A090970 Number of primes strictly between T(n) and T(n+1), where T(n) = n-th triangular number.

1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 4, 5, 3, 6, 6, 7, 5, 5, 6, 4, 8, 5, 6, 6, 8, 6, 8, 5, 7, 5, 11, 4, 6, 9, 7, 8, 9, 8, 7, 7, 9, 7, 8, 7, 12, 5, 9, 9, 11, 9, 7, 7, 12, 10, 10, 9, 9, 9, 6, 11, 10, 11, 9, 12, 11, 12, 9, 10, 11, 12, 10, 13, 9, 11, 10, 12

Offset

1,4

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Example

a(8)=3 because between T(8)=36 and T(9)=45 we have the prime numbers 37,41 and 43.

Maple

a:= proc(n) local ct, j: ct:=0: for j from n*(n+1)/2+1 to (n+1)*(n+2)/2-1 do if isprime(j)=true then ct:=ct+1 else ct:=ct fi: od: end: seq(a(n), n=1..103); # Emeric Deutsch, Feb 23 2005

Mathematica

With[{trs=Partition[Accumulate[Range[100]], 2, 1]}, Join[{1}, Rest[ PrimePi[ #[[2]]]- PrimePi[#[[1]]]&/@trs]]] (* Harvey P. Dale, Aug 25 2015 *)

Prog

(Python)
from sympy import primerange
def A090970(n): return sum(1 for p in primerange((n*(n+1)>>1)+1, (n+2)*(n+1)>>1)) # Chai Wah Wu, May 22 2025

Crossrefs

Essentially the same as A065382 and A066888.

Sequence in context: A238325 A238885 A023566 * A091972 A025833 A200647

Adjacent sequences: A090967 A090968 A090969 * A090971 A090972 A090973

Keyword

nonn

Author

Amarnath Murthy, Jan 03 2004

Extensions

More terms from Emeric Deutsch, Feb 23 2005

Status

approved