A103221 -id:A103221 - cp's OEIS frontend

A008676 Expansion of 1/((1-x^3)*(1-x^5)).

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

Offset: 0

N. J. A. Sloane

a(n) gives the number of partitions of n using only the parts 3 and 5.  e.g. a(25)=2: 5+5+5+5+5 and 5+5+3+3+3+3+3+3. - Andrew Baxter, Jun 20 2011
a(n) gives the number of partitions of n+8 involving both a 3 and a 5. e.g. a(25)=2 and we may write 33 as 5+5+5+5+5+5+3 and 5+5+5+3+3+3+3+3+3. 11*3 doesn't count as no 5 is involved. - Jon Perry, Jul 03 2004
Conjecture: a(n) = Floor(2*(n + 3)/3) - Floor(3*(n + 3)/5). - John W. Layman, Sep 23 2009
Also, it appears that a(n) gives the number of distinct multisets of n-1 integers, each of which is -2, +3, or +4, such that the sum of the members of each multiset is 2. E.g., for n=5, the multiset {-2,-2,3,3}, and no others, of n-1=4 members, sums to 2, so a(5)=1. - John W. Layman, Sep 23 2009
Appears to be the number of ordered triples summing to n such that 2x = 3y + 4z, ranked by A357489. An unordered version appears to be A357849, ranked by A358102. - Gus Wiseman, Nov 04 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 217
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,0,-1).

Cf. A103221.

a:=[1,0,0,1,0,1,1,0];; for n in [9..100] do a[n]:=a[n-3]+a[n-5]-a[n-8]; od; a; # G. C. Greubel, Sep 08 2019

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019

a := proc (n) option remember; if n < 0 then return 0 elif n = 0 then return 1 else return a(n-3)+a(n-5)-a(n-8) end if end proc

CoefficientList[Series[1/((1-x^3)(1-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)

Vec(O(x^99)+1/(1-x^3)/(1-x^5)) \\ Charles R Greathouse IV, Jun 20 2011

def A008676_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^5))).list()
A008676_list(100) # G. C. Greubel, Sep 08 2019

G.f.: 1/( (1-x^3) * (1-x^5) ).
a(n) = a(n-3) + a(n-5) - a(n-8), a(0)=a(3)=a(5)=a(6)=1, a(1)=a(2)=a(4) =a(6)=a(7)=0.
a(n) = floor((2*n+5)/5) - floor((n+2)/3). - Tani Akinari, Aug 07 2013

Edited by Andrew Baxter, Jun 20 2011

Typo in name fixed by Vincenzo Librandi, Jun 23 2013

A105637 a(n) = a(n-2) + a(n-3) - a(n-5).

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 3, 4, 5, 4, 6, 6, 6, 7, 8, 7, 9, 9, 9, 10, 11, 10, 12, 12, 12, 13, 14, 13, 15, 15, 15, 16, 17, 16, 18, 18, 18, 19, 20, 19, 21, 21, 21, 22, 23, 22, 24, 24, 24, 25, 26, 25, 27, 27, 27, 28, 29, 28, 30, 30, 30, 31, 32, 31, 33, 33, 33, 34, 35, 34, 36, 36, 36, 37, 38, 37

Offset: 0

Paul Barry, Apr 16 2005

a(n+6) = a(n) + 3; convolution of A000035(n) with A010872(n). - Reinhard Zumkeller, Mar 08 2009
Let B be the periodic sequence that repeats (1,2,1,3,3,3,4,5,4,6,6,6). Then the sequence a(1), a(2), ... is obtained by adding 6*(i-1) to every term of the i-th period of B. - Vladimir Shevelev, May 31 2011
Also for n > 0: number of partitions of n into parts 1 or 2 with distinct multiplicities, cf. A211858, A098859. - Reinhard Zumkeller, Dec 27 2012

Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A000035, A010872, A103221.

