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Number of positive integers with n digits that are the product of two integers ending with 7.
+10
4
0, 1, 20, 255, 3064, 34743, 380939, 4089499, 43282317, 453472867, 4715695283, 48760330737, 501941505404, 5148657883067, 52659616820819
COMMENTS
a(n) is the number of n-digit numbers in A348054.
FORMULA
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.
PROG
(Python)
def a(n):
lo, hi = 10**(n-1), 10**n
return len(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))
Number of positive integers with n digits that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.
+10
2
0, 0, 8, 129, 1771, 21802, 252793, 2826973, 30872783
COMMENTS
a(n) is the number of n-digit numbers in A348544.
FORMULA
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.
MATHEMATICA
Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Intersection[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}]
PROG
(Python)
def a(n):
lo, hi = 10**(n-1), 10**n
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))
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.
+10
2
1, 4, 49, 524, 5596, 58706, 608886, 6267854, 64180304, 654605898, 6656849267, 67539297095, 683989985496, 6916722312963, 69859080168037
COMMENTS
a(n) is the number of n-digit numbers in A348545.
FORMULA
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.
MATHEMATICA
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}]
PROG
(Python)
def a(n):
lo, hi = 10**(n-1), 10**n
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))
Positive integers that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.
+10
1
189, 399, 459, 609, 729, 819, 969, 999, 1029, 1239, 1269, 1449, 1479, 1539, 1659, 1729, 1809, 1869, 1989, 2079, 2109, 2289, 2349, 2499, 2619, 2639, 2679, 2709, 2889, 2919, 3009, 3059, 3129, 3159, 3219, 3249, 3339, 3429, 3519, 3549, 3699, 3759, 3819, 3969, 4029
FORMULA
Lim_{n->infinity} a(n)/a(n-1) = 1.
EXAMPLE
189 = 7*27 = 3*63, 399 = 3*133 = 7*57, 459 = 3*153 = 17*27, 609 = 3*203 = 7*87, ...
MATHEMATICA
max=4050; Select[Intersection[Union@Flatten@Table[a*b, {a, 3, Floor[max/3], 10}, {b, a, Floor[max/a], 10}], Union@Flatten@Table[a*b, {a, 7, Floor[max/7], 10}, {b, a, Floor[max/a], 10}]], 0<#<max&]
PROG
(PARI) isok(m) = my(ok3=0, ok7=0); fordiv(m, d, if (((d % 10) == 3) && ((m/d % 10) == 3), ok3++); if (((d % 10) == 7) && ((m/d % 10) == 7), ok7++); if (ok3 && ok7, return(1))); \\ Michel Marcus, Oct 22 2021 (Python)
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)))
Positive integers with final digit 9 that are equal to the product of two integers ending with the same digit.
+10
1
9, 39, 49, 69, 99, 119, 129, 159, 169, 189, 219, 249, 259, 279, 289, 299, 309, 329, 339, 369, 399, 429, 459, 469, 489, 519, 529, 539, 549, 559, 579, 609, 629, 639, 669, 679, 689, 699, 729, 749, 759, 789, 799, 819, 849, 879, 889, 909, 939, 949, 959, 969, 989, 999
FORMULA
Lim_{n->infinity} a(n)/a(n-1) = 1.
EXAMPLE
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, ...
MATHEMATICA
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
PROG
(PARI) isok(m) = ((m%10) == 9) && sumdiv(m, d, (d % 10) == (m/d % 10)); \\ Michel Marcus, Oct 22 2021 (Python)
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)))
Positive integers that are the product of two integers ending with 8.
+10
1
64, 144, 224, 304, 324, 384, 464, 504, 544, 624, 684, 704, 784, 864, 944, 1024, 1044, 1064, 1104, 1184, 1224, 1264, 1344, 1404, 1424, 1444, 1504, 1584, 1624, 1664, 1744, 1764, 1824, 1904, 1944, 1984, 2064, 2124, 2144, 2184, 2204, 2224, 2304, 2384, 2464, 2484, 2544
FORMULA
Lim_{n->infinity} a(n)/a(n-1) = 1.
EXAMPLE
64 = 8*8, 144 = 8*18, 224 = 8*28, 304 = 8*38, 324 = 18*18, 384 = 8*48, ...
MATHEMATICA
a={}; For[n=0, n<=260, n++, For[k=0, k<=n, k++, If[Mod[10*n+4, 10*k+8]==0 && Mod[(10*n+4)/(10*k+8), 10]==8 && 10*n+4>Max[a], AppendTo[a, 10*n+4]]]]; a
PROG
(Python)
def aupto(lim): return sorted(set(a*b for a in range(8, lim//8+1, 10) for b in range(a, lim//a+1, 10)))
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