A370678 a(n) is the number of pairs x <= y of n-digit numbers such that the number of distinct digits in their product is less than in their concatenation.

10, 1395, 147718, 15187437, 1530456465, 152653821364

Offset: 1

Author

Hugo Pfoertner, Feb 26 2024

Keywords

• nonn
• base
• more

Examples

			a(1) = 10: 8 products 1*2, ..., 1*9, 2*3, 2*4 with 1 digit in x*y and 2 digits in x|y.
		

Crossrefs

Cf. A370675, A370676, A370679, A370680.

Programs

PARI

\\ returns [number of products, [a(n), A370679(n), A370680(n)]]
 a370678_80(n) = {my (m=0, c=vector(3), n1=10^(n-1), n2=10*n1-1); for (k1=n1, n2, my (s1=digits(k1)); for (k2=k1, n2, my (s2=digits(k2), cs=#Set(digits(k1*k2)), d=cs-#Set(concat(s1,s2))); c[sign(d)+2]++; m++)); [m,c]}

Python

def A370678(n):
    a = 10**(n-1)
    b, c = 10*a, 0
    for x in range(a,b):
        s = set(str(x))
        for y in range(x,b):
            if len(s|set(str(y))) > len(set(str(x*y))):
                c += 1
    return c # Chai Wah Wu, Feb 28 2024

Formula

a(n) = 9 * 2^(n-3) * 5^(n-2) * (10 + 9*10^n) - A370679(n) - A370680(n).

Extensions

a(6) from Martin Ehrenstein, Feb 29 2024