A151918 a(n) = k! - prime(n) where k is the smallest number for which prime(n) <= k!.

0, 3, 1, 17, 13, 11, 7, 5, 1, 91, 89, 83, 79, 77, 73, 67, 61, 59, 53, 49, 47, 41, 37, 31, 23, 19, 17, 13, 11, 7, 593, 589, 583, 581, 571, 569, 563, 557, 553, 547, 541, 539, 529, 527, 523, 521, 509, 497, 493, 491, 487, 481, 479, 469, 463, 457, 451, 449, 443, 439, 437

Offset: 0

Author

Ctibor O. Zizka, Apr 06 2008

Keywords

• easy
• nonn

Comments

How many times does each prime number appear in this sequence?

Are there infinitely many solutions of the form

(k!-p(n)) = p(r_1)*...*p(r_i); r_i < n for all i?

Examples

			a(1)  = 2! - p(1)  =   2 -  2 =  0;
a(2)  = 3! - p(2)  =   6 -  3 =  3;
a(3)  = 3! - p(3)  =   6 -  5 =  1;
a(4)  = 4! - p(4)  =  24 -  7 = 17;
a(5)  = 4! - p(5)  =  24 - 11 = 13;
a(6)  = 4! - p(6)  =  24 - 13 = 11;
a(7)  = 4! - p(7)  =  24 - 17 =  7;
a(8)  = 4! - p(8)  =  24 - 19 =  5;
a(9)  = 4! - p(9)  =  24 - 23 =  1;
a(10) = 5! - p(10) = 120 - 29 = 91;
etc.
		

Crossrefs

Cf. A000040, A000142.

Programs

Maple

A048765 := proc(n) for i from 1 do if i! >= n then return i! ; end if; end do: end proc:
A151918 := proc(n) p := ithprime(n) ; A048765(p)-p ; end proc:
seq(A151918(n),n=1..80) ; # R. J. Mathar, Aug 25 2010

Mathematica

Module[{fs=Range[10]!,p},Join[{0},Flatten[Table[p=Prime[n];Select[ fs,#>p&,1]-p,{n,2,70}]]]] (* Harvey P. Dale, Oct 04 2013 *)

Formula

a(n) = A048765(prime(n)) - prime(n). - R. J. Mathar, Aug 25 2010

Extensions

More terms from R. J. Mathar, Aug 25 2010