A329344 Number of times most frequent primorial is present in the greedy sum of primorials adding to A108951(n).

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 6, 2, 1, 2, 1, 4, 6, 2, 1, 3, 4, 2, 1, 4, 1, 5, 1, 1, 6, 2, 8, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 6, 8, 6, 4, 1, 2, 4, 8, 6, 2, 1, 3, 1, 2, 3, 2, 13, 12, 1, 4, 6, 5, 1, 3, 1, 2, 5, 4, 16, 12, 1, 2, 6, 2, 1, 2, 11, 2, 6, 8, 1, 10, 12, 4, 6, 2, 7, 6, 1, 12, 10, 6, 1, 12, 1, 8, 4

Offset: 1

Author

Antti Karttunen, Nov 11 2019

Keywords

• nonn

Comments

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

Examples

			For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 30 + 6 + 6 + 6, and as the most frequent primorial in the sum is 6 = A002110(2), we have a(24) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to primorial base
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A002110, A034386, A051903, A108951, A276086, A324886, A324888, A328114, A329040, A329045, A329343, A329348, A329349.

Programs

Mathematica

With[{b = Reverse@ Prime@ Range@ 120}, Array[Max@ IntegerDigits[#, MixedRadix[b]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] ] (* Michael De Vlieger, Nov 18 2019 *)

PARI

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A329344(n) = A328114(A108951(n));

PARI

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A329344(n) = A051903(A324886(n));

Formula

a(n) = A328114(A108951(n)) = A051903(A324886(n)).