A376129 Run lengths of the most significant decimal digit in the primes (A077648).

1, 1, 1, 1, 4, 2, 2, 3, 2, 2, 3, 2, 1, 21, 16, 16, 17, 14, 16, 14, 15, 14, 135, 127, 120, 119, 114, 117, 107, 110, 112, 1033, 983, 958, 930, 924, 878, 902, 876, 879, 8392, 8013, 7863, 7678, 7560, 7445, 7408, 7323, 7224, 70435, 67883, 66330, 65367, 64336, 63799, 63129, 62712, 62090

Offset: 1

Author

Stuart Coe, Sep 11 2024

Keywords

• nonn
• base

Examples

			The primes and the run lengths of their initial digits begin
  primes   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...
  runs                 \------------/  \----/  \----/
  lengths  1  1  1  1         4          2       2     ...
		

Crossrefs

Cf. A000720, A037124, A077648.

Programs

Python

from sympy import primepi
def A376129(n):
    if n<5: return 1
    def f(m): return (lambda x:primepi(10**x[0]*(x[1]+1)))(divmod(m,9))
    return int(f(n+5)-f(n+4)) # Chai Wah Wu, Oct 16 2024

Formula

a(n) = A000720(A037124(n+6)) - A000720(A037124(n+5)) for n >= 5. - Pontus von Brömssen, Oct 07 2024