 24. - _Omar E. Pol_, May 08 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

T. D. Noe, Table of n, a(n) for n = 1..1000
J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.
Jean Bourgain, Sergei Konyagin and Igor Shparlinski. Distribution on elements of cosets of small subgroups and applications, arXiv:1103.0567 [math.NT], Mar 2 2011.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147. [Broken link?]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147.

Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.
Cf. A127648, A127093, A127528.
Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).
Column 1 of A329323.

a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory): A038040 := n->tau(n)*n;

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

n*numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])

a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(x^k-1)^2,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020

from sympy import divisor_count as d
def a(n): return n*d(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 15 2022

[n*sigma(n,0) for n in range(1, 60)] # Stefano Spezia, Jul 20 2025

Dirichlet g.f.: zeta(s-1)^2.
G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - Vladeta Jovovic, Dec 30 2001
Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - Vladeta Jovovic, Oct 30 2001
Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
Equals row sums of triangle A127528. - Gary W. Adamson, May 21 2007
a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - Omar E. Pol, Jan 18 2013
a(n) = A000203(n) * A240471(n) + A106315(n). - Reinhard Zumkeller, Apr 06 2014
L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017
a(n) = Sum_{d|n} A018804(d). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{d|n} phi(d)*sigma(n/d). - Ridouane Oudra, Jan 21 2021
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 + 2*n*q^n - n^2*q^(2*n))/(1 - q^n)^2. - Peter Bala, Jan 22 2021
a(n) = Sum_{k=1..n} sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/sqrt(log x). That is, there are 0 < A < B such that Ax/sqrt(log x) < f(x) < Bx/sqrt(log x). - Charles R Greathouse IV, Mar 15 2022
Sum_{k=1..n} a(k) ~ n^2*log(n)/2 + (gamma - 1/4)*n^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 25 2022
Mobius transform of A060640. - R. J. Mathar, Feb 07 2023

A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, 1947270476915296449559703445493848930452791205, 3791862310265926082868235028027893277370233152247388584761734150717768254410341175325352026

Offset: 0

N. J. A. Sloane and Richard Stanley

Number of binary trees of height less than or equal to n. [Corrected by Orson R. L. Peters, Jan 03 2020]
The rightmost digits cycle (0,1,2,5,6,7,0,1,2,5,6,7,...). - Jonathan Vos Post, Jul 21 2005
Apart from the initial term, a subsequence of A008318. - Reinhard Zumkeller, Jan 17 2008
Partial sums of A001699. - Jonathan Vos Post, Feb 17 2010
Corresponds to the second and second last diagonals of A119687. - John M. Campbell, Jul 25 2011
This is a divisibility sequence. - Michael Somos, Jan 01 2013
Sum_{n>=1} 1/a(n) = 1.739940825174794649210636285335916041018367182486941... . - Vaclav Kotesovec, Jan 30 2015
From Vladimir Vesic, Oct 03 2015: (Start)
Forming Herbrand's domains of formula: (∃x)(∀y)(∀z)(∃k)(P(x)∨Q(y)∧R(k))
where: x->a
       k->f(y,z)
we get:
  H0 = {a}
  H1 = {a, f(a,a)}
  H2 = {a, f(a,a), f(a,f(a,a)), f(f(a,a),a), f(f(a,a),f(a,a))}
  ...
The number of elements in each domain follows this sequence.
(End)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
This is a strong divisibility sequence; see A329429. - Clark Kimberling, Nov 13 2019
From Peter Bala, Oct 31 2022: (Start)
Let k be a positive integer. Clearly, the sequence obtained by reducing a(n) modulo k is eventually periodic. Conjectures:
1) The sequence obtained by reducing a(n) modulo 2^k is eventually periodic with period 2.
2) The sequence obtained by reducing a(n) modulo 10^k is eventually periodic with period 6 (the case k = 1 is noted above).
3) The sequence obtained by reducing a(n) modulo 20^k is eventually periodic with period 6.
4) For n >= floor(k/2) and for 1 <= i <= 6, the value of a(6*n+i) mod 10^k is a constant independent of n. The digits of these 6 constant integers, when read from right to left, are the first k digits of the 10-adic numbers A318135 (i = 1), A318136 (i = 2), A318137 (i = 3), A318138 (i = 4), A318139 (i = 5) and A318140 (i = 6), respectively. An example is given below.
  n      a(6*n+1) mod 10^11
1           10066388901
2           72084948901
3           67988948901
4           61588948901
5           01588948901
6           01588948901
7           01588948901
...             ...
A318135 begins 1, 0, 9, 8, 4, 9, 8, 8, 5, 1, 0, 2, .... (End)

