A329343 Difference between the indices of the smallest and the largest primorial in the greedy sum of primorials adding to A108951(n).

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 0, 0, 2

Offset

1,27

Comments

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

Positions of the records (and conjecturally, the positions of the first occurrences of each n) begin as 1, 8, 27, 162, 289, 529, 841, 1369, 1681, 2209, 2809, 3481, 4489, 5041, 5329, 6889, ..., that after 162 all seem to be squares of certain primes. See also A329051.

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

Formula

a(n) = A243055(A324886(n)).

Example

For n = 18 = 2 * 3^2, A108951(18) = A034386(2) * A034386(3)^2 = 2 * 6^2 = 72 = 30 + 30 + 6 + 6, and as the largest primorial in the sum is 30 = A002110(3), and the least primorial is 6 = A002110(2), we have a(18) = 3-2 = 1.

Prog

(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
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));
A243055(n) = if(1==n, 0, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (primepi(gpf)-primepi(lpf)));
A329343(n) = A243055(A324886(n));

Crossrefs

Cf. A002110, A034386, A108951, A243055, A276086, A324886, A324888, A329040, A329051.

Sequence in context: A321912 A329921 A092303 * A063725 A084888 A091400

Adjacent sequences: A329340 A329341 A329342 * A329344 A329345 A329346

Keyword

nonn

Author

Antti Karttunen, Nov 11 2019

Status

approved