A260187 a(n) = n modulo the greatest primorial <= n.

0, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0

Offset: 1

Author

Jean-Marc Rebert, Jul 18 2015

Keywords

• nonn

Comments

I think this can help in finding prime numbers.

If n>2 and a(n)=0 then n is not prime.

If n>2 and 2 <= a(n) <= p with p = A006530(A260188(n)) (i.e., greatest prime factor of A260188(n)) then n is not prime.

If n>2 and (a(n) mod k) = 0 and 2 <= k <= p with p = A006530(A260188(n)) (i.e., greatest prime factor of A260188(n)) then n is not prime.

Alternative definition: count up from 0 to primorial(n)-1, prime(n+1)-1 times, where primorial(n) is A002110(n). - Franklin T. Adams-Watters, Jul 20 2015

Examples

			a(5) = 1 because 5 modulo 2# = 1 and 2# = 2 is the greatest primorial <= 5. (3# = 2*3 = 6 > 5)
		

Links

• Jean-Marc Rebert, Table of n, a(n) for n = 1..40000 (first 10000 terms from Charles R Greathouse IV)

Crossrefs

Cf. A034386 and A002110 (primorials), A260188, A257687.

Programs

Maple

N:= 100: # to get a(1) to a(N)
P:= 1: p:= 2: R:= 2:
for n from 1 to N do
  if n >= R then
    P:= R; p:= nextprime(p); R:= P*p;
  fi;
  A[n]:= n mod P;
od:
seq(A[i],i=1..N); # Robert Israel, Jul 20 2015

Mathematica

s = Product[Prime@ n, {n, #}] & /@ Range[0, 6]; Table[Mod[n, Last@ Select[s, # <= n &]], {n, 120}] (* Michael De Vlieger, Jul 20 2015 *)
f[n_] := Block[{m = p = 1}, While[p*(m + 1) <= n, p = p*m; m = NextPrime@ m]; Mod[n, p]]; Array[f, 101] (* Robert G. Wilson v, Jul 21 2015 *)

PARI

a(n)=my(t=1, k); forprime(p=2, , k=t*p; if(k>n, return(n%t), t=k)) \\ Charles R Greathouse IV, Jul 20 2015

Formula

a(n) = n mod A260188(n).

a(n) <= (n+1)/2. - Charles R Greathouse IV, Jul 20 2015