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 443-448.
R. K. Guy, How to factor a number, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 49-89.
R. Penrose, The Emperor's New Mind, Oxford, 1989, p. 122.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..13
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Transactions (2021) Vol. 1, No. 1, Article 1.
P. Flajolet and A. M. Odlyzko, Limit distributions of coefficients of iterates of polynomials with applications to combinatorial enumerations, Math. Proc. Camb. Phil. Soc., 96 (1984), 237-253.
Claudio Gentile, Fabio Vitale, and Anand Rajagopalan, Flattening a Hierarchical Clustering through Active Learning, arXiv:1906.09458 [cs.LG], 2019.
Spencer Hamblen, Rafe Jones, and Kalyani Madhu, The density of primes in orbits of z^d + c, arXiv:1303.6513 [math.NT], 2013; to appear, Int. Math. Res. Not., c. 2015.
Dimitur Krustev, Simple Programs on Binary Trees Testing and Decidable Equivalence, 2016.
Robin Lamarche-Perrin, An Information-theoretic Framework for the Lossy Compression of Link Streams, arXiv:1807.06874 [cs.DS], 2018.
R. Lamarche-Perrin, Y. Demazeau, and J.-M. Vincent, A Generic Algorithmic Framework to Solve Special Versions of the Set Partitioning Problem, Preprint 105, Max-Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig, 2014.
C. Lenormand, Arbres et permutations II, see p. 6.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, Hunter College, CUNY, 2019.
Michael Penn, a stylish proof that..., YouTube video, 2021.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
M. Tainiter, Algebraic approach to stopping variable problems: Representation theory and applications, J. Combinatorial Theory 9 1970 148-161.
P. Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv preprint arXiv:1507.06944 [cs.LO], 2015.
Stephan Wagner and Volker Ziegler, Irrationality of growth constants associated with polynomial recursions, arXiv:2004.09353 [math.NT], 2020.
Wikipedia, Herbrand Structure
Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid.
Index entries for sequences of form a(n+1)=a(n)^2 + ...
Index to divisibility sequences
Index entries for sequences related to Benford's law

Cf. A001699, A004019, A038044, A056207, A076949, A077496, A091980, A143848.
Cf. A143849, A213437, A247981, A248218, A248219, A318135, A318136, A318137.
Cf. A318138, A318139, A318140, A355108.
Cf. A137560, which enumerates binary trees of height less than n and exactly j leaf nodes. - Robert Munafo, Nov 03 2009

function A003095(n)
  if n eq 0 then return 0;
  else return 1 + A003095(n-1)^2;
  end if; return A003095;
end function;
[A003095(n): n in [0..12]]; // G. C. Greubel, Nov 29 2022

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)^2+1) end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 11 2019

NestList[#^2+1&,0,10] (* Harvey P. Dale, Dec 17 2010 *)

{a(n) = if( n<1, 0, 1 + a(n-1)^2)}; /* Michael Somos, Jan 01 2013 */

def A003095(n): return 0 if (n==0) else 1 + A003095(n-1)^2
[A003095(n) for n in range(13)] # G. C. Greubel, Nov 29 2022

a(n) = B_{n-1}(1) where B_n(x) = 1 + x*B_{n-1}(x)^2 is the generating function of trees of height <= n.
a(n) is asymptotic to c^(2^n) where c=1.2259024435287485386279474959130085213... (see A076949). - Benoit Cloitre, Nov 27 2002
c = b^(1/4) where b is the constant in Bottomley's formula in A004019. a(n) appears very asymptotic to c^(2^n) - Sum_{k>=1} A088674(k)/(2*c^(2^n))^(2*k-1). - Gerald McGarvey, Nov 17 2007
a(n) = Sum_{i=1..n} A001699(i). - Jonathan Vos Post, Feb 17 2010
G.f. = x + 2*x^2 + 5*x^3 + 26*x^4 + 677*x^5 + 458330*x^6 + 210066388901*x^7 + ... . - Michael Somos, Jan 01 2013
a(2n) mod 2 = 0 ; a(2n+1) mod 2 = 1. - Altug Alkan, Oct 04 2015
a(n) + a(n-1) = A213437(n). - Peter Bala, Feb 03 2017
0 = a(n)^2*(+a(n+1) + a(n+2)) + a(n+1)^2*(-a(n+1) - a(n+2) - a(n+3)) + a(n+2)^3 for all n>=0. - Michael Somos, Feb 10 2017
a(n) = A091980(2^(n-1)) for n > 0. - Alois P. Heinz, Jul 11 2019

