A247884 Number of positive integers < 10^n divisible by their first digit.

9, 41, 327, 3158, 31450, 314349, 3143320, 31433005, 314329833, 3143298089, 31432980631, 314329806030, 3143298060001, 31432980599686, 314329805996514, 3143298059964770, 31432980599647312, 314329805996472711, 3143298059964726682, 31432980599647266367

Offset: 1

Author

Derek Orr, Sep 25 2014

Keywords

• nonn
• easy
• base

Comments

a(n)/10^n seems to converge to a number around .3143...

a(n)/10^n converges to 7129/22680. - Hiroaki Yamanouchi, Sep 26 2014

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100
• Beyond Solutions Blog, Multiple of the first digit
• J. J. O'Connor and E. F. Robertson, Pietro Mengoli

Crossrefs

Cf. A034837, A028569, A119947, A119948.

Programs

PARI

a(n)=c=0;for(k=1,10^n-1,d=digits(k);if(k%d[1]==0,c++));c
n=1;while(n<10,print1(a(n),", ");n++)

Python

count = 9 # Start with the first 9 digits
print(1, 9)
n = 2
while n < 101:
    for a in range(1, 10):
        count += 10**(n-1)//a
        if 10**(n-1) % a != 0:
            count += 1
    print(n, count)
    n += 1
# David Consiglio, Jr., Sep 26 2014

Formula

G.f.: x*(9 - 67*x + 24*x^2 + 14*x^3 - 56*x^4 + 21*x^5 + 7*x^6 + 5*x^7)/((1 - x)^2*(1 + x)*(1 - 10*x)*(1 - x + x^2)). - Robert Israel, Mar 10 2025

Extensions

a(9)-a(20) from Hiroaki Yamanouchi, Sep 26 2014