A356549 a(n) is the number of divisors of 10^n whose first digit is 1.

1, 2, 3, 5, 8, 11, 15, 20, 25, 31, 38, 45, 52, 60, 69, 78, 88, 99, 110, 122, 135, 148, 161, 175, 190, 205, 221, 238, 255, 273, 292, 311, 330, 350, 371, 392, 414, 437, 460, 484, 509, 534, 559, 585, 612, 639, 667, 696, 725, 755, 786, 817, 848, 880, 913, 946, 980, 1015, 1050, 1086

Offset: 0

Author

Michel Marcus, Sep 23 2022

Keywords

• nonn
• base

Examples

			The divisors of 1000 with initial digit 1 are: 1, 10, 100, 125 and 1000, so a(3)=5.
		

Links

• Michel Marcus, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A011557, A357299.

Programs

Maple

a:= n-> add(`if`((""||d)[1]="1", 1, 0), d=numtheory[divisors](10^n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 23 2022

Mathematica

a[n_] := DivisorSum[10^n, 1 &, IntegerDigits[#][[1]] == 1 &]; Array[a, 60, 0] (* Amiram Eldar, Sep 23 2022 *)

PARI

a(n) = sumdiv(10^n, d, digits(d)[1] == 1);

Python

from sympy import divisors
def a(n): return sum(1 for d in divisors(10**n, generator=True) if str(d)[0]=="1")
print([a(n) for n in range(60)]) # Michael S. Branicky, Sep 23 2022

Python

def A356549(n): return n+1+sum(n-m+1 for m in range(1,n+2) for d in (2,5) if str(d**m).startswith('1')) # Chai Wah Wu, Sep 23 2022

Formula

a(n) = A357299(A011557(n)).