This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
29 is in the sequence because A000586(29) = 7 is prime.
135 is in the sequence because A000586(135) = 5 * 135. 1040 is in the sequence because A000586(1040) = 4201384 * 1040.
n = 10 has a(10) = 5 partitions into prime parts: 10 = 2 + 2 + 2 + 2 + 2 = 2 + 2 + 3 + 3 = 2 + 3 + 5 = 3 + 7 = 5 + 5. n = 15 has a(15) = 12 partitions into prime parts: 15 = 2 + 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 5 = 2 + 3 + 5 + 5 = 2 + 3 + 3 + 7 = 2 + 2 + 11 = 2 + 13 = 3 + 3 + 3 + 3 + 3 = 3 + 5 + 7 = 5 + 5 + 5.
a000607 = p a000040_list where p _ 0 = 1 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m -- Reinhard Zumkeller, Aug 05 2012
[1] cat [#RestrictedPartitions(n,{p:p in PrimesUpTo(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019
with(gfun): t1:=mul(1/(1-q^ithprime(n)),n=1..51): t2:=series(t1,q,50): t3:=seriestolist(t2); # fixed by Vaclav Kotesovec, Sep 14 2014
CoefficientList[ Series[1/Product[1 - x^Prime[i], {i, 1, 50}], {x, 0, 50}], x]
f[n_] := Length@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]; Array[f, 57] (* Robert G. Wilson v, Jul 23 2010 *)
Table[Length[Select[IntegerPartitions[n],And@@PrimeQ/@#&]],{n,0,60}] (* Harvey P. Dale, Apr 22 2012 *)
a[n_] := a[n] = If[PrimeQ[n], 1, 0]; c[n_] := c[n] = Plus @@ Map[# a[#] &, Divisors[n]]; b[n_] := b[n] = (c[n] + Sum[c[k] b[n - k], {k, 1, n - 1}])/n; Table[b[n], {n, 1, 20}] (* Thomas Vogler, Dec 10 2015: Uses Euler transform, caches computed values, faster than IntegerPartitions[] function. *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j + 1]] -= poly[[j + 1 - p]], {j, nmax, p, -1}];, {k, 2, pmax}]; s = Sum[poly[[k + 1]]*x^k, {k, 0, Length[poly] - 1}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2021 *)
N=66;x='x+O('x^N); Vec(1/prod(k=1,N,1-x^prime(k))) \\ Joerg Arndt, Sep 04 2014
from sympy import primefactors
l = [1, 0]
for n in range(2, 101):
l.append(sum(sum(primefactors(k)) * l[n - k] for k in range(1, n + 1)) // n)
l # Indranil Ghosh, Jul 13 2017
[Partitions(n, parts_in=prime_range(n + 1)).cardinality() for n in range(100)] # Giuseppe Coppoletta, Jul 11 2016
a(11) = 3 since 11 = 1+2+3+5=1+3+7 has 3 partitions of distinct primes-including-1. - _Ron Knott_, Aug 27 2016
s:= proc(n) option remember;
`if`(n<1, n+1, ithprime(n)+s(n-1))
end:
b:= proc(n, i) option remember; (p-> `if`(n=0, 1,
`if`(n>s(i), 0, b(n, i-1)+ `if`(p>n, 0,
b(n-p, i-1)))))(`if`(i<1, 1, ithprime(i)))
end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100); # Alois P. Heinz, Aug 27 2016
myprime[ n_ ] := If[ n===0, 1, Prime[ n ] ]; ta1=Table[ Product[ 1+z^myprime[ k ], {k, 0, n} ]~CoefficientList~z, {n, 31, 32} ]; leveled=Count[ Take[ Last@ta1, Length@ta1[ [ -2 ] ] ]-ta1[ [ -2 ] ], 0 ]; Take[ Last@ta1, leveled ]
Table[Length@ DeleteCases[DeleteCases[IntegerPartitions@ n, {_, a_, _} /; CompositeQ@ a], w_ /; MemberQ[Differences@ w, 0]], {n, 0, 60}] (* Michael De Vlieger, Aug 27 2016 *)
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.
