cp's OEIS Frontend

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.

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A219182 Maximal number of partitions of n into any number k of distinct prime parts or 0 if there are no such partitions.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 4, 3, 5, 4, 5, 4, 6, 4, 6, 4, 6, 6, 6, 6, 9, 6, 9, 8, 8, 10, 11, 10, 11, 11, 11, 13, 13, 14, 13, 16, 13, 18, 14, 19, 15, 21, 15, 22, 18, 25, 18, 26, 22, 29, 22
Offset: 0

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Author

Alois P. Heinz, Nov 13 2012

Keywords

Comments

a(n) is maximal element of row n of triangle A219180 or 0 if the row is empty. a(n) = 0 iff n in {1,4,6}.

Examples

			a(31) = 4 because there are 4 partitions of 31 into 3 distinct prime parts ([3,5,23], [3,11,17], [5,7,19], [7,11,13]) but not more than 4 partitions of 31 into k distinct prime parts for any other k.

Links

Crossrefs

Cf. A219180.

Programs

  • Maple
    with(numtheory):
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [0], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= n-> max(b(n, pi(n))[]):
seq(a(n), n=0..120);
  • Mathematica
    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, {0}, zip[b[n, i-1], Join[{0}, If[Prime[i]>n, {}, b[n-Prime[i], i-1]]]]]]; a[n_] := Max[b[n, PrimePi[n]]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Feb 12 2017, translated from Maple *)

Formula

a(n) = max_{k>=0} A219180(n,k).

A358009 Number of partitions of n into at most 4 distinct prime parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 5, 7, 5, 9, 7, 9, 7, 9, 9, 11, 9, 12, 8, 13, 11, 14, 13, 13, 12, 16, 14, 18, 17, 16, 17, 20, 17, 23, 19, 21, 19, 24, 23, 28, 24, 26, 25, 26, 30, 30, 29, 29, 29, 32, 36, 37, 36, 32, 38, 35, 43, 41, 43, 20
Offset: 0

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Author

Ilya Gutkovskiy, Oct 24 2022

Keywords

Crossrefs

Cf. A000040, A000586, A219180, A219198, A223893, A347550, A347578, A347586, A347741, A358010, A358011.

A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

Original entry on oeis.org

2, 16, 26, 33, 55, 59, 0, 0, 124, 159, 233, 227, 276, 0, 372, 480, 0, 0, 0, 752, 0, 920, 0, 1011, 0, 1211, 1425, 0, 0, 0, 0, 0, 2050, 2336, 2495, 0, 0, 0, 0, 3340, 0, 3712, 0, 0, 4303, 0, 0, 0, 0, 5195, 0, 5669, 0, 6163, 6673, 0, 0, 0, 7504, 0, 0, 8670, 0, 9304, 9623, 0, 0, 0, 10638, 10981, 0, 12062, 0
Offset: 1

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Author

Metin Sariyar, Jun 04 2021

Keywords

Comments

From David A. Corneth, Aug 21 2025: (Start)
How to prove a 0? I used the heuristic:
a(n) = 0 if 2*n consecutive integers can be written in strictly more than n ways as a sum of n distinct primes and up to that point no positive integer has exactly n such ways.
What other rules where used? (End)

Examples

			a(2) = 16 because 16 is the smallest number whose number of partitions into 2 distinct primes is 2; 16 = 3+13 = 5+11.

Links

Crossrefs

Cf. A364692 asks for the largest number with the same properties.
Cf. A000586, A077914, A117929, A125688, A219180, A219198, A219199, A219200, A219201, A219202, A219203, A219204.

Extensions

a(12)-a(20) from Alois P. Heinz, Jun 04 2021
More terms from David A. Corneth, Aug 21 2025

A382330 a(n) is the number of positive integers k for which Sum_{i=1..j} (p_i+e_i) = n, where p_1^e_1*...*p_j^e_j is the prime factorization of k.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 27, 36, 47, 61, 79, 104, 133, 170, 215, 272, 343, 433, 542, 678, 845, 1050, 1300, 1608, 1981, 2437, 2988, 3655, 4460, 5433, 6603, 8014, 9705, 11731, 14155, 17055, 20509, 24624, 29512, 35313, 42184, 50315, 59916, 71248, 84598
Offset: 1

Views

Author

Felix Huber, Mar 23 2025

Keywords

Comments

a(n) is the number of positive integers k for A008474(k) = n.

Examples

			The a(7) = 4 positive integers k are 32 = 2^5, 81 = 3^4, 25 = 5^2, 6 = 2^1*3^1 because 2 + 5 = 3 + 4 = 5 + 2 = 2 + 1 + 3 + 1 = 7 and there is no further positive integer with that property.
The a(11) = 15 positive integers k are 512 = 2^9, 6561 = 3^8, 15625 = 5^6, 2401 = 7^4, 96 = 2^5*3^1, 144 = 2^4*3^2, 216 = 2^3*3^3, 324 = 2^2*3^4, 486 = 2^1*3^5, 40 = 2^3*5^1, 100 = 2^2*5^2, 250 = 2^1*5^3, 14 = 2^1*7^1, 45 = 3^2*5^1, 75 = 3^1*5^2 because 2 + 9 = 3 + 8 = 5 + 6 = 7 + 4 = 2 + 5 + 3 + 1 = 2 + 4 + 3 + 2 = 2 + 3 + 3 + 3 = 2 + 2 + 3 + 4 = 2 + 1 + 3 + 5 = 2 + 3 + 5 + 1 = 2 + 2 + 5 + 2 = 2 + 1 + 5 + 3 = 2 + 1 + 7 + 1 = 3 + 2 + 5 + 1 = 3 + 1 + 5 + 2 = 11 and there is no further positive integer with that property.

Links

Crossrefs

Cf. A008474, A219180, A377505, A377537.

Programs

  • Maple
    # processes b and T from Alois P. Heinz (A219180).
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,NumberTheory:-pi(n));
       while nops(l)>0 and l[-1]=0 do l:=subsop(-1=NULL,l) od; l[]
    end:
A382330:=proc(n)
    local a,k,s,i,j,L;
    a:=0;k:=1;s:=0;
    while s+k<=n do
        s:=s+ithprime(k);k:=k+1
    od;
    for i to k-1 do
        for j to n-i do
            L:=[T(j)];
            if nops(L)>=i+1 then
                a:=a+L[i+1]*binomial(n-j-1,n-j-i);
            fi
        od
    od;
    return a
end proc;
seq(A382330(n),n=1..51);
Previous Showing 21-24 of 24 results.

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