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Positive integers with final digit 9 that are equal to the product of two integers ending with the same digit.
1

%I #8 Oct 23 2021 00:09:24

%S 9,39,49,69,99,119,129,159,169,189,219,249,259,279,289,299,309,329,

%T 339,369,399,429,459,469,489,519,529,539,549,559,579,609,629,639,669,

%U 679,689,699,729,749,759,789,799,819,849,879,889,909,939,949,959,969,989,999

%N Positive integers with final digit 9 that are equal to the product of two integers ending with the same digit.

%C Union of A346950 and A348054.

%F Lim_{n->infinity} a(n)/a(n-1) = 1.

%e 9 = 3*3, 39 = 3*13, 49 = 7*7, 69 = 3*23, 99 = 3*33, 119 = 7*17, 129 = 3*43, 159 = 3*53, 169 = 13*13, 189 = 3*63 = 7*27, ...

%t a={}; For[n=0, n<=100, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+3]==0 && Mod[(10*n+9)/(10*k+3), 10]==3 && 10*n+9>Max[a] || Mod[10*n+9, 10*k+7]==0 && Mod[(10*n+9)/(10*k+7), 10]==7 && 10*n+9>Max[a], AppendTo[a, 10*n+9]]]]; a

%o (PARI) isok(m) = ((m%10) == 9) && sumdiv(m, d, (d % 10) == (m/d % 10)); \\ _Michel Marcus_, Oct 22 2021

%o (Python)

%o def aupto(lim): return sorted(set(a*b for a in range(3, lim//3+1, 10) for b in range(a, lim//a+1, 10)) | set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))

%o print(aupto(999)) # _Michael S. Branicky_, Oct 22 2021

%Y Cf. A017377 (supersequence), A346950, A348054, A348547.

%K nonn,base

%O 1,1

%A _Stefano Spezia_, Oct 22 2021