A085303 Positions of 2 in A085301.

1, 2, 7, 10, 14, 17, 20, 23, 27, 30, 33, 37, 40, 43, 47, 50, 53, 57, 60, 63, 67, 70, 73, 77, 80, 84, 87, 90, 94, 97, 101, 104, 108, 111, 114, 118, 121, 125, 128, 132, 135, 139, 142, 146, 149, 153, 156, 160, 164, 167, 171, 174, 178, 181, 185, 188, 192, 196, 199, 203

Offset: 1

Author

Labos Elemer, Jun 26 2003

Keywords

• nonn

Comments

Numbers k such that A085301(k) = 2, i.e., between primorial(k-1) and primorial(k) there are two distinct factorial numbers.

Examples

			10 is a term since between the 9th and the 10th primorials there are two factorials: 12! and 13!.
14 is a term since between the 13th and the 14th primorials there are two factorials: 17! and 18!.
584 is a term since between the 583rd and the 584th primorials there are two factorials: 745! and 746!.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A085301, A002110, A000142.

Programs

Mathematica

f[n_] := f[n] = Module[{k = 1, r = Times @@ Prime[Range[n]]}, While[r >= 1, k++; r /= k]; k - 1]; q[n_] := f[n] - f[n - 1] == 2; q[1] = q[2] = True; Select[Range[210], q] (* Amiram Eldar, Feb 18 2025 *)

PARI

f(n) = {my(k = 1); while(n >= 1, k++; n /= k); k-1;}
list(lim) = {my(c = 1, f1 = 1, r = 1, k = 0); print1("1, 2, "); forprime(p = 2, lim, k++; r* = p; f2 = f(r); if(f2 == f1 + 2, print1(k, ", ")); f1 = f2);} \\ Amiram Eldar, Feb 18 2025

Formula

Solutions x to A085301(x) = 2.