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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.
5
10, 1395, 147718, 15187437, 1530456465, 152653821364
OFFSET
1,1
FORMULA
a(n) = 9 * 2^(n-3) * 5^(n-2) * (10 + 9*10^n) - A370679(n) - A370680(n).
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Hugo Pfoertner, Feb 26 2024
EXTENSIONS
a(6) from Martin Ehrenstein, Feb 29 2024
STATUS
approved