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%I #26 May 11 2022 13:59:12
%S 0,3,4,11,6,13,6,14,25,8,27,16,8,16,29,28,12,29,18,12,28,19,32,46,21,
%T 8,20,12,22,81,20,36,8,59,12,38,34,18,39,32,18,58,14,21,12,80,70,25,
%U 12,24,34,20,56,43,34,38,16,40,26,8,65,96,24,16,22,99,40,62,12,32,30,61,40,44
%N Sum of the number of divisors of the numbers between prime(n) and prime(n+1).
%H David A. Corneth, <a href="/A133760/b133760.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{j=prime(n)+1..prime(n+1)-1} A000005(j).
%F Sum_{j = 1..m} (a(j)+2) + 1 = A006218(prime(m+1) - 1). - _David A. Corneth_, May 11 2022
%e a(1) = 0 because there is no composite number between the primes 2 and 3.
%e a(4) = 11 = A000005(8)+A000005(9)+A000005(10), indices delimited by prime(4) = 7 and prime(5) = 11.
%p A000005 := proc(n) numtheory[tau](n) ; end: A133760 := proc(n) add( A000005(i),i=ithprime(n)+1..ithprime(n+1)-1) ; end: seq(A133760(n),n=1..80);
%t f[n_] := Plus @@ Flatten@ DivisorSigma[0, { Range[ Prime[n] + 1, Prime[n + 1] - 1]}]; Array[f, 74] (* _Robert G. Wilson v_, Jan 06 2008 *)
%t Total[DivisorSigma[0,Range[First[#]+1,Last[#]-1]]]&/@Partition[ Prime[ Range[ 80]],2,1] (* _Harvey P. Dale_, Jul 19 2013 *)
%o (PARI) a(n) = my(p=prime(n)); sum(k=p+1, nextprime(p+1)-1, numdiv(k)); \\ _Michel Marcus_, May 11 2022
%Y Cf. A000005, A006218.
%K easy,nonn
%O 1,2
%A _Giovanni Teofilatto_, Jan 05 2008
%E Edited, corrected and extended by _Robert G. Wilson v_ and _R. J. Mathar_, Jan 06 2008
%E Name corrected by _Jon Perry_, Nov 23 2012
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