PARI
```
a(n)=1+floor(n/2)-if(n%3==0,1,0)
```

G.f.: x*(1+2*x)/((1-x^2)*(1-x^3)).
a(n) = Sum_{k=0..n} (k mod 3)*(1-(-1)^(n+k-1))/2.
a(n) = Sum_{k=0..floor(n/2)} (n-2k mod 3).
a(n) = 1 + floor(n/2) - [3 divides n]. - Ralf Stephan, Nov 15 2010
a(n) = A103221(n-1) + 2*A103221(n-2). - R. J. Mathar, Jun 30 2011
a(n) = floor(n/2) + floor((n+2)/3) - floor(n/3). - Mircea Merca, May 20 2013

A025795 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 58, 60, 62, 64, 66, 68, 71, 72, 75, 77, 79, 82, 84, 86, 89, 91, 94, 96, 99, 101, 104

Offset: 0

N. J. A. Sloane

a(n) is the number of ways to pay n dollars with coins of two, three and five dollars. E.g., a(0)=1 because there is one way to pay: with no coin; a(1)=0 no possibility; a(2)=1 (2=1*2); a(3)=1 (3=1*3); a(4)=1 (4=2*2) a(5)=2 (5=3+2=1*5) ... - Richard Choulet, Jan 20 2008

a(n) is the number of partitions of n into parts 2, 3, and 5. See the preceding comment by R. Choulet. - Wolfdieter Lang, Mar 15 2012

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,-1,-1,0,1).

Cf. A008676, A078495, A103221.

a[ n_] := Quotient[n^2 + 10 n + 1 - 13 Mod[n, 2], 60] + 1; (* Michael Somos, Nov 17 2017 *)

{a(n) = (n^2 + 10*n + 1 - n%2 * 13) \60 + 1} /* Michael Somos, Feb 05 2008 */

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^5)).
Let [b(1); b(2); ...; b(p)] denote a periodic sequence: e.g., [0; 1] defines the sequence c such that c(0)=c(2)=...=c(2*k)=0 and c(1)=c(3)=...=c(2*k+1)=1. Then a(n)=0.25*[0; 1] - (1/3)*[1; 0; 0] + (1/5)*[0; 1; 1; 0; 3] + ((n+1)*(n+2)/60) + (7*(n+1)/60). - Richard Choulet, Jan 20 2008
If ||A|| is the nearest number to A (A not a half-integer) we also have a(n) = ||((n+1)*(n+9)/60) + (1/5)[0; 1; 1; 0; 3]. - Richard Choulet, Jan 20 2008
a(n) = 77/360 + 7*(n+1)/60 + (n+2)*(n+1)/60 + (-1)^n/8 - (2/9)*cos(2*(n+2)*Pi/3) + (4/(5*sqrt(5)+25))*cos(2*n*Pi/5) - (4/(5*sqrt(5)-25))*cos(4*n*Pi/5). - Richard Choulet, Jan 20 2008
Euler transform of length 5 sequence [0, 1, 1, 0, 1]. - Michael Somos, Feb 05 2008
a(n) = a(-10-n) for all n in Z. - Michael Somos, Feb 25 2008
a(n) - a(n-2) = A008676(n). a(n) - a(n-5) = A103221(n) = A008615(n+2). A078495(n) = 2^(a(n-7) + a(n-9)) * 3^a(n-8) for all n in Z. - Michael Somos, Nov 17 2017, corrected Jun 23 2021
a(n)-a(n-3) = A008616(n). - R. J. Mathar, Jun 23 2021
a(n) = floor((n^2 + 10*n + 6*(9+(-1)^n))/60). - Hoang Xuan Thanh, Jun 15 2025

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

Original entry on oeis.org

1, 3, 9, 20, 42, 78, 139, 231, 372, 573, 861, 1254, 1791, 2499, 3432, 4629, 6162, 8085, 10492, 13455, 17094, 21503, 26832, 33201, 40795, 49764, 60333, 72687, 87096, 103785, 123075, 145236, 170646, 199626, 232617, 269997, 312277, 359898, 413448, 473438

Offset: 1

Alford Arnold, Aug 21 2009

Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),
or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,
or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.
If we apply the enumeration of Molien series as described in A139672,
this is row 45=9*5 of a table of values related to Molien series, i.e., the
product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).
This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:
      1
      |
      2
      |
1--2--3--2--1

			To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Cf. A139672 (row 21).
For G2, the corresponding sequence is A001399.
For F4, the corresponding sequence is A115264.
For E7, the corresponding sequence is A210068.
For E8, the corresponding sequence is A045513.
See A210634 for a closely related sequence.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^3*(1-x^2)^3*(1-x^3)) )); // G. C. Greubel, Jan 13 2020

seq(coeff(series(x/((1-x)^3*(1-x^2)^3*(1-x^3)), x, n+1), x, n), n = 1..40); # G. C. Greubel, Jan 13 2020

Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* Harvey P. Dale, Aug 03 2025 *)

Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

x=PowerSeriesRing(QQ, 'x', 40).gen()
1/((1-x)^3*(1-x^2)^3*(1-x^3))

a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - R. J. Mathar, Mar 19 2012

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

Corrected link to index entries - R. J. Mathar, Aug 26 2009

A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 18, 18, 19, 21, 22, 23, 23, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 35, 37, 38, 38, 39, 40, 40, 42, 43, 44, 45, 45, 46, 47, 47, 49, 51, 52, 52, 53, 55, 56, 58, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n-th prime > 3 into a sum of 2's or 3's (inclusive or).
From Wolfdieter Lang, Mar 13 2012: (Start)
The primes of the form 6*k+1 are given in A002476.
For n >= 1 such that prime(n+2) is from A002476, one has 8*T(prime(n+2)-1) + 1 = r(n)^2, n >= 1, with the triangular numbers T(n) = A000217(n) and r(n) = A208296(n). Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=2: prime(4)=7, a(2)=1, 8*21 + 1 = 13^2 = A208296(2)^2 = 24*7*1 + 1.
The primes of the form 6*k-1 are given in A007528.
For n >= 1 such that prime(n+2) is from A007528, one has 8*T(prime(n+2)) + 1 = r(n)^2. For T and r see the preceding comment. Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=1, prime(3)=5, a(1)=1, 8*15 + 1 = 11^2 = A208296(1)^2 = 24*5*1 + 1.
(End)

[(NthPrime(n+2)+3) div 6: n in [1..80]]; // Vincenzo Librandi, Sep 06 2016

From R. J. Mathar, May 02 2010: (Start)
A103221 := proc(n) a := 0 ; for t from 0 do if 2*t > n then return a; end if; if n-2*t mod 3 = 0 then a := a+1 ; end if; end do : end proc:
A024699 := proc(n) A103221(ithprime(n+2)) ; end proc: seq(A024699(n),n=1..120) ; (End)

pi6[n_]:=Module[{p=Prime[n+2],c},c=(p-1)/6;If[IntegerQ[c],c,(p+1)/6]]; Array[pi6,80] (* Harvey P. Dale, Aug 19 2013 *)
Table[Floor[(Prime[n + 2] + 3) / 6], {n, 100}] (* Vincenzo Librandi, Sep 06 2016 *)

a(n) = (prime(n+2)+3)\6; \\ Michel Marcus, Sep 06 2016; after Wolfdieter Lang

a(n) = A103221(prime(n+2)). - R. J. Mathar, May 02 2010

a(n) = floor((prime(n+2)+3)/6), n >= 1, prime(n)=A000040(n). Consider the two cases prime(n+2) == 1 (mod 6) and == -1 (mod 6) separately. See the formula above. - Wolfdieter Lang, Mar 15 2012

A090184 Number of partitions of the n-th 3-smooth number into parts 2 and 3.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 22, 25, 28, 33, 37, 41, 43, 49, 55, 65, 73, 82, 86, 97, 109, 122, 129, 145, 163, 171, 193, 217, 244, 257, 289, 325, 342, 365, 385, 433, 487, 513, 577, 649, 683, 730, 769, 865, 973, 1025, 1094, 1153

Offset: 1

Reinhard Zumkeller, Jan 21 2004

nonn

			n=11: A003586(11) = 2^3 * 3 = 24: 3+3+3+3+3+3+3+3 = 3+3+3+3+3+3+2+2+2 = 3+3+3+3+2+2+2+2+2+2 = 3+3+2+2+2+2+2+2+2+2+2 = 2+2+2+2+2+2+2+2+2+2+2+2: a(11)=5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022328, A022329.
Cf. A003586, A008615.
Cf. A103221, A117222, A117220, A117221.

