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

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

3,8

Comments

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)

Links

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(PARI) 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)}

Crossrefs

Cf. A053669, A245689.

Sequence in context: A266227 A043285 A353756 * A146291 A327538 A346087

Adjacent sequences: A245687 A245688 A245689 * A245691 A245692 A245693

Keyword

nonn

Author

K. Spage, Jul 29 2014

Status

approved