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A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n). 8

%I #42 May 31 2022 12:51:50

%S 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,

%T 77,79,83,85,89,7,11,17,19,29,31,37,43,47,53,59,61,71,73,77,79,91,103,

%U 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

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

%C How many times does each prime appear in this sequence?

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

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

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

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

%C From _Bernard Schott_, Jul 16 2021: (Start)

%C 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.

%C 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).

%C 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)

%H Jinyuan Wang, <a href="/A136437/b136437.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = A212598(prime(n)). - _Michel Marcus_, Feb 19 2019

%F a(n) = A000040(n) - A346425(n). - _Bernard Schott_, Jul 16 2021

%e a(1) = prime(1) - 2! = 2 - 2 = 0;

%e a(2) = prime(2) - 2! = 3 - 2 = 1;

%e a(3) = prime(3) - 2! = 5 - 2 = 3;

%e a(4) = prime(4) - 3! = 7 - 6 = 1;

%e a(5) = prime(5) - 3! = 11 - 6 = 5;

%e a(6) = prime(6) - 3! = 13 - 6 = 7;

%e a(7) = prime(7) - 3! = 17 - 6 = 11;

%e a(8) = prime(8) - 3! = 19 - 6 = 13;

%e a(9) = prime(9) - 3! = 23 - 6 = 17;

%e a(10) = prime(10) - 4! = 29 - 24 = 5.

%p 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;

%p [seq(f(n),n=1..70)]; # _N. J. A. Sloane_, May 22 2012

%o (PARI) a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p - (k-1)!; \\ _Michel Marcus_, Feb 19 2019

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

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

%K easy,nonn

%O 1,3

%A _Ctibor O. Zizka_, Apr 02 2008

%E More terms from _Jinyuan Wang_, Feb 18 2019

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Last modified March 29 08:53 EDT 2024. Contains 371268 sequences. (Running on oeis4.)