Additional comments from Cyril Banderier, Jun 05 2000
Minor edits by Vaclav Kotesovec, Oct 04 2014
Initial term clarified by Clark Kimberling, Nov 13 2019

A144324 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution (DC:(b,b)->a) applied k times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 9, 1, 1, 16, 64, 70, 18, 1, 1, 32, 256, 540, 280, 40, 1, 1, 64, 1024, 4216, 4320, 1168, 80, 1, 1, 128, 4096, 33264, 67456, 35008, 4672, 168, 1, 1, 256, 16384, 264160, 1064448, 1083136, 280064, 18884, 340, 1, 1, 512, 65536

Offset: 1

Alois P. Heinz, Sep 17 2008

			Square array begins:
  1,  1,   1,    1,     1, ...
  1,  1,   1,    1,     1, ...
  2,  4,   8,   16,    32, ...
  4, 16,  64,  256,  1024, ...
  9, 70, 540, 4216, 33264, ...

Alois P. Heinz, Antidiagonals n = 1..65, flattened
N. J. A. Sloane, Transforms

Columns 1-9 give: A038044, A144316, A144317, A144318, A144319, A144320, A144321, A144322, A144323.

Rows 1+2, 3-4 give: A000012, A000079(k+1), A000302(k+1).

with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],b[t-1]) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: seq(seq(A(n, 1+d-n), n=1..d), d=1..11);

dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum [b[d] *c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc]; A [n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 11}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A144823 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution with a_k (DC:(b,a_k)->a) applied k times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 30, 18, 1, 1, 6, 25, 70, 90, 40, 1, 1, 7, 36, 135, 280, 288, 80, 1, 1, 8, 49, 231, 675, 1168, 864, 168, 1, 1, 9, 64, 364, 1386, 3475, 4672, 2647, 340, 1, 1, 10, 81, 540, 2548, 8496, 17375, 18884, 7968, 698, 1, 1, 11, 100

Offset: 1

Alois P. Heinz, Sep 21 2008

			Square array A(n,k) begins:
   1,   1,    1,     1,     1,      1,      1,      1, ...
   1,   1,    1,     1,     1,      1,      1,      1, ...
   2,   3,    4,     5,     6,      7,      8,      9, ...
   4,   9,   16,    25,    36,     49,     64,     81, ...
   9,  30,   70,   135,   231,    364,    540,    765, ...
  18,  90,  280,   675,  1386,   2548,   4320,   6885, ...
  40, 288, 1168,  3475,  8496,  18130,  35008,  62613, ...
  80, 864, 4672, 17375, 50976, 126910, 280064, 563517, ...

Alois P. Heinz, Antidiagonals n = 1..100, flattened
N. J. A. Sloane, Transforms

Columns 1-9 give: A038044, A144817, A144316, A144818, A144819, A144820, A144317, A144821, A144822.

Rows 1+2, 3-4 give: A000012, A000027, A000290, A002414.

with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum [b[d] *c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc]; A [n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table [A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A144374 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k begins with (k+1) 1's and a_k(n) shifts k places down under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 2, 2, 1, 1, 18, 5, 2, 2, 1, 1, 40, 4, 3, 2, 2, 1, 1, 80, 12, 4, 3, 2, 2, 1, 1, 168, 8, 6, 2, 3, 2, 2, 1, 1, 340, 28, 6, 6, 2, 3, 2, 2, 1, 1, 698, 17, 10, 4, 4, 2, 3, 2, 2, 1, 1, 1396, 60, 13, 8, 4, 4, 2, 3, 2, 2, 1, 1, 2844, 34, 16, 5, 6, 2, 4, 2, 3, 2, 2, 1, 1, 5688

Offset: 1

Alois P. Heinz, Sep 18 2008

Sequence a_k of column k begins with k terms from A000012 (only the last is in the triangle), followed by the first (k+1) terms from A000005.

