A329349 Number of occurrences of the largest primorial 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, 1, 4, 2, 1, 4, 1, 1, 1, 1, 6, 2, 2, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 1, 8, 6, 4, 1, 2, 2, 8, 6, 2, 1, 3, 1, 2, 3, 2, 1, 12, 1, 4, 6, 5, 1, 1, 1, 2, 2, 4, 16, 12, 1, 2, 6, 2, 1, 2, 1, 2, 6, 8, 1, 10, 12, 4, 6, 2, 1, 6, 1, 2, 2, 1, 1, 12, 1, 8, 1

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 = 21 = 3 * 7, A108951(21) = A034386(3) * A034386(7) = 6 * 210, so the factor of the largest primorial present (210) in the greedy sum is 6 (as 1260 = 210 + 210 + 210 + 210 + 210 + 210), thus a(21) = 6.
For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 1*30 + 3*6, and as the factor of the largest primorial in the sum is 1, we have a(24) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed 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, A071178, A108951, A276086, A276153, A324886, A324888, A329040, A329343, A329344, A329345, A329348.

Programs

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
A276153(n) = { my(e=0, p=2); while(n, e=n%p; n = n\p; p = nextprime(1+p)); (e); };
A329349(n) = A276153(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));
A071178(n) = if(1==n,0,my(es=factor(n)[,2]); es[#es]);
A329349(n) = A071178(A324886(n));

Formula

a(n) = A276153(A108951(n)) = A071178(A324886(n)).

a(n) <= A324888(n).