A245689 -id:A245689 - cp's OEIS frontend

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, 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, 1, 1, 1, 1, 1

Offset: 3

K. Spage, Jul 29 2014

nonn

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.
The first occurrence of a(n) = 5 seems to be at n=23768741896345550770650537601358310. - Robert Israel, Jul 31 2014
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:
f(3) = A002110(3) = 2*3*5 = 30.
f(4) = A002110(6) = 2*3*5*7*11*13 = 30030.
f(5) = A002110(24) = 23768741896345550770650537601358310.
f(6) = A002110(347).
f(7) = A002110(51).
f(8) = A002110(3022).
The values of n for f(x), x>5 are extremely large. For example n has 11926 digits for f(8).
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:
f(4) notdiv (2, 3, 17, 19),
f(5) notdiv (3, 7, 10, 97, 101),
f(6) notdiv (2, 5, 27, 2347, 2351),
f(7) notdiv (2, 7, 11, 81, 239, 241),
f(8) notdiv (2, 3, 7, 15, 43, 27733, 27737, 27739)

			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.
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.

K. Spage, Table of n, a(n) for n = 3..1000

Cf. A053669, A245689.

a:= proc(n)
uses numtheory;
local F,p,j;
if n::odd then p:= 2
  else
  F:= map(pi,factorset(n));
  p:= ithprime(min(map(`+`,F,1) minus F));
fi;
for j from p+1 do if n mod j = 0 then return j mod p fi od;
end proc:
seq(a(n),n=3..100); # Robert Israel, Jul 31 2014

A053669[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[CoprimeQ[n, p], Return[p]]]];
A245689[n_] := SelectFirst[Divisors[n], # > A053669[n]&];
A245690[n_] := Mod[A245689[n], A053669[n]];
Table[A245690[n], {n, 3, 100}] (* Jean-François Alcover, May 15 2023 *)

A053669(n) = {forprime(p=2, ,if(n%p, return(p)))}
A245689(n) = {my(c=A053669(n)+1);if(isprime(n),n,while(n%c,c++);c)}
A245690(n) = {A245689(n) % A053669(n)}

cp's OEIS Frontend

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

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