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A329343
Difference between the indices of the smallest and the largest primorial in the greedy sum of primorials adding to A108951(n).
9
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.
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); };
A243055(n) = if(1==n, 0, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (primepi(gpf)-primepi(lpf)));
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 11 2019
STATUS
approved