A332740 Prime numbers p such that the set of composite numbers in the range [p+1, nextprime(p)-1] has more than one element and all the elements have the same number of divisors.

229, 8293, 9829, 14887, 16087, 20389, 21493, 44983, 50581, 53887, 57943, 63463, 64663, 72223, 81547, 93253, 108343, 134917, 138727, 143239, 157207, 192613, 199669, 203653, 206407, 210853, 218839, 244837, 248749, 251287, 255049, 262693, 280183, 296437, 300319

Offset

1,1

Comments

The corresponding numbers of divisors are 8, 16, 8, 8, 8, 8, 8, 8, 8, 16, 8, 8, 16, 24, 24, ... and the number of divisors in the order of their first appearance are 8, 16, 24, 20, 12, 32, 48, ...

The number of composites between a(n) and its next prime are 3, 3, 3, 3, 3, 3, 5, 3, 5, 3, ... Are there terms with number of composites larger than 5?

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

229 is a term since between 229 and its next prime, 233, there are 3 composite numbers, 230, 231 and 232 and all of them have the same number of divisors, 8.

Mathematica

seqQ[n_] := PrimeQ[n] && (nx=NextPrime[n]) > n + 2 && Length @ Union @ DivisorSigma[0, Range[n+1, nx-1]] == 1; Select[Range[10^6], seqQ]

Crossrefs

Cf. A000005, A075580, A075583, A075584, A075585, A075586, A075587, A075588, A075589.

Sequence in context: A210514 A142665 A124684 * A178673 A028452 A072020

Adjacent sequences: A332737 A332738 A332739 * A332741 A332742 A332743

Keyword

nonn

Author

Amiram Eldar, Feb 21 2020

Status

approved