|
|
A257994
|
|
Number of prime parts in the partition having Heinz number n.
|
|
26
|
|
|
0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 1, 0, 1, 2, 0, 3, 0, 0, 2, 1, 0, 2, 1, 1, 2, 0, 0, 1, 1, 1, 1, 0, 1, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 1, 1, 0, 2, 0, 0, 3, 0, 1, 1, 0, 1, 4, 1, 1, 1, 2, 0, 1, 1, 0, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,9
|
|
COMMENTS
|
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
|
|
REFERENCES
|
George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.
|
|
LINKS
|
|
|
FORMULA
|
Additive with a(p^e) = e if primepi(p) is prime, and 0 otherwise. - Amiram Eldar, Nov 03 2023
|
|
EXAMPLE
|
a(30) = 2 because the partition with Heinz number 30 = 2*3*5 is [1,2,3], having 2 prime parts.
|
|
MAPLE
|
with(numtheory): a := proc (n) local B, ct, s: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for s to nops(B(n)) do if isprime(B(n)[s]) = true then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 130);
|
|
MATHEMATICA
|
B[n_] := Module[{nn, j, m}, nn = FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[ m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
a[n_] := Module[{ct, s}, ct = 0; For[s = 1, s <= Length[B[n]], s++, If[ PrimeQ[B[n][[s]]], ct++]]; ct];
Table[Total[Cases[FactorInteger[n], {p_, k_}/; PrimeQ[PrimePi[p]]:>k]], {n, 30}] (* Gus Wiseman, Jan 17 2020 *)
|
|
PROG
|
(PARI) a(n) = my(f = factor(n)); sum(i=1, #f~, if(isprime(primepi(f[i, 1])), f[i, 2], 0)); \\ Amiram Eldar, Nov 03 2023
|
|
CROSSREFS
|
Positions of positive terms are A331386.
Products of primes of prime index are A076610.
The number of nonprime prime indices is A330944.
Cf. A000040, A000720, A001222, A007821, A018252, A056239, A112798, A215366, A279952, A302242, A320628, A330945.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|