A133760 Sum of the number of divisors of the numbers between prime(n) and prime(n+1).

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, 8, 20, 12, 22, 81, 20, 36, 8, 59, 12, 38, 34, 18, 39, 32, 18, 58, 14, 21, 12, 80, 70, 25, 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

Offset

1,2

Links

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

Formula

a(n) = Sum_{j=prime(n)+1..prime(n+1)-1} A000005(j).

Sum_{j = 1..m} (a(j)+2) + 1 = A006218(prime(m+1) - 1). - David A. Corneth, May 11 2022

Example

a(1) = 0 because there is no composite number between the primes 2 and 3.
a(4) = 11 = A000005(8)+A000005(9)+A000005(10), indices delimited by prime(4) = 7 and prime(5) = 11.

Maple

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);

Mathematica

f[n_] := Plus @@ Flatten@ DivisorSigma[0, { Range[ Prime[n] + 1, Prime[n + 1] - 1]}]; Array[f, 74] (* Robert G. Wilson v, Jan 06 2008 *)
Total[DivisorSigma[0, Range[First[#]+1, Last[#]-1]]]&/@Partition[ Prime[ Range[ 80]], 2, 1] (* Harvey P. Dale, Jul 19 2013 *)

Prog

(PARI) a(n) = my(p=prime(n)); sum(k=p+1, nextprime(p+1)-1, numdiv(k)); \\ Michel Marcus, May 11 2022

Crossrefs

Cf. A000005, A006218.

Sequence in context: A290517 A295824 A180696 * A335887 A335888 A328851

Adjacent sequences: A133757 A133758 A133759 * A133761 A133762 A133763

Keyword

easy,nonn

Author

Giovanni Teofilatto, Jan 05 2008

Extensions

Edited, corrected and extended by Robert G. Wilson v and R. J. Mathar, Jan 06 2008

Name corrected by Jon Perry, Nov 23 2012

Status

approved