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.

Previous Showing 11-16 of 16 results.

A139264 a(n) = 70*n - 63.

Original entry on oeis.org

7, 77, 147, 217, 287, 357, 427, 497, 567, 637, 707, 777, 847, 917, 987, 1057, 1127, 1197, 1267, 1337, 1407, 1477, 1547, 1617, 1687, 1757, 1827, 1897, 1967, 2037, 2107, 2177, 2247, 2317, 2387, 2457, 2527, 2597, 2667, 2737, 2807, 2877, 2947, 3017, 3087, 3157, 3227
Offset: 1

Views

Author

Odimar Fabeny, Jun 06 2008

Keywords

Comments

Multiples of 7 with unit digit equal to 7.

Links

Crossrefs

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139279 and A139280.

Programs

Formula

a(n) = a(n-1) + 70.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 7*x*(1+9*x)/(1-x)^2.
E.g.f.: 7*(exp(x)*(10*x - 9) + 9).
a(n) = 7*A017281(n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Extensions

More terms from Reinhard Zumkeller, Jun 22 2008

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.

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.

A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

Original entry on oeis.org

24, 84, 144, 204, 264, 324, 384, 444, 504, 564, 624, 684, 744, 804, 864, 924, 984, 1044, 1104, 1164, 1224, 1284, 1344, 1404, 1464, 1524, 1584, 1644, 1704, 1764, 1824, 1884, 1944, 2004, 2064, 2124, 2184, 2244, 2304, 2364, 2424, 2484, 2544, 2604, 2664, 2724, 2784, 2844, 2904, 2964
Offset: 1

Views

Author

Emeric Deutsch, May 08 2018

Keywords

Comments

The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of the linear phenylene G[n] is M(G[n];x,y) = 6*x^2*y^2 + 4*(n - 1)*x^2*y^3 + 4(n - 1)*x^3*y^3.

Examples

			a(1) = 24; indeed, G[1] is a hexagon; we have 6 edges, each with end vertices of degree 2; then the second Zagreb index is 6*2*2 =24.

Links

Crossrefs

Cf. A016873, A304157.
Subsequence of A121024.

Programs

  • Julia
    [60*n-36 for n in 1:50] |> println # Bruno Berselli, May 09 2018
  • Maple
    seq(60*n - 36, n = 1 .. 40);
  • PARI
    a(n) = 60*n-36; \\ Altug Alkan, May 09 2018
  • PARI
    Vec(12*x*(2 + 3*x)/(1 - x)^2 + O(x^40)) \\ Colin Barker, May 23 2018

Formula

a(n) = 60*n - 36.
a(n) = 12 * A016873(n-1). - Alois P. Heinz, May 09 2018
From Bruno Berselli, May 09 2018: (Start)
O.g.f.: 12*x*(2 + 3*x)/(1 - x)^2.
E.g.f.: 12*(3 - 3*exp(x) + 5*x*exp(x)).
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008594(5*n-3) = A017317(6*n-4) = A072710(4*n-2) = A139245(3*n-1). (End)

A346629 Number of n-digit positive integers that are the product of two integers ending with 2.

Original entry on oeis.org

1, 4, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000
Offset: 1

Views

Author

Stefano Spezia, Jul 25 2021

Keywords

Comments

a(n) is the number of n-digit numbers in A139245.
After initial 1 or 2 values the same as A137233. - R. J. Mathar, Aug 23 2021

Links

Crossrefs

Cf. A011557 (powers of 10), A017293 (positive integers ending with 2), A052268 (number of n-digit integers), A139245 (product of two integers ending with 2), A093143, A337855, A337856.
Cf. A137233.

Programs

  • Mathematica
    LinearRecurrence[{10},{1,4,45},25]

Formula

O.g.f.: x*(1 - 6*x + 5*x^2)/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 + 110*x - 50*x^2)/200.
a(n) = 10*a(n-1) for n > 3, with a(1) = 1, a(2) = 4 and a(3) = 45.
a(n) = 45*10^(n-3) for n > 2.
a(n) = 45*A011557(n-3) for n > 2.
Sum_{i=1..n} a(n) = A093143(n-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.
Previous Showing 11-16 of 16 results.

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