A220378 With p as the n-th prime, a(n) is the smallest number such that all sums of a(n) squarefree addends having product (p#)^(a(n)-1) are composite.

5, 21, 134, 1365, 1627177

Offset: 1

Author

James G. Merickel, Feb 19 2013

Keywords

• nonn
• hard

Comments

Each term a(n) is the resolution of a simultaneous composites problem involving A000110(n) different numbers all contrived to be relatively prime to the n-th primorial. 203 numbers all need to be determined composite for a(6).

Constructing some number of addends producing only composites is elementary; but computing the smallest seems inconceivable beyond a(6), with a(6)'s computation being hard.

A perhaps dubious comparison* of the collection of sums for each potential value to independent randomly selected numbers relatively prime to 13#=30030 and numerical integrations of

log{1-[1-(1001/192)/log(30030*x)]^203} (See following note for a slight clarification)

suggest a(6) is almost certainly less than 14 digits in length and fairly unlikely to be other than 13 digits. This might be at the margins of computability at time of submission (with strong hardware, nuanced programming, perhaps an exceptional pre-computed prime database (?), and not an inordinate amount of time). Similar computations for a(7), however, suggest it is 28 or 29 digits long.

Note: In the above integrand (where differences between addends in any particular sum are treated as insignificant), 1001/192 is the product (2/1)*(3/2)*(5/4)*(7/6)*(11/10)*(13/12), an adjustment factor; and the whole integrand is the logarithm of the approximate expression for the factors in a product of (quasi-)probabilities that a prime will be found for each number tested over a range. As long as exponentiation of the integral remains near 1, the likelihood is that no number in a given range will have solved the problem.

This problem was conceived while deliberating upon A187749.

Note, as a coincidence apropos of nothing, that the (prime) a(5) is a minor permutation of the digits of 8^8=16777216 with its final digit removed, the only known multidigit prime obtained by right-truncation of a number n^n at its last nonzero digit.

*Disclaimer: If in the search for a(6) one is dependent upon a positive outcome under 10^13, it's advised to devise a stronger--empirically-based--integrand covering some primes beyond 13.

Examples

			a(1)=5, since a sum of one 1 and a number of 2s produces a prime for 2 through 4 addends and a composite for 5.
Two possibilities present for n=2: A) where one addend lacks factors 2 and 3 and B) where two different addends lack one or the other (i.e., 6(k-1)+1=6k-5 and 6(k-2)+5=6k-7, where k is the number of addends). Not until there are 21 addends are both sums--121 and 119--composite. So a(2)=21.
		

Crossrefs

Cf. A000110, A187749.