A131996 -id:A131996 - cp's OEIS frontend

A106244 Number of partitions into distinct prime powers.

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 21, 24, 27, 30, 33, 37, 41, 46, 50, 56, 62, 68, 75, 82, 91, 99, 108, 118, 129, 141, 152, 166, 180, 196, 211, 229, 248, 267, 288, 310, 335, 360, 387, 415, 447, 479, 513, 549, 589, 630, 672, 719, 768, 820, 873, 930

Offset: 0

Reinhard Zumkeller, Apr 26 2005

nonn

A054685(n) < a(n) < A023893(n) for n>2.

			a(10) = #{3^2+1,2^3+2,7+3,7+2+1,5+2^2+1,5+3+2,2^2+3+2+1} = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)

Cf. A000586, A000607, A000961.
Cf. A062051, A105420, A131996.
Cf. A023893, A051613, A054685.

import Data.MemoCombinators (memo2, integral)
a106244 n = a106244_list !! n
a106244_list = map (p' 1) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < pp then 0 else p' (k + 1) (m - pp) + p' (k + 1) m
           where pp = a000961 k
-- Reinhard Zumkeller, Nov 24 2015

g:=(1+x)*(product(product(1+x^(ithprime(k)^j),j=1..5),k=1..20)): gser:=series(g,x=0,68): seq(coeff(gser,x,n),n=1..63); # Emeric Deutsch, Aug 27 2007

m = 64; gf = (1+x)*Product[1+x^(Prime[k]^j), {j, 1, 5}, {k, 1, 18}] + O[x]^m; CoefficientList[gf, x] (* Jean-François Alcover, Mar 02 2019, from Maple *)

lista(m) = {x = t + t*O(t^m); gf = (1+x)*prod(k=1, m, if (isprimepower(k),(1+x^k), 1)); for (n=0, m, print1(polcoeff(gf, n, t), ", "));} \\ Michel Marcus, Mar 02 2019

a(n) = A054685(n-1)+A054685(n). - Vladeta Jovovic, Apr 28 2005

G.f.: (1+x)*Product(Product(1+x^(p(k)^j), j=1..infinity),k=1..infinity), where p(k) is the k-th prime (offset 0). - Emeric Deutsch, Aug 27 2007

Offset corrected and a(0)=1 added by Reinhard Zumkeller, Nov 24 2015

A081601 Numbers m such that 3 does not divide Sum_{k=0..m} binomial(2k,k) = A006134(m).

Original entry on oeis.org

0, 3, 9, 12, 27, 30, 36, 39, 81, 84, 90, 93, 108, 111, 117, 120, 243, 246, 252, 255, 270, 273, 279, 282, 324, 327, 333, 336, 351, 354, 360, 363, 729, 732, 738, 741, 756, 759, 765, 768, 810, 813, 819, 822, 837, 840, 846, 849, 972, 975, 981, 984, 999, 1002, 1008, 1011

Offset: 1

Benoit Cloitre, Apr 22 2003

Apparently a(n)/3 mod 2 = A010060(n-1), the Thue-Morse sequence.

a(n+1) is the smallest number with exactly n+1 partitions into distinct powers of 2 or of 3: A131996(a(n+1)) = n+1 and A131996(m) < n+1 for m < a(n+1). - Reinhard Zumkeller, Aug 06 2007

			For n=0, A006134(0) = 1, hence 0 is a term.

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A005836, A006134, A006996, A010060, A083096, A131996.

Equals A089118(n-2) + 1, n > 1.

Select[Range[0, 1020], Mod[Sum[Binomial[2 k, k], {k, 0, #}], 3] != 0 &] (* Michael De Vlieger, Nov 28 2015 *)

for(n=0, 1e3, if(sum(k=0, n, binomial(2*k, k)) % 3 > 0, print1(n,", "))) \\ Altug Alkan, Nov 26 2015

Apparently a(n) = 3*A005836(n).

G.f.: (x/(1 - x))*Sum_{k>=0} 3^(k+1)*x^(2^k)/(1 + x^(2^k)) (conjecture). - Ilya Gutkovskiy, Jul 23 2017

Zero prepended to the sequence and formulas modified accordingly by L. Edson Jeffery, Nov 25 2015

A131995 Number of partitions of n into powers of 2 or of 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 16, 20, 26, 32, 42, 50, 62, 74, 92, 108, 131, 153, 184, 213, 251, 288, 339, 387, 448, 511, 589, 666, 761, 857, 976, 1095, 1237, 1384, 1561, 1737, 1946, 2161, 2415, 2672, 2971, 3281, 3640, 4007, 4425, 4860, 5359, 5869, 6446, 7049, 7729, 8428

Offset: 0

Reinhard Zumkeller, Aug 06 2007

			a(10) = #{9+1, 8+2, 8+1+1, 4+4+2, 4+4+1+1, 4+3+3, 4+3+2+1,
4+3+1+1+1, 4+2+2+2, 4+2+2+1+1, 4+2+1+1+1+1, 4+1+1+1+1+1+1, 3+3+3+1,
3+3+2+2, 3+3+2+1+1, 3+3+1+1+1+1, 3+2+2+2+1, 3+2+2+1+1+1,
3+2+1+1+1+1+1, 3+1+1+1+1+1+1+1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1,
2+2+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 26.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A018819, A062051, A023893, A000041, A131996.

g:=(1-x)/(product((1-x^(2^k))*(1-x^(3^k)),k=0..10)): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..53); # Emeric Deutsch, Aug 26 2007

G.f.: (1-x)/Product_{k>=0} (1-x^(2^k))*(1-x^(3^k)). - Emeric Deutsch, Aug 26 2007

Prepended a(0) = 1, Joerg Arndt and David A. Corneth, Sep 06 2020

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A106244 Number of partitions into distinct prime powers.

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A081601 Numbers m such that 3 does not divide Sum_{k=0..m} binomial(2k,k) = A006134(m).

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A131995 Number of partitions of n into powers of 2 or of 3.

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