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If a given prime p is less than a(n) then the numbers of distinct prime factors of p+1 and p-1 have a difference less than n.

From Daniel Suteu, May 11 2022: (Start)

a(13) <= 693386350578511591,

a(14) <= 42296567385289206991,

a(15) <= 3291505006196194517729,

a(16) <= 222099275340153625904489,

a(17) <= 12592092354842984193179971,

a(18) <= 873339227295479848905071071,

a(19) <= 54536351988824964540662450069,

a(20) <= 5513390541916364286137713664909. (End)

From Jon E. Schoenfield, May 11 2022: (Start)

For n > 1, if there exists any prime p < 2*prime(n+2)# such that the absolute difference of the numbers of distinct prime factors of p+1 and p-1 is exactly n, then (since a(n) <= p) it follows that a(n)+1 and a(n)-1, in some order, are either (1) a power of 2, and 2 times an odd number with n distinct prime factors, or (2) an odd prime times a power of 2, and the product of n+2 distinct primes. (In the first case, the numbers of distinct prime factors are 1 and n+1; in the second, they are 2 and n+2.)

E.g., for n = 6, given that p = 18888871 is a prime < 19399380 = 2*(2*3*5*7*11*13*17*19) and the prime factorizations of p+1 and p-1 are 2^3 * 2361109 and 2*3*5*7*11*13*17*37, then a(6)+1 must be too small to have more than 8 distinct prime factors, and also too small to have exactly 8 distinct prime factors with any factor having a multiplicity greater than 1. Thus, a(6) is the smallest prime p such that p+1 and p-1, in some order, are either (1) a power of 2, and 2 times an odd number with 6 distinct prime factors, or (2) a prime times a power of 2, and the product of 8 distinct primes. (As it turns out, a(6) = 16546531, so a(6) + 1 = 2^2 * 4136633 (2 distinct prime factors) and a(6) - 1 = 2*3*5*7*11*13*19*29 (8 distinct prime factors).)

For each n in 2..100, there exists such a prime p < 2*prime(n+2)#, so the numbers of distinct prime factors of a(n)+1 and a(n)-1 are, in some order, 1 and n+1 or 2 and n+2.

For n <= 100, the maximum value of a(n)/prime(n+2)# is a(16)/prime(18)# = 1.893617....

(End)

This sequence is a map from the set of nonnegative integers into the set of all integers. It is clearly not one-to-one. It is not known if it is onto.

Comments from N. J. A. Sloane, Nov 17 2019: (Start)

For m >= 0, A329002 gives an expression for the first time that m appears in this sequence (if it does appear), and A329492 plays a similar role for negative m.

In all these sequences it is safer to say "sum of digits" (which is A007953) rather than "digital sum" (which is also A007953), because many people confuse the latter term with the "digital root" (A010888). (End)