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A245690 a(n) = A245689(n) mod A053669(n).

%I A245690 #29 May 15 2023 08:43:48
%S A245690 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,2,1,1,
%T A245690 1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,1,2,1,3,1,1,1,1,1,1,1,1,1,2,
%U A245690 1,1,1,1,1
%N A245690 a(n) = A245689(n) mod A053669(n).
%C A245690 The interesting aspect of this sequence is the sparsity of values of a(n) greater than 3. The first occurrence of a(n)=4 occurs at n=30030. Values of a(n)=4 appear to occur only at 30030*k where k is a positive integer that is not divisible by 2,3,17 or 19, but a proof is required.
%C A245690 The first occurrence of a(n) = 5 seems to be at n=23768741896345550770650537601358310. - _Robert Israel_, Jul 31 2014
%C A245690 Conjecture - Let f(x) be the value of n at the first occurrence of a(n) = x. It seems that f(x) for x>2 is always a primorial number (See A002110) and that subsequent values of a(n)=x occur at multiples of n = f(x). If this conjecture is true then:
%C A245690 f(3) = A002110(3) = 2*3*5 = 30.
%C A245690 f(4) = A002110(6) = 2*3*5*7*11*13 = 30030.
%C A245690 f(5) = A002110(24) = 23768741896345550770650537601358310.
%C A245690 f(6) = A002110(347).
%C A245690 f(7) = A002110(51).
%C A245690 f(8) = A002110(3022).
%C A245690 The values of n for f(x), x>5 are extremely large. For example n has 11926 digits for f(8).
%C A245690 Using f(x) notdiv (a,b,c...) as shorthand for multiples of f(x) that are not divisible by a, b, c ... it seems that a(n) = x occurs at:
%C A245690 f(4) notdiv (2, 3, 17, 19),
%C A245690 f(5) notdiv (3, 7, 10, 97, 101),
%C A245690 f(6) notdiv (2, 5, 27, 2347, 2351),
%C A245690 f(7) notdiv (2, 7, 11, 81, 239, 241),
%C A245690 f(8) notdiv (2, 3, 7, 15, 43, 27733, 27737, 27739)
%H A245690 K. Spage, <a href="/A245690/b245690.txt">Table of n, a(n) for n = 3..1000</a>
%e A245690 For n = 10, the smallest prime non-divisor of 10 is 3. The smallest divisor of 10 that is greater than 3 is 5. 5 mod 3 is 2 so a(10) = 2.
%e A245690 For n = 12, the smallest prime non-divisor of 12 is 5. The smallest divisor of 12 that is greater than 5 is 6. 6 mod 5 is 1 so a(12) = 1.
%p A245690 a:= proc(n)
%p A245690 uses numtheory;
%p A245690 local F,p,j;
%p A245690 if n::odd then p:= 2
%p A245690   else
%p A245690   F:= map(pi,factorset(n));
%p A245690   p:= ithprime(min(map(`+`,F,1) minus F));
%p A245690 fi;
%p A245690 for j from p+1 do if n mod j = 0 then return j mod p fi od;
%p A245690 end proc:
%p A245690 seq(a(n),n=3..100); # _Robert Israel_, Jul 31 2014
%t A245690 A053669[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[CoprimeQ[n, p], Return[p]]]];
%t A245690 A245689[n_] := SelectFirst[Divisors[n], # > A053669[n]&];
%t A245690 A245690[n_] := Mod[A245689[n], A053669[n]];
%t A245690 Table[A245690[n], {n, 3, 100}] (* _Jean-François Alcover_, May 15 2023 *)
%o A245690 (PARI) A053669(n) = {forprime(p=2, ,if(n%p, return(p)))}
%o A245690 A245689(n) = {my(c=A053669(n)+1);if(isprime(n),n,while(n%c,c++);c)}
%o A245690 A245690(n) = {A245689(n) % A053669(n)}
%Y A245690 Cf. A053669, A245689.
%K A245690 nonn
%O A245690 3,8
%A A245690 _K. Spage_, Jul 29 2014