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Number of positive integers with n digits and final digit 9 that are equal to the product of two integers ending with the same digit.
2

%I #11 Nov 06 2021 06:36:30

%S 1,4,49,524,5596,58706,608886,6267854,64180304,654605898,6656849267,

%T 67539297095,683989985496,6916722312963,69859080168037

%N Number of positive integers with n digits and final digit 9 that are equal to the product of two integers ending with the same digit.

%C a(n) is the number of n-digit numbers in A348545.

%F a(n) < A052268(n).

%F a(n) = A346952(n) + A348055(n) - A348546(n).

%F Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

%t Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Union[Union@Flatten@Table[a*b, {a, 3, Floor[hi/3], 10}, {b, a, Floor[hi/a], 10}], Union@Flatten@Table[a*b, {a, 7, Floor[hi/7], 10}, {b, a, Floor[hi/a], 10}]], lo<#<hi&], {n, 8}]

%o (Python)

%o def a(n):

%o lo, hi = 10**(n-1), 10**n

%o return len(set(a*b for a in range(3, hi//3+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) | set(a*b for a in range(7, hi//7+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))

%o print([a(n) for n in range(1, 9)]) # _Michael S. Branicky_, Oct 22 2021

%Y Cf. A052268, A346950, A346952, A348054, A348055, A348545, A348546.

%K nonn,base,hard,more

%O 1,2

%A _Stefano Spezia_, Oct 22 2021

%E a(9) from _Michael S. Branicky_, Oct 22 2021

%E a(10)-a(15) from _Martin Ehrenstein_, Nov 06 2021