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 1-5 of 5 results.

A347253 Positive integers that are the product of two integers ending with 4.

Original entry on oeis.org

16, 56, 96, 136, 176, 196, 216, 256, 296, 336, 376, 416, 456, 476, 496, 536, 576, 616, 656, 696, 736, 756, 776, 816, 856, 896, 936, 976, 1016, 1036, 1056, 1096, 1136, 1156, 1176, 1216, 1256, 1296, 1316, 1336, 1376, 1416, 1456, 1496, 1536, 1576, 1596, 1616, 1656
Offset: 1

Views

Author

Stefano Spezia, Aug 24 2021

Keywords

Examples

			16 = 4*4, 56 = 4*14, 96 = 4*24, 136 = 4*34, 176 = 4*44, 196 = 14*14, 216 = 4*54, ...

Crossrefs

Cf. A017341 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347254, A347255.

Programs

  • Mathematica
    a={}; For[n=0, n<=200, n++, For[k=0, k<=n, k++, If[Mod[10*n+6, 10*k+4]==0 && Mod[(10*n+6)/(10*k+4), 10]==4 && 10*n+6>Max[a], AppendTo[a, 10*n+6]]]]; a
  • Python
    def aupto(lim): return sorted(set(a*b for a in range(4, lim//4+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(1660)) # Michael S. Branicky, Aug 24 2021

Formula

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

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

A347748 Number of positive integers with n digits that are equal both to the product of two integers ending with 4 and to that of two integers ending with 6.

Original entry on oeis.org

0, 1, 12, 159, 1859, 20704, 223525, 2370684, 24842265, 258128126, 2665475963
Offset: 1

Views

Author

Stefano Spezia, Sep 12 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A347746.

Crossrefs

Cf. A017341, A052268, A324297, A337856, A347253, A347255, A347746, A347749.

Programs

  • Mathematica
    Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Intersection[Union@Flatten@Table[a*b, {a, 4, Floor[hi/4], 10}, {b, a, Floor[hi/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[hi/6], 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(4, hi//4+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) & set(a*b for a in range(6, hi//6+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 06 2021

Formula

a(n) < A052268(n).
a(n) = A337856(n) + A347255(n) - A347749(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(10) from Michael S. Branicky, Oct 06 2021
a(11) from Frank A. Stevenson, Jan 06 2024

A347749 Number of positive integers with n digits and final digit 6 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

0, 4, 33, 352, 3597, 36781, 374071, 3790993, 38326689, 386782889
Offset: 1

Views

Author

Stefano Spezia, Sep 12 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A347747.

Crossrefs

Cf. A017341, A052268, A324297, A337856, A347253, A347255, A347747, A347748.

Programs

  • Mathematica
    Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Union[Union@Flatten@Table[a*b, {a, 4, Floor[hi/4], 10}, {b, a, Floor[hi/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[hi/6], 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(4, hi//4+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) | set(a*b for a in range(6, hi//6+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 06 2021

Formula

a(n) < A052268(n).
a(n) = A337856(n) + A347255(n) - A347748(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(10) from Michael S. Branicky, Oct 06 2021

A348549 Number of positive integers with n digits that are the product of two integers ending with 8.

Original entry on oeis.org

0, 1, 14, 195, 2200, 24013, 255969, 2687317, 27934809, 288342379, 2960920297, 30285890402, 308834717932, 3141625339760, 31895159990436
Offset: 1

Views

Author

Stefano Spezia, Oct 22 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A348548.

Crossrefs

Cf. A052268, A348548.
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), A348055 (ending with 7).

Programs

  • Python
    def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(8, hi//8+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).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

Extensions

a(9)-a(10) from Michael S. Branicky, Oct 22 2021
a(11)-a(15) from Martin Ehrenstein, Nov 06 2021
Showing 1-5 of 5 results.

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

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