A093451 - OEIS

%I #18 Jan 09 2024 16:19:10

%S 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,

%T 26,26,27,30,28,30,31,32,33,33,36,35,36,38,39,39,39,43,41,43,44,44,47,

%U 45,49,48,48,52,49,53,53,54,54,55,58,55,60,59,59,62,60,65,64,64,65,66,68

%N 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, ...).

%H David A. Corneth, <a href="/A093451/b093451.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001221(A057003(n)). - _Michel Marcus_, Jul 29 2017

%e 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.

%p 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

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

%o (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

%Y Cf. A001221, A057003.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Apr 03 2004

%E Corrected and extended by _Emeric Deutsch_, Feb 05 2006