A245690 - OEIS

%I #29 May 15 2023 08:43:48

%S 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 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 1,1,1,1,1

%N a(n) = A245689(n) mod A053669(n).

%C 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 The first occurrence of a(n) = 5 seems to be at n=23768741896345550770650537601358310. - _Robert Israel_, Jul 31 2014

%C 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 f(3) = A002110(3) = 2*3*5 = 30.

%C f(4) = A002110(6) = 2*3*5*7*11*13 = 30030.

%C f(5) = A002110(24) = 23768741896345550770650537601358310.

%C f(6) = A002110(347).

%C f(7) = A002110(51).

%C f(8) = A002110(3022).

%C The values of n for f(x), x>5 are extremely large. For example n has 11926 digits for f(8).

%C 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 f(4) notdiv (2, 3, 17, 19),

%C f(5) notdiv (3, 7, 10, 97, 101),

%C f(6) notdiv (2, 5, 27, 2347, 2351),

%C f(7) notdiv (2, 7, 11, 81, 239, 241),

%C f(8) notdiv (2, 3, 7, 15, 43, 27733, 27737, 27739)

%H K. Spage, <a href="/A245690/b245690.txt">Table of n, a(n) for n = 3..1000</a>

%e 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 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 a:= proc(n)

%p uses numtheory;

%p local F,p,j;

%p if n::odd then p:= 2

%p   else

%p   F:= map(pi,factorset(n));

%p   p:= ithprime(min(map(`+`,F,1) minus F));

%p fi;

%p for j from p+1 do if n mod j = 0 then return j mod p fi od;

%p end proc:

%p seq(a(n),n=3..100); # _Robert Israel_, Jul 31 2014

%t A053669[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[CoprimeQ[n, p], Return[p]]]];

%t A245689[n_] := SelectFirst[Divisors[n], # > A053669[n]&];

%t A245690[n_] := Mod[A245689[n], A053669[n]];

%t Table[A245690[n], {n, 3, 100}] (* _Jean-François Alcover_, May 15 2023 *)

%o (PARI) A053669(n) = {forprime(p=2, ,if(n%p, return(p)))}

%o A245689(n) = {my(c=A053669(n)+1);if(isprime(n),n,while(n%c,c++);c)}

%o A245690(n) = {A245689(n) % A053669(n)}

%Y Cf. A053669, A245689.

%K nonn

%O 3,8

%A _K. Spage_, Jul 29 2014