			Triangle begins:
   1;
   1,  1;
   2,  1, 1;
   4,  2, 1, 1;
   9,  2, 2, 1, 1;
  18,  5, 2, 2, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
N. J. A. Sloane, Transforms

Columns 1-9 give: A038044, A144366, A144367, A144368, A144369, A144370, A144371, A144372, A144373.

Cf. A000012, A000005.

with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: seq(seq(T(n, k), k=1..n), n=1..14);

dck[b_, c_][n_, k_] := dck[b, c][n, k] = Sum[b[d, k]*c[n/d, k], {d, If[n < 0, {}, Divisors[n]]}]; B = dck[T, T]; T[n_, k_] := If[n <= k, 1, B[n-k, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

A341697 a(1) = a(2) = 1; a(n+1) = Sum_{d|n, d < n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 6, 7, 11, 11, 17, 17, 25, 29, 38, 38, 54, 54, 72, 80, 102, 102, 136, 140, 174, 186, 228, 228, 300, 300, 366, 388, 464, 480, 594, 594, 702, 736, 874, 874, 1068, 1068, 1250, 1324, 1528, 1528, 1828, 1844, 2144, 2220, 2534, 2534, 2982, 3026, 3464, 3572, 4028, 4028

Offset: 1

Ilya Gutkovskiy, Feb 17 2021

nonn

Cf. A038044, A122698, A333051, A341698.

a[1] = a[2] = 1; a[n_] := a[n] = Sum[If[d < (n - 1), a[(n - 1)/d] a[d], 0], {d, Divisors[n - 1]}]; Table[a[n], {n, 60}]

A341697(n) = if(n<3, 1, sumdiv(n-1,d,if(d<(n-1), A341697((n-1)/d)*A341697(d), 0))); \\ Antti Karttunen, Feb 17 2021

A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 2, 5, 11, 27, 55, 131, 263, 571, 1168, 2445, 4891, 10113, 20227, 40979, 82229, 165632, 331265, 665365, 1330731, 2666729, 5334769, 10679319, 21358639, 42740683, 85482096, 171004645, 342015001, 684113793, 1368227587, 2736633741, 5473267483, 10946869669, 21893763789, 43788190107

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

nonn

Cf. A038044, A068336, A097558, A122698, A277120, A325303.

a:= proc(n) option remember; uses numtheory;
      1+add(a(d)*a((n-1)/d), d=divisors(n-1))
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Dec 15 2020

a[1] = 1; a[n_] := a[n] = 1 + Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 35}]

G.f.: x * (1/(1 - x) + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j)).

a(n) ~ c * 2^n, where c = 1.27442410710035207761153205319824525254716841098942446508584158048310907298... - Vaclav Kotesovec, Dec 16 2020

A325303 a(1) = 1; a(n+1) = -Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, -1, 2, -4, 7, -14, 32, -64, 120, -244, 502, -1004, 1996, -3992, 8048, -16124, 32104, -64208, 128712, -257424, 514416, -1028960, 2058924, -4117848, 8233832, -16467713, 32939418, -65879316, 131750904, -263501808, 527020884, -1054041768, 2108050776, -4216103560, 8432271328

Offset: 1

Ilya Gutkovskiy, Sep 05 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..3000

Cf. A038044, A122698.

a[n_] := a[n] = -Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 35}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=1, #v-1, v[n+1] = -sumdiv(n, d, v[d]*v[n/d])); v} \\ Andrew Howroyd, Sep 05 2019

a(n) ~ -(-1)^n * c * 2^n, where c = 0.245410823583396667908354210407104718986708517177206856531763635090205896729... - Vaclav Kotesovec, Sep 09 2019

A062177 Shifts left when MASKCONVolved with itself.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 72, 192, 720, 1440, 4320, 11520, 43200, 103680, 362880, 1105920, 4665600, 9331200, 27993600, 74649600, 279936000, 671846400, 2351462400, 7166361600, 30233088000, 67184640000, 221709312000, 644972544000

Offset: 0

Antti Karttunen, Jun 12 2001

Note that the factorials 1!, 2!, 4!, 6!, 9! can be found from the positions 1,3,6,9,15 (or 2,4,7,10,16 if zero-based indexing is used) of this sequence. I do not know whether any larger factorials occur in the sequence.

