A340802
Number of composite numbers between the largest noncomposite number <= n! and the smallest noncomposite number >= n!.
Original entry on oeis.org
0, 0, 1, 5, 13, 7, 11, 53, 29, 21, 13, 29, 89, 19, 89, 75, 89, 77, 189, 59, 61, 103, 185, 203, 189, 95, 43, 167, 253, 107, 187, 79, 37, 289, 173, 257, 97, 43, 169, 135, 131, 175, 179, 155, 291, 189, 311, 155, 141, 157, 449, 119, 129, 349, 131, 609, 383, 391, 429
Offset: 1
a(4) = 5: 24, 25, 26, 27, 28.
a(5) = 13: 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126.
a(6) = 7: 720, 721, 722, 723, 724, 725, 726.
-
prevprime(2):= 1:
a:= n-> (f-> max(nextprime(f-1)-prevprime(f+1)-1, 0))(n!):
seq(a(n), n=1..64);
-
a[n_] := If[n<3, 0, NextPrime[n!] - NextPrime[n!, -1] - 1];
Array[a, 100] (* Jean-François Alcover, Jan 29 2021 *)
A340041
The prime gap, divided by two, which surrounds p#.
Original entry on oeis.org
1, 1, 6, 1, 9, 24, 23, 40, 51, 37, 60, 36, 68, 87, 66, 84, 99, 95, 115, 88, 117, 143, 51, 177, 182, 168, 139, 243, 221, 193, 204, 516, 260, 154, 182, 306, 239, 216, 191, 211, 303, 263, 672, 303, 615, 417, 312, 378, 275, 375, 322, 445, 312, 294, 354, 492, 399, 348, 461
Offset: 2
For a(1), there are two contiguous primes {2, 3} with 2 being 2#. The prime gap is 1. However, the two primes do not surround 2#, so a(1) like A340013(2) is undefined.
For a(2), the prime gap contains {5, 6, 7}, with 3# = 6 in the middle. The prime gap is 2, therefore a(2) = 1;
For a(3), the prime gap contains {29, 30, 31}, with 5# = 30 in the middle. The prime gap is 2, therefore a(3) = 1.
For a(4), the prime gap contains {199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211}, with 7# = 205 in the middle. The prime gap is 12, therefore a(4) = 6. etc.
-
a[n_] := Block[{p = Times @@ Prime@ Range@ n}, (NextPrime[p, 1] - NextPrime[p, -1])/2]; a[1] = 0; Array[a, 60]
Showing 1-2 of 2 results.
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