

A136437


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


8



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
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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(nt), 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?
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



FORMULA

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


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;
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.


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(i1)!; end;


PROG

(PARI) a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p  (k1)!; \\ Michel Marcus, Feb 19 2019


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



