A337054
Numbers that have at least 3 different representations as the product of a number and of its decimal digits.
Original entry on oeis.org
0, 549504, 1798848, 4193856, 4804128, 5827584, 7426944, 14397696, 34324992, 39401250, 39611040, 42856128, 45312750, 62593440, 81575424, 86171040, 92348928, 140184576, 151600896, 196475328, 221695488, 251584704, 263680704, 271165104, 287945280, 475388928, 499654656
Offset: 1
a(43) = 1578092544 = 342468*(3*4*2*4*6*8) = 913248*(9*1*3*2*4*8) = 97848*(9*7*8*4*8) = 86976*(8*6*9*7*6).
A337100
Numbers that have at least 4 different representations as the product of a number and of its decimal digits.
Original entry on oeis.org
0, 549504, 1578092544, 12276847296, 28961412480, 35998381440, 87012926784, 118082893824, 259456659840, 335449175040, 397315715328, 579305502720, 672777778176, 712539265536, 741360356352, 863562591360, 1138944651264, 1264664088576, 1276070713344, 1300488037632
Offset: 1
a(3) = 12276847296 = 676634*(6*7*6*6*3*4) = 773296*(7*7*3*2*9*6) = 2368219*(2*3*6*8*2*1*9) = 12179412*(1*2*1*7*9*4*1*2).
A337732
Least positive number that has exactly n different representations as the sum of a number and the product of its decimal digits.
Original entry on oeis.org
1, 0, 10, 50, 150, 1014, 9450, 8305, 283055, 931395, 92441055, 84305555, 28322235955
Offset: 0
10 = 5 + 5 = 10 + 1*0 and as 10 is the smallest number with 2 such representations, so, a(2) = 10.
50 = 35 + 3*5 = 42 * 4*2 = 50 + 5*0 and as 50 is the smallest number with 3 such representations, so, a(3) = 50.
Cf.
A337051 (similar for Bogotá numbers).
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f[n_] := n + Times @@ IntegerDigits[n]; m = 10^6; v = Table[0, {m}]; Do[i = f[n] + 1; If[i <= m, v[[i]]++], {n, 0, m}]; s = {1}; k = 1; While[(p = Position[v, k]) != {}, AppendTo[s, p[[1, 1]] - 1]; k++]; s (* Amiram Eldar, Sep 18 2020 *)
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f(n) = if (n==0, return(1)); sum(k=1, n, k+vecprod(digits(k)) == n); \\ A230103
a(n) = my(k=0); while(f(k) !=n, k++); k; \\ Michel Marcus, Sep 18 2020
A337817
Smallest nonnegative number that has exactly n different representations as the product of a number and the sum of its decimal digits.
Original entry on oeis.org
2, 0, 36, 900, 138600, 25336080, 3732276240, 240277237200
Offset: 0
2 is the smallest number that is not possible to write as (m * sum of digits of m) for some m, hence a(0) = 2.
0 = 0 * 0, hence a(1) = 0
36 = 6 * 6 = 12 * (1+2) and 36 is the smallest number with 2 such representations, hence a(2) = 36.
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f[n_] := n*Plus @@ IntegerDigits[n]; m = 2*10^5; v = Table[0, {m}]; Do[i = f[n] + 1; If[i <= m, v[[i]]++], {n, 0, m}]; s = {}; k = 0; While[(p = Position[v, k]) != {}, AppendTo[s, p[[1, 1]] - 1]; k++]; s (* Amiram Eldar, Sep 23 2020 *)
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a(n)={if(n==1, 0, for(k=1, oo, if(sumdiv(k, d, d*sumdigits(d)==k) == n, return(k))))} \\ Andrew Howroyd, Sep 23 2020
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