A093451 Number of distinct prime divisors of Product_{k=1+(n-1)n/2..n(n+1)/2} k (i.e., of 1, 2*3, 4*5*6, 7*8*9*10, ...).

0, 2, 3, 4, 6, 6, 7, 8, 10, 10, 11, 13, 13, 14, 16, 15, 18, 17, 20, 19, 22, 21, 22, 24, 24, 26, 26, 27, 30, 28, 30, 31, 32, 33, 33, 36, 35, 36, 38, 39, 39, 39, 43, 41, 43, 44, 44, 47, 45, 49, 48, 48, 52, 49, 53, 53, 54, 54, 55, 58, 55, 60, 59, 59, 62, 60, 65, 64, 64, 65, 66, 68

Offset: 1

Author

Amarnath Murthy, Apr 03 2004

Keywords

• nonn

Examples

			a(7) = 7 as the prime divisors of the product 22*23*24*25*26*27*28 are 2,3,5,7,11,13 and 23.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A057003.

Programs

Maple

with(numtheory): a:=n->nops(factorset(product(k,k=1+n*(n-1)/2..n*(n+1)/2))): seq(a(n),n=1..80); # Emeric Deutsch, Feb 05 2006

Mathematica

With[{nn=75},PrimeNu[#]&/@Times@@@TakeList[Range[(nn(nn+1))/2],Range[ nn]]] (* Harvey P. Dale, Sep 01 2021 *)

PARI

a(n) = { my(b=binomial(n,2)+1, bp1=binomial(n+1,2), res = primepi(n)); forprime(p = n + 1, bp1, bp = b%p; if(bp > bp1 % p || bp == 0, res++ ) ); res } \\ David A. Corneth, Sep 01 2021

Formula

a(n) = A001221(A057003(n)). - Michel Marcus, Jul 29 2017

Extensions

Corrected and extended by Emeric Deutsch, Feb 05 2006