0,2

If it exists, a(5) > 2^35. - Antti Karttunen, Feb 19 2025

Conjecture: Sequence is finite and a(4) is the last term. This is equivalent to the claim that no arithmetic derivative (A003415) of a product of five or more distinct primes, (i.e., the value of the (n-1)-st elementary symmetric polynomial formed from those n distinct primes) can be formed as a carry-free sum of those n summands in primorial base (A049345). See also A380476 and A380528, A380530.

Wikipedia, Elementary symmetric polynomial

Index entries for sequences related to primorial base.

a(n) = Min_{k in A380468} for which A001221(k) = n.

186 = 2*3*31 and A276086(186/2) = 2058 = 2 * 3 * 7^3, A276086(186/3) = 3 * 7^2, A276086(186/31) = 5, whose product = 2^1 * 3^2 * 5^1 * 7^5 = 1512630 = A380459(186), and as all the exponents are less than the corresponding primes, the product is in A048103, and because there are no any smaller number with three prime factors satisfying the same condition (of A380468), 186 is the term a(3) of this sequence. Note that A049345(A003415(186)) = 5121, where the digits are the exponents in the product read from the largest to the smallest prime factor.

See also the example in A380476 about 4686.

Cf. A001221, A001222, A003415, A048103, A049345, A276086, A380459, A380468, A380476 (terms of A380468 with more than three prime factors).

Cf. A047247, A047257.

Cf. also A317836, A358235, A380525, A380528, A380530.

Sequence in context: A298883 A252740 A055696 * A158096 A345726 A302344

Adjacent sequences: A380472 A380473 A380474 * A380476 A380477 A380478

nonn,hard,more

Antti Karttunen, Feb 03 2025

approved