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(10) = #{10, 6+4} = 2;
a(11) = #{ } = 0;
a(12) = #{12, 8+4} = 2;
a(13) = #{9+4} = 1;
a(14) = #{14, 10+4, 8+6} = 3;
a(15) = #{15, 9+6} = 2;
a(16) = #{16, 12+4, 10+6} = 3;
a(17) = #{9+8} = 1;
a(18) = #{18, 14+4, 12+6, 10+8, 8+6+4} = 5;
a(19) = #{15+4, 10+9, 9+6+4} = 3;
a(20) = #{20, 16+4, 14+6, 12+8, 10+6+4} = 5.
a204389 = p a002808_list where p _ 0 = 1 p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..70); # Alois P. Heinz, May 29 2013
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)
T(0,0) = 1: [], the empty partition. T(2,1) = 1: [2]. T(5,1) = 1: [5], T(5,2) = 1: [2,3]. T(16,2) = 2: [5,11], [3,13]. Triangle T(n,k) begins: 1; ; 0, 1; 0, 1; ; 0, 1, 1; ; 0, 1, 1; 0, 0, 1; 0, 0, 1; 0, 0, 1, 1; 0, 1; 0, 0, 1, 1; ...
b:= proc(n, i) option remember;
`if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
[0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
end:
T:= proc(n) local l; l:= b(n, numtheory[pi](n));
while nops(l)>0 and l[-1]=0 do l:= subsop(-1=NULL, l) od; l[]
end:
seq(T(n), n=0..50);
nn=20;a=Table[Prime[n],{n,1,nn}];CoefficientList[Series[Product[1+y x^a[[i]],{i,1,nn}],{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Nov 21 2012 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]]; b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, zip[Plus, b[n, i-1], Join[{0}, If[Prime[i] > n, {}, b[n-Prime[i], i-1]]], 0]]]; T[n_] := Module[{l}, l = b[n, PrimePi[n]]; While[Length[l]>0 && l[[-1]] == 0, l = ReplacePart[l, -1 -> Sequence[]]]; l]; Table[T[n], {n, 0, 50}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)
T(n)={ Vec(prod(k=1, n, 1 + isprime(k)*y*x^k + O(x*x^n))) }
{ my(t=T(20)); for(n=1, #t, print(if(t[n]!=0, Vecrev(t[n]), []))) } \\ Andrew Howroyd, Dec 22 2017
a(5) = 3: [2,3], [3,2], [5]. a(7) = 3: [2,5], [5,2], [7]. a(8) = 2: [3,5], [5,3]. a(9) = 2: [2,7], [7,2]. a(10) = 8: [2,3,5], [2,5,3], [3,2,5], [3,5,2], [5,2,3], [5,3,2], [3,7], [7,3].
with(numtheory):
b:= proc(n, i) b(n, i):=
`if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
[0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
end:
a:= proc(n) local l; l:= b(n, pi(n));
a(n):= add(l[i]*(i-1)!, i=1..nops(l))
end:
seq(a(n), n=0..70);
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..70); # Alois P. Heinz, Jan 30 2020
zip = With[{m=Max[Length[#1], Length[#2]]}, PadRight[#1, m]+PadRight[#2, m] ]&;
b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, b[n, i-1] ~zip~ Join[{0}, If[Prime[i] > n, {}, b[n - Prime[i], i-1]]], {0}]];
a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 24 2017, adapted from Maple *)
The a(10) = 2 through a(15) = 10 partitions:
(8,2) (11) (9,3) (13) (9,5) (8,7)
(9,1) (6,5) (10,2) (7,6) (12,2) (10,5)
(7,4) (6,4,2) (8,5) (8,4,2) (11,4)
(8,3) (10,3) (9,4,1) (12,3)
(9,2) (12,1) (14,1)
(10,1) (6,4,3) (6,5,4)
(6,4,1) (8,4,1) (8,4,3)
(8,6,1)
(9,4,2)
(10,4,1)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k)) * prod(k=4, n, 1 + if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024
The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]
Comments