A061117 Maximum number of divisors for any composite between prime(n) and prime(n+1).

3, 4, 4, 6, 5, 6, 6, 8, 8, 9, 8, 8, 6, 10, 8, 12, 8, 8, 12, 8, 10, 12, 12, 9, 8, 8, 12, 10, 16, 8, 12, 8, 15, 12, 12, 12, 8, 16, 10, 18, 8, 14, 9, 12, 16, 16, 12, 12, 8, 12, 20, 8, 18, 12, 16, 16, 12, 16, 8, 18, 18, 12, 16, 12, 16, 20, 12, 12, 12, 8, 24, 12, 16, 12, 16, 18, 15, 16, 12

Offset: 2

Author

Labos Elemer, May 29 2001

Keywords

• nonn

Examples

			p(30)=113 is followed by 13 composites; numbers of divisors are {8, 4, 6, 6, 4, 4, 16, 3, 4, 4, 6, 4, 12}; the smallest is 4=a(30) and the largest is 16.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..10000 (terms up to n = 1000 by Harry J. Smith)

Crossrefs

Cf. A000005, A061112.

Programs

Mathematica

Max /@ DivisorSigma[0, Select[SplitBy[Range@ Prime@ 81, PrimeQ], CompositeQ@ First@ # &]] (* Michael De Vlieger, Nov 02 2017 *)
Table[Max[DivisorSigma[0,Range[p+1,NextPrime[p]-1]]],{p,Prime[Range[2,80]]}] (* Harvey P. Dale, Feb 02 2025 *)

PARI

{ n=-1; q=3; forprime (p=5, prime(1003), a=0; for (i=q + 1, p - 1, a=max(numdiv(i), a)); q=p; write("b061117.txt", n++, " ", a) ) } \\ Harry J. Smith, Jul 18 2009

Formula

a(n) = Max{d(c); p(n+1) > c > p(n)}, c is composite, p(n) is the n-th prime and d=A000005().