A191648 - OEIS

%I #16 Dec 25 2022 23:13:28

%S 1,1,2,3,23,3

%N a(1)=1, a(2)=1. For n > 2, start with n and iterate the map (k -> concatenation of anti-divisors of k) until we reach a prime q; then a(n) = q. If we never reach a prime, a(n) = 0.

%C Similar to A120716, which uses the proper divisors of n. Other known values include a(10) = 347, a(14) = 349, and a(16) = 311. See also A191859.

%e The anti-divisors of 5 are 2, 3, and 23 is prime, hence a(5) = 23.

%e The anti-divisors of 7 are 2, 3, 5, and 235 is composite; the anti-divisors of 235 are 2, 3, 7, 10, 67, 94, 157, and 237106794157 = 59*547*7346909 is composite; the anti-divisors of 237106794157 start 2, 3, 5, 15, 118, 1094, 1709, 4519, 61403, 64546, 7722971, 14693818, 104937727, but the others are unknown, hence a(7) is also unknown.

%e Note that the example speaks of the anti-divisors of 237106794157 being incomplete.  As of this date, those are well-established (see code of A066272).  It is the primality evaluation chain from anti-divisors for n=7 from 23515118109417094519614036454677229711469381810493772727748015786693526280375184463161423922194842717663158071196105 that is incomplete. - _Bill McEachen_, Dec 14 2022

%p antidivisors := proc(n) local a, k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:

%p A130846 := proc(n) digcatL(sort(convert(antidivisors(n),list))) ; end proc:

%p A191648 := proc(n) if n <=2 then 1; else m := A130846(n) ; while not isprime(m) do m := A130846(m) ; end do: return m; end if; end proc: # _R. J. Mathar_, Jun 30 2011

%Y Cf. A130846, A066272, A120716, A191647.

%K nonn,base,more,hard

%O 1,3

%A _Paolo P. Lava_, Jun 10 2011