cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A348055 Number of positive integers with n digits that are the product of two integers ending with 7.

Original entry on oeis.org

0, 1, 20, 255, 3064, 34743, 380939, 4089499, 43282317, 453472867, 4715695283, 48760330737, 501941505404, 5148657883067, 52659616820819
Offset: 1

Views

Author

Stefano Spezia, Sep 26 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A348054.

Crossrefs

Cf. A017377, A052268, A348054.
Cf. A346509 (ending with 1), A346629 (ending with 2), A346952 (ending with 3), A347255 (ending with 4), A337855 (ending with 5), A337856 (ending with 6), A348549 (ending with 8).

Programs

  • 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))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Sep 26 2021

Formula

a(n) < A052268(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(11) from Michael S. Branicky, Sep 26 2021
a(12)-a(15) from Martin Ehrenstein, Oct 25 2021

A348546 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.

Original entry on oeis.org

0, 0, 8, 129, 1771, 21802, 252793, 2826973, 30872783
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A348544.

Crossrefs

Cf. A052268, A346950, A346952, A348054, A348055, A348544, A348547.

Programs

  • 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<#
  • 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))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Oct 22 2021

Formula

a(n) < A052268(n).
a(n) = A346952(n) + A348055(n) - A348547(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9) from Michael S. Branicky, Oct 22 2021

A348547 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.

Original entry on oeis.org

1, 4, 49, 524, 5596, 58706, 608886, 6267854, 64180304, 654605898, 6656849267, 67539297095, 683989985496, 6916722312963, 69859080168037
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Comments

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

Crossrefs

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

Programs

  • 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<#
  • 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))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Oct 22 2021

Formula

a(n) < A052268(n).
a(n) = A346952(n) + A348055(n) - A348546(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9) from Michael S. Branicky, Oct 22 2021
a(10)-a(15) from Martin Ehrenstein, Nov 06 2021

A348544 Positive integers that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

Original entry on oeis.org

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
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Comments

Intersection of A346950 and A348054.

Examples

			189 = 7*27 = 3*63, 399 = 3*133 = 7*57, 459 = 3*153 = 17*27, 609 = 3*203 = 7*87, ...

Crossrefs

Cf. A017377 (supersequence), A346950, A348054, A348546.

Programs

  • 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<#
  • 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)))
print(aupto(4029)) # Michael S. Branicky, Oct 22 2021

Formula

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

A348545 Positive integers with final digit 9 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

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
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Comments

Union of A346950 and A348054.

Examples

			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, ...

Crossrefs

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

Programs

  • 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
  • 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)))
print(aupto(999)) # Michael S. Branicky, Oct 22 2021

Formula

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

A348548 Positive integers that are the product of two integers ending with 8.

Original entry on oeis.org

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
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Examples

			64 = 8*8, 144 = 8*18, 224 = 8*28, 304 = 8*38, 324 = 18*18, 384 = 8*48, ...

Crossrefs

Cf. A017317 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347253 (ending with 4), A348054 (ending with 7), A348549.

Programs

  • 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
  • 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)))
print(aupto(2550)) # Michael S. Branicky, Oct 22 2021

Formula

Lim_{n->infinity} a(n)/a(n-1) = 1.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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