A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

0, 1, 3, 1, 5, 7, 11, 13, 17, 5, 7, 13, 17, 19, 23, 29, 35, 37, 43, 47, 49, 55, 59, 65, 73, 77, 79, 83, 85, 89, 7, 11, 17, 19, 29, 31, 37, 43, 47, 53, 59, 61, 71, 73, 77, 79, 91, 103, 107, 109, 113, 119, 121, 131, 137, 143, 149, 151, 157, 161, 163, 173, 187, 191, 193, 197, 211, 217, 227, 229, 233, 239, 247

Offset: 1

Author

Ctibor O. Zizka, Apr 02 2008

Keywords

• easy
• nonn

Comments

How many times does each prime appear in this sequence?

The only value (prime(n) - k!) = 0 is at n=1, where k=2.

Are n=2, k=2 and n=4, k=3 the only occurrences of (prime(n) - k!) = 1?

There exist infinitely many solutions of the form (prime(n) - k!) = prime(n-t), t < n.

Are there infinitely many solutions of the form (prime(n) - k!) = prime(r_1)*...*prime(r_i); r_i < n for all i?

From Bernard Schott, Jul 16 2021: (Start)

Answer to the second question is no: 18 other occurrences (n,k) of (prime(n) - k!) = 1 are known today; indeed, every k > 1 in A002981 that satisfies k! + 1 is prime gives an occurrence, but only a third pair (n, k) is known exactly; and this comes for n = 2428957, k = 11 because (prime(2428957) - 11!) = 1.

The next occurrence corresponds to k = 27 and n = X where prime(X) = 1+27! = 10888869450418352160768000001 but index X is not yet available (see A062701).

For the occurrences of (prime(m) - k!) = 1, integers k are in A002981 \ {0, 1}, corresponding indices m are in A062701 \ {1} (only 3 indices are known today) and prime(m) are in A088332 \ {2}. (End)

Examples

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

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A135996, A000040, A000142, A212598, A212266.

Cf. also A002981, A062701, A088332, A346425 (gives k).

Programs

Maple

f:=proc(n) local p,i; p:=ithprime(n); for i from 0 to p do if i! > p then break; fi; od; p-(i-1)!; end;
[seq(f(n),n=1..70)]; # N. J. A. Sloane, May 22 2012

Mathematica

a[n_] := Module[{p, k},p = Prime[n];k = 1;While[Factorial[k] <= p, k++];p - Factorial[k - 1]] (* James C. McMahon, May 05 2025 *)

PARI

a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p - (k-1)!; \\ Michel Marcus, Feb 19 2019

Formula

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

a(n) = A212598(prime(n)). - Michel Marcus, Feb 19 2019

a(n) = A000040(n) - A346425(n). - Bernard Schott, Jul 16 2021

Extensions

More terms from Jinyuan Wang, Feb 18 2019