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 1-8 of 8 results.

A347254 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 4 (A347253).

Original entry on oeis.org

1, 5, 9, 13, 17, 19, 21, 25, 29, 33, 37, 41, 45, 47, 49, 53, 57, 61, 65, 69, 73, 75, 77, 81, 85, 89, 93, 97, 101, 103, 105, 109, 113, 115, 117, 121, 125, 129, 131, 133, 137, 141, 145, 149, 153, 157, 159, 161, 165, 169, 173, 177, 181, 183, 185, 187, 189, 193, 197
Offset: 1

Views

Author

Stefano Spezia, Aug 24 2021

Keywords

Comments

Since an integer 10*k + 6 = (10*a + 4)*(10*b + 4) implies that k = 10*a*b + 4*(a + b) + 1, all the terms of this sequence are odd.

Examples

			13 is a term because 4*34 = 136 = 13*10 + 6.

Crossrefs

Cf. A016873 (ending with 5), A017341, A324298 (ending with 6), A346951 (ending with 3), A347253.

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 && n>Max[a], AppendTo[a, n]]]]; a
  • PARI
    isok(k) =  my(x=10*k+6); sumdiv(x, d, (Mod(d, 10)==4) && Mod(x/d, 10)==4); \\ Michel Marcus, Oct 04 2021
  • Python
    def aupto(lim): return sorted(set(a*b//10 for a in range(4, 10*lim//4+3, 10) for b in range(a, 10*lim//a+3, 10) if a*b//10 <= lim))
print(aupto(197)) # Michael S. Branicky, Aug 24 2021

Formula

a(n) = (A347253(n) - 6)/10.
Lim_{n->infinity} a(n)/a(n-1) = 1.

A348054 Positive integers that are the product of two integers ending with 7.

Original entry on oeis.org

49, 119, 189, 259, 289, 329, 399, 459, 469, 539, 609, 629, 679, 729, 749, 799, 819, 889, 959, 969, 999, 1029, 1099, 1139, 1169, 1239, 1269, 1309, 1369, 1379, 1449, 1479, 1519, 1539, 1589, 1649, 1659, 1729, 1739, 1799, 1809, 1819, 1869, 1939, 1989, 2009, 2079, 2109
Offset: 1

Views

Author

Stefano Spezia, Sep 26 2021

Keywords

Examples

			49 = 7*7, 119 = 7*17, 189 = 7*27, 259 = 7*37, 289 = 17*17, 329 = 7*47, 399 = 7*57, ...

Crossrefs

Cf. A017377 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347253 (ending with 4), A348055.

Programs

  • Mathematica
    a={}; For[n=0, n<=210, n++, For[k=0, k<=n, k++, If[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
  • Python
    def aupto(lim): return sorted(set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2110)) # Michael S. Branicky, Sep 26 2021

Formula

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

A347255 Number of positive integers with n digits that are the product of two integers ending with 4.

Original entry on oeis.org

0, 3, 25, 281, 2941, 30596, 315385, 3231664, 32972224, 335346193, 3402373313, 34454358909, 348373701706, 3518101287286, 35491654274101
Offset: 1

Views

Author

Stefano Spezia, Aug 24 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A347253.

Crossrefs

Cf. A017341, A052268, A347253.
Cf. A346509 (ending with 1), A346952 (ending with 3), A337855 (ending with 5), A337856 (ending with 6).

Programs

  • 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))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Aug 24 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, Aug 25 2021
a(12)-a(15) from Martin Ehrenstein, Sep 29 2021

A347747 Positive integers with final digit 6 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

16, 36, 56, 96, 136, 156, 176, 196, 216, 256, 276, 296, 336, 376, 396, 416, 456, 476, 496, 516, 536, 576, 616, 636, 656, 676, 696, 736, 756, 776, 816, 856, 876, 896, 936, 976, 996, 1016, 1036, 1056, 1096, 1116, 1136, 1156, 1176, 1196, 1216, 1236, 1256, 1296, 1316
Offset: 1

Views

Author

Stefano Spezia, Sep 12 2021

Keywords

Comments

Union of A324297 and A347253.

Examples

			16 = 4*4, 36 = 6*6, 56 = 4*14, 96 = 4*24 = 6*16, 136 = 4*34, 156 = 6*26, ...

Links

Crossrefs

Cf. A017341 (supersequence), A324297, A347253, A347749.

Programs

  • Mathematica
    a={}; For[n=0, n<=150, 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] || Mod[10*n+6,10*k+6]==0 && Mod[(10*n+6)/(10*k+6),10]==6 && 10*n+6>Max[a], AppendTo[a, 10*n+6]]]]; a
tisdQ[n_]:=AnyTrue[{Mod[#,10],Mod[n/#,10]}&/@Divisors[n],#[[1]] == #[[2]]&]; Select[10 Range[150]+6,tisdQ] (* Harvey P. Dale, Dec 27 2021 *)
  • PARI
    isok(m) = if ((m % 10) == 6, fordiv(m, d, if ((d % 10) == (m/d % 10), return(1)))); \\ Michel Marcus, Oct 06 2021
  • 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)) | set(a*b for a in range(6, lim//6+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(1317)) # Michael S. Branicky, Sep 12 2021

Formula

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

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

A347746 Positive integers 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

96, 216, 256, 336, 416, 456, 576, 696, 736, 756, 816, 896, 936, 1056, 1176, 1216, 1296, 1376, 1416, 1456, 1536, 1596, 1656, 1696, 1776, 1836, 1856, 1896, 1976, 2016, 2136, 2176, 2256, 2336, 2376, 2436, 2496, 2576, 2616, 2656, 2736, 2816, 2856, 2916, 2976, 3016
Offset: 1

Views

Author

Stefano Spezia, Sep 12 2021

Keywords

Comments

Intersection of A324297 and A347253.

Examples

			96 = 4*24 = 6*16, 216 = 4*54 = 6*36, 256 = 4*64 = 16*16, 336 = 4*84 = 6*56, ...

Crossrefs

Cf. A017341 (supersequence), A324297, A347253, A347748.

Programs

  • Mathematica
    max=3050;Select[Intersection[Union@Flatten@Table[a*b, {a, 4, Floor[max/4], 10}, {b, a, Floor[max/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[max/6], 10}, {b, a, Floor[max/a], 10}]], 0<#
  • 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)) & set(a*b for a in range(6, lim//6+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(3017)) # Michael S. Branicky, Sep 12 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 1-8 of 8 results.

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

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