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 11-15 of 15 results.

A346507 Positive integers k that are the product of two integers greater than 1 and ending with 1.

Original entry on oeis.org

121, 231, 341, 441, 451, 561, 651, 671, 781, 861, 891, 961, 1001, 1071, 1111, 1221, 1271, 1281, 1331, 1441, 1491, 1551, 1581, 1661, 1681, 1701, 1771, 1881, 1891, 1911, 1991, 2091, 2101, 2121, 2201, 2211, 2321, 2331, 2431, 2501, 2511, 2541, 2601, 2651, 2751, 2761
Offset: 1

Views

Author

Stefano Spezia, Jul 21 2021

Keywords

Comments

All the terms end with 1 (A017281).

Examples

			121 = 11*11, 231 = 11*21, 341 = 11*31, 441 = 21*21, 451 = 11*41, ...

Links

Crossrefs

Cf. A017281 (supersequence), A053742 (ending with 5), A324297 (ending with 6), A346508, A346509, A346510.

Programs

  • Mathematica
    a={}; For[n=1, n<=300, n++, For[k=1, kMax[a], AppendTo[a, 10n+1]]]]; a
  • PARI
    isok(k) = fordiv(k, d, if ((d>1) && (dMichel Marcus, Jul 28 2021
  • Python
    def aupto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2761)) # Michael S. Branicky, Jul 22 2021

Formula

Conjecture: lim_{n->infinity} a(n)/a(n-1) = 1.
The conjecture is true since it can be proved that a(n) = (sqrt(a(n-1)) + g(n-1))^2 where [g(n): n > 1] is a bounded sequence of positive real numbers. - Stefano Spezia, Aug 21 2021

A053741 Sum of even numbers in range 10*n to 10*n+9.

Original entry on oeis.org

20, 70, 120, 170, 220, 270, 320, 370, 420, 470, 520, 570, 620, 670, 720, 770, 820, 870, 920, 970, 1020, 1070, 1120, 1170, 1220, 1270, 1320, 1370, 1420, 1470, 1520, 1570, 1620, 1670, 1720, 1770, 1820, 1870, 1920, 1970, 2020, 2070, 2120, 2170, 2220, 2270
Offset: 0

Views

Author

Odimar Fabeny, Feb 13 2000

Keywords

Examples

			20 is sum of 0, 2, 4, 6, 8;
70 is sum of 10, 12, 14, 16, 18;
120 is sum of 20, 22, 24, 26, 28; etc.

Links

Crossrefs

Cf. A053742, A053743.

Programs

Formula

a(n) = 50*n + 20.
From Colin Barker, Jun 27 2012: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: 10*(2+3*x)/(1-x)^2. (End)
E.g.f.: 10*(2+5*x)*exp(x). - G. C. Greubel, Sep 06 2019

Extensions

More terms from James Sellers, Feb 22 2000

A053743 Sum of numbers in range 10*n to 10*n+9.

Original entry on oeis.org

45, 145, 245, 345, 445, 545, 645, 745, 845, 945, 1045, 1145, 1245, 1345, 1445, 1545, 1645, 1745, 1845, 1945, 2045, 2145, 2245, 2345, 2445, 2545, 2645, 2745, 2845, 2945, 3045, 3145, 3245, 3345, 3445, 3545, 3645, 3745, 3845, 3945, 4045, 4145, 4245, 4345
Offset: 0

Views

Author

Odimar Fabeny, Feb 13 2000

Keywords

Examples

			45 = sum of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
145 = sum of 10, 11, 12, 13, 14, 15, 16, 17, 18, 19.

Links

Crossrefs

Cf. A053741, A053742.

Programs

  • GAP
    List([0..50], n-> 5*(20*n+9)); # G. C. Greubel, Sep 06 2019
  • Magma
    [5*(20*n+9): n in [0..50]]; // G. C. Greubel, Sep 06 2019
  • Maple
    seq(5*(20*n+9), n=0..50); # G. C. Greubel, Sep 06 2019
  • Mathematica
    LinearRecurrence[{2,-1},{45,145},50] (* or *) 100*Range[50]-55 (* Harvey P. Dale, Feb 03 2019 *)
  • PARI
    vector(50, n, 5*(20*n -11)) \\ G. C. Greubel, Sep 06 2019
  • Sage
    [5*(20*n+9) for n in (0..50)] # G. C. Greubel, Sep 06 2019

Formula

a(n) = 100*n + 45.
From Colin Barker, Jun 26 2012: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: 5*(9+11*x)/(1-x)^2. (End)
E.g.f.: 5*(9 + 20*x)*exp(x). - G. C. Greubel, Sep 06 2019

Extensions

More terms from James Sellers, Feb 22 2000

A346526 Positive integers k that are the product of two integers greater than 1 and ending with the same digit as k.

Original entry on oeis.org

25, 36, 75, 96, 100, 121, 125, 156, 175, 200, 216, 225, 231, 256, 275, 276, 300, 325, 336, 341, 375, 396, 400, 416, 425, 441, 451, 456, 475, 500, 516, 525, 561, 575, 576, 600, 625, 636, 651, 671, 675, 676, 696, 700, 725, 736, 756, 775, 781, 800, 816, 825, 861, 875
Offset: 1

Views

Author

Stefano Spezia, Jul 22 2021

Keywords

Comments

Union of 100*A000027, A053742, A324297 and A346507.

Examples

			25 = 5*5, 36 = 6*6, 75 = 5*15, 96 = 6*16, 100 = 10*10, 121 = 11*11, 125 = 5*25, 156 = 6*26, 175 = 5*35, 200 = 10*20, 216 = 6*36, 225 = 15*15, 231 = 11*21, ...

Links

Crossrefs

Cf. A000027, A053742, A324297, A346507.

Programs

  • Lisp
    (setf candidates (list 25)) (setf result nil)
(defun factor (num small-num) (equalp 0 (mod num small-num)))
(defun same-end-digit (num1 num2 num3) (and (equalp (mod num1 10) (mod num2 10)) (equalp (mod num2 10) (mod num3 10))))
(defun good-factor-p (num) (loop for i from 5 to (sqrt num) do ( if (factor num i) ( if (same-end-digit num i (/ num i) ) (return T) ))))
(loop for i from 26 to 9000 do ( if (or (equalp 0 (mod i 10)) (equalp 1 (mod i 10)) (equalp 5 (mod i 10)) (equalp 6 (mod i 10))) (push i candidates)))
(dolist (element candidates) (if (good-factor-p element) (push element result)))
(format t (write-to-string result)) \\ FUNG Cheok Yin, Aug 12 2021
  • PARI
    isok(k) = my(u=k%10); sumdiv(k, d, (d>1) && (d 0; \\ Michel Marcus, Jul 23 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.
Previous Showing 11-15 of 15 results.

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