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
Length[IntegerPartitions[#, All, {2, 3}]]& /@ Select[Range[10000], smooth3Q] (* Jean-François Alcover, Oct 13 2021 *)
With[{nn = 6^5}, Map[Floor[#/2] - Floor[#/3] &, Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}] &[nn] + 2]] (* Michael De Vlieger, Oct 13 2021 *)

from sympy import integer_log
def A090184(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return ((m:=bisection(f,n,n)+2)>>1)-m//3 # Chai Wah Wu, Oct 22 2024

a(2^i * 3^j) = floor(2^(i-1) * 3^(j-1) + 1), i*j>0.

a(n) = A103221(A003586(n)).

Offset changed to 1 by Alois P. Heinz, Oct 15 2021

A266777 Molien series for invariants of finite Coxeter group A_8.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 34, 44, 51, 66, 75, 95, 110, 134, 155, 189, 215, 258, 296, 349, 398, 468, 529, 617, 698, 804, 907, 1042, 1167, 1332, 1492, 1690, 1886, 2130, 2366, 2660, 2951, 3298, 3649, 4069, 4484, 4981, 5482, 6064, 6657, 7347, 8041, 8849, 9670, 10605, 11565, 12659, 13769, 15034, 16330, 17782, 19278, 20955

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i).

Note that this is the root system A_k, not the alternating group Alt_k.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,0,-1,-1,-1,-2,-2,0,1,2,3,3,2,1,0,-2,-3,-4,-3,-2,0,1,2,3,3,2,1,0,-2,-2,-1,-1,-1,0,0,1,1,1,0,-1).
Index entries for Molien series

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (&*[1/(1-x^j): j in [2..9]]) )); // G. C. Greubel, Feb 01 2020

seq(coeff(series( mul(1/(1-x^j), j=2..9), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 01 2020

CoefficientList[Series[Product[1/(1-x^j), {j,2,9}], {x,0,70}], x] (* G. C. Greubel, Feb 01 2020 *)

Vec( prod(j=2,9, 1/(1-x^j)) + O('x^70) ) \\ G. C. Greubel, Feb 01 2020

def A266777_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( product(1/(1-x^j) for j in (2..9)) ).list()
A266777_list(70) # G. C. Greubel, Feb 01 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)).

A266778 Molien series for invariants of finite Coxeter group A_9.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 20, 22, 31, 36, 48, 55, 73, 83, 107, 123, 154, 177, 220, 251, 306, 351, 422, 481, 575, 652, 771, 875, 1024, 1158, 1348, 1518, 1754, 1973, 2265, 2538, 2901, 3241, 3684, 4109, 4646, 5167, 5823, 6457, 7246, 8020, 8965, 9898, 11031, 12150, 13495, 14837, 16428, 18022, 19905, 21789, 23999, 26228, 28813

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i).

Note that this is the root system A_k not the alternating group Alt_k.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -2, -1, 0, 1, 3, 3, 3, 2, 1, 0, -1, -4, -4, -4, -3, -2, 0, 2, 3, 4, 4, 4, 1, 0, -1, -2, -3, -3, -3, -1, 0, 1, 2, 1, 1, 1, 1, 0, 0, -1, -1, -1, 0, 1).
Index entries for Molien series

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( &*[1/(1-x^j): j in [2..10]] )); // G. C. Greubel, Feb 02 2020

seq(coeff(series( mul(1/(1-x^j), j=2..10), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 02 2020

CoefficientList[Series[Product[1/(1-x^j), {j,2,10}], {x,0,70}], x] (* G. C. Greubel, Feb 02 2020 *)
LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1,-1,-2,-1,0,1,3,3,3,2,1,0,-1,-4,-4,-4,-3,-2,0,2,3,4,4,4,1,0,-1,-2,-3,-3,-3,-1,0,1,2,1,1,1,1,0,0,-1,-1,-1,0,1},{1,0,1,1,2,2,4,4,7,8,12,13,20,22,31,36,48,55,73,83,107,123,154,177,220,251,306,351,422,481,575,652,771,875,1024,1158,1348,1518,1754,1973,2265,2538,2901,3241,3684,4109,4646,5167,5823,6457,7246,8020,8965,9898},70] (* Harvey P. Dale, Aug 10 2021 *)

Vec( prod(j=2,10, 1/(1-x^j)) +O('x^70) ) \\ G. C. Greubel, Feb 02 2020

def A266778_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( product(1/(1-x^j) for j in (2..10)) ).list()
A266778_list(70) # G. C. Greubel, Feb 02 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)*(1-t^10)).

A335106 Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.

