A136437 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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)

LINKS

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

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

EXAMPLE

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;