N. J. A. Sloane, Transforms

Other self-convolved sequences: A000108, A007460 - A007464, A025192, A038044, A061922.

EIGENbyMASKCONV := proc(upto_n) local n,a,j,i,s,m; a := [1]; for i from 0 to upto_n do s := 0; m := maskees(i); n := nops(m); for j from 1 to n do s := s+(a[m[j]+1]*a[m[(n-j)+1]+1]); od; a := [op(a),s]; od; RETURN(a); end;
maskees := proc(n) local a,b,u,i; a := []; b := list_mask_bits(n); u := (2^nops(b))-1; for i from 0 to u do a := [op(a),sum_by_mask_list(i,b)]; od; RETURN(a); end;
list_mask_bits := proc(nn) local n,a,x; n := nn; x := 1; a := []; while(n > 0) do if(1 = (n mod 2)) then a := [op(a),x]; fi; n := floor(n/2); x := 2*x; od; RETURN(a); end;
sum_by_mask_list := proc(nn,a) local n,i,s; n := nn; s := 0; i := 1; while(n > 0) do if(1 = (n mod 2)) then s := s + a[i]; fi; n := floor(n/2); i := i+1; od; RETURN(s); end;

A097417 a(1)=1; a(n+1) = Sum_{k=1..n} a(k) a(floor(n/k)).

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 90, 236, 621, 1629, 4274, 11193, 29337, 76818, 201173, 526730, 1379178, 3610804, 9453695, 24750281, 64798235, 169644626, 444138288, 1162770238, 3044180080, 7969770106, 20865148382, 54625676431, 143011928942

Offset: 1

Leroy Quet, Aug 19 2004

4 is the only composite number n such that a(n+1) = 3a(n) - a(n-1) and if n is a composite number greater than 4 then a(n+1) > 3a(n) - a(n-1). - Farideh Firoozbakht, Feb 05 2005

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

Cf. A038044, A078346, A097919.

a[1]:=1: for n from 1 to 50 do: a[n+1]:=sum(a[k]*a[floor(n/k)],k=1..n): od: seq(a[i],i=1..51) # Mark Hudson, Aug 21 2004

a[1] = 1; a[n_] := a[n] = Sum[ a[k]*a[Floor[(n - 1)/k]], {k, n - 1}]; Table[ a[n], {n, 29}] (* Robert G. Wilson v, Aug 21 2004 *)

{m=29;a=vector(m);print1(a[1]=1,",");for(n=1,m-1,print1(a[n+1]=sum(k=1,n,a[k]*a[floor(n/k)]),","))} \\ Klaus Brockhaus, Aug 21 2004

Ratio a(n+1)/a(n) seems to tend to 1 + Golden Ratio = 2.61803398... = 1 + A001622. - Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 23 2004
Satisfies the "partial linear recursion": a(prime(n)+1) = 3*a(prime(n)) -  a(prime(n)-1). This explains why we get a(n+1)/a(n) -> 1 + phi. Also, lim_{n->oo} a(n)/(1 + phi)^n exists but should not have a simple closed form. - Benoit Cloitre, Aug 29 2004
Limit_{n->oo} a(n)/(1 + phi)^n = 0.108165624886204570982244311730754895284041534583990405146651275318889227986... - Vaclav Kotesovec, May 28 2021

More terms from Klaus Brockhaus, Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 21 2004

cp's OEIS Frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

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A003095 a(n) = a(n-1)^2 + 1 for n >= 1, with a(0) = 0.

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A144324 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution (DC:(b,b)->a) applied k times.

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A144823 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution with a_k (DC:(b,a_k)->a) applied k times.

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A144374 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k begins with (k+1) 1's and a_k(n) shifts k places down under Dirichlet convolution.

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A341697 a(1) = a(2) = 1; a(n+1) = Sum_{d|n, d < n} a(n/d) * a(d).

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A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

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