Original entry on oeis.org

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

Offset: 0

David James Sycamore, Michael De Vlieger, May 23 2020

Let L(n) be the set of least part primes of all prime partitions of n, and let G(n) be corresponding set of greatest prime parts. All prime partitions, commencing with p in L(n) and terminating with q in G(n) can be shown as unique pathways on a partition tree of n; see link for details. |G(n)| = A000607(n).

			A000607(10) = 5 and the prime partitions of 10 are: (2,2,2,2,2), (2,2,3,3), (2,3,5), (5,5) and (3,7). Thus G(10) = {2,3,5,5,7}, and consequently row 10 is [1,1,2,1]. In the table below, for n >= 2,  0 is used to indicate when prime(k) is not in G(n) and is less than the greatest member of G(n), otherwise the entry for prime(k) not in G(n) is left empty. For n >= 2 the sum of entries in the n-th row is |G(n)| = A000607(n). Triangle T(n,k) begins:
0;
0;
1;
0, 1;
1;
0, 1, 1;
1, 1;
0, 1, 1, 1;
1, 1, 1;
0, 2, 1, 1;
1, 1, 2, 1;
0, 2, 2, 1, 1;
1, 2, 2, 2;
0, 2, 3, 2, 1, 1;
1, 2, 3, 3, 1;
0, 3, 4, 3, 1, 1;
1, 2, 4, 4, 2, 1;
0, 3, 5, 5, 2, 1, 1;
...

David James Sycamore, Prime Partition Trees

Cf. A000040, A000607, A000720, A333365, A331634, A335285.
Row sums gives A000607 for n > 1. Length of n-th row is A000720(A335285(n)) for n >1.
Number of partition of n in the first k primes: A059841 (k = 1), A103221 (k = 2), A025795 (k = 3), A029144 (k = 4), A140952 (k = 5), A140953 (k = 6).

Flatten@ Block[{nn = 22, t}, t = Block[{s = {Prime@ PrimePi@ nn}}, KeySort@ Merge[#, Identity] &@ Join[{0 -> {}, 1 -> {}}, Reap[Do[If[# <= nn, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1] ] ] &@Total[s], {i, Infinity}]][[-1, -1]] ] ]; Array[Function[p, If[! IntegerQ@ First@ p, {0}, Array[Count[p, Prime@ #] &, PrimePi@ Max@ p]]]@ Map[Max, t[[#]]] &, Max@ Keys@ t]] (* Michael De Vlieger, May 23 2020 *)
row[0]={0}; row[k_] := Join[If[OddQ@k, {0}, {}], Last /@ Tally@ Sort[ First /@ IntegerPartitions[k, All, Prime@ Range@ PrimePi@ k]]]; Join @@ Array[row, 20, 0] (* Giovanni Resta, May 31 2020 *)

More terms from Giovanni Resta, May 31 2020

A140952 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 31, 35, 39, 43, 48, 53, 58, 64, 70, 77, 84, 91, 100, 108, 117, 127, 137, 148, 159, 172, 184, 198, 212, 227, 243, 259, 277, 295, 314, 334, 355, 377, 400, 424, 449, 475, 502, 531, 560

Offset: 0

Alois P. Heinz, Jul 25 2008

Number of partitions of n into the first 5 primes.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1).

Cf. A000040, A025795, A029144, A103221, A140953, A335106.

M := Matrix(28, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 15, 19, 20, 28])) then 1 elif j=1 and member(i, [8, 9, 13, 17, 25, 26]) then -1 else 0 fi):
a:= n-> (M^(n))[1,1]:
seq(a(n), n=0..50);

CoefficientList[Series[1/Times@@(1-x^Prime[Range[5]]),{x,0,70}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1},{1,0,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,28,31,35,39,43,48},70] (* Harvey P. Dale, Jun 18 2021 *)

cp's OEIS Frontend

A008676 Expansion of 1/((1-x^3)*(1-x^5)).

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A105637 a(n) = a(n-2) + a(n-3) - a(n-5).

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A025795 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)).

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A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

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A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

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A090184 Number of partitions of the n-th 3-smooth number into parts 2 and 3.

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A266777 Molien series for invariants of finite Coxeter group A_8.

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A025795 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)).

A164680 Expansion of x/((1-x)^3(1-x^2)^3(1-x^3)).

A140952 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)).