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.

A228844 Smallest sets of 3 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

24, 42, 80, 100, 104, 114, 120, 126, 144, 162, 180, 196, 200, 220, 228, 234, 240, 246, 272, 282, 288, 304, 324, 348, 350, 364, 392, 402, 420, 426, 440, 460, 504, 572, 582, 588, 594, 608, 616, 624, 640, 654, 660, 666, 684, 700, 708, 714, 728, 736, 740, 786
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Examples

			24, 30, 36 is the smallest set of 3 consecutive abundant numbers in arithmetic progression so 24 is in the list.

Links

Crossrefs

Cf. A005101, A228433.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 2, AppendTo[a, n - 2*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000}]; a

A231623 Smallest starting deficient number for n consecutive deficient numbers in arithmetic progression.

Original entry on oeis.org

1, 1, 1, 2987, 801339, 221355125, 221355125
Offset: 3

Views

Author

Shyam Sunder Gupta, Nov 11 2013

Keywords

Comments

The next term is a(10) > 10^9. The common difference for first 3 terms is 1 and for next 4 terms is 2.

Examples

			First and smallest occurrence of n, n >= 3, consecutive deficient numbers in arithmetic progression:
a(3) = 1: (1, 2, 3);
a(4) = 1: (1, 2, 3, 4);
a(5) = 1: (1, 2, 3, 4, 5);
a(6) = 2987: (2987, 2989, 2991, 2993, 2995, 2997);
a(7) = 801339: (801339, 801341, 801343, 801345, 801347, 801349, 801351);
a(8) = 221355125: (221355125, 221355127, 221355129, 221355131, 221355133, 221355135, 221355137, 221355139);
a(9) = 221355125: (221355125, 221355127, 221355129, 221355131, 221355133, 221355135, 221355137, 221355139, 221355141);

Crossrefs

Cf. A005100, A228433.

A228961 Smallest sets of 4 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

114, 120, 228, 234, 240, 282, 348, 420, 582, 588, 654, 660, 708, 840, 1002, 1008, 1014, 1068, 1122, 1242, 1248, 1254, 1260, 1380, 1434, 1542, 1548, 1674, 1794, 1800, 1908, 1914, 1998, 2094, 2100, 2208, 2214, 2220, 2262, 2268, 2328, 2334, 2340, 2502, 2508
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Examples

			114, 120, 126, 132 is the smallest set of 4 consecutive abundant numbers in arithmetic progression so 114 is in the list.

Links

Crossrefs

Cf. A005101, A228433, A228844.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 3, AppendTo[a, n - 3*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000}];a

A228962 Smallest sets of 5 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

114, 228, 234, 582, 654, 1002, 1008, 1242, 1248, 1254, 1542, 1794, 1908, 2094, 2214, 2262, 2328, 2334, 2502, 2634, 2754, 2988, 3054, 3102, 3348, 3354, 3642, 4182, 4188, 4314, 4362, 4368, 4428, 4434, 4482, 4728, 4788, 4902, 5202, 5208, 5268, 5274, 5742, 5862
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Examples

			114, 120, 126, 132, 138 is the smallest set of 5 consecutive abundant numbers in arithmetic progression so 114 is in the list.

Links

Crossrefs

Cf. A005101, A228433, A228844, A228961.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 4, AppendTo[a, n - 4*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000}]; a

A228963 Smallest sets of 6 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

228, 1002, 1242, 1248, 2328, 3348, 4182, 4362, 4428, 5202, 5268, 6702, 6708, 6882, 7962, 7968, 8142, 8382, 8982, 9822, 9888, 10242, 11568, 11922, 11988, 12162, 12168, 12588, 13248, 13422, 13842, 13848, 14022, 14088, 14508, 15282, 15522, 15528, 16362, 16368
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Examples

			228, 234, 240, 246, 252, 258 is the smallest set of 6 consecutive abundant numbers in arithmetic progression so 228 is in the list.

Links

Crossrefs

Cf. A005101, A228433, A228844, A228961, A228962.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 5, AppendTo[a, n - 5*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000}]; a

A228964 Smallest sets of 7 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

1242, 6702, 7962, 12162, 13842, 15522, 16362, 18042, 18882, 19722, 24762, 26442, 27282, 27702, 28122, 28962, 36942, 38202, 39462, 43662, 44922, 45762, 48282, 48702, 51222, 55842, 56682, 60042, 62562, 63402, 66762, 69282, 69702, 70962, 71802, 73062, 73482
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Examples

			1242, 1248, 1254, 1260, 1266, 1272, 1278 is the smallest set of 7 consecutive abundant numbers in arithmetic progression so 1242 is in the list.

Links

Crossrefs

Cf. A005101, A228433, A228844, A228961, A228962, A228963.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 6, AppendTo[a, n - 6*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000}]; a
Select[Partition[Select[Range[80000],DivisorSigma[1,#]>2#&],7,1], Length[ Union[ Differences[#]]] ==1&][[All,1]] (* Harvey P. Dale, Oct 15 2017 *)

A228965 Smallest sets of 8 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed.

Original entry on oeis.org

221355126, 402640540, 668862580, 739577140
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 10 2013

Keywords

Comments

Is this a duplicate of A231093? - R. J. Mathar, Nov 15 2013
No; there is some element of this sequence not in A231093 below approximately 10^10^72. In fact A228965 \ A231093 has positive lower density (though presumably quite small). Capsule proof: choose n such that an appropriately large number of primes divide n, n+1, ..., n+7. Since the reciprocal of the primes diverges, you can get sigma(n+i)/(n+i) arbitrarily large. - Charles R Greathouse IV, Nov 15 2013

Examples

			221355126, 221355128, 221355130, 221355132, 221355134, 221355136, 221355138, 221355140  is the smallest set of 8 consecutive abundant numbers in arithmetic progression so 221355126 is in the list.

Crossrefs

Cf. A005101, A228433, A228844, A228961, A228962, A228963, A228964.

Programs

  • Mathematica
    AbundantQ[n_] := DivisorSigma[1, n] > 2 n; m = 2; z1 = 18; cd = 6; a = {}; Do[If[AbundantQ[n], If[n - z1 == cd, m = m + 1; If[m > 7, AppendTo[a, n - 7*cd]], m = 2; cd = n - z1]; z1 = n], {n, 19, 1000000000}]; a

A350005 a(n) is the smallest number that starts an arithmetic progression of n consecutive ludic numbers (A003309), or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 71, 6392047
Offset: 1

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

a(n) is the smallest ludic number A003309(k), such that A260723(k) = A260723(k+1) = ... = A260723(k+n-2).
a(6) > 10^8 (unless a(6) = 0).

Examples

			The first arithmetic progression of 3 consecutive ludic numbers is (1, 2, 3), so a(3) = 1.
The first arithmetic progression of 4 consecutive ludic numbers is (71, 77, 83, 89), so a(4) = 71.
The first arithmetic progression of 5 consecutive ludic numbers is (6392047, 6392077, 6392107, 6392137, 6392167), so a(5) = 6392047.

Crossrefs

From n = 3, first row of A350007.
Cf. A003309, A260723.
Counterparts for other sequences than ludic numbers: A006560 (primes), A228433 (abundant numbers), A231623 (deficient numbers), A276821 (Sophie Germain primes), A330362 (lucky numbers).
Showing 1-8 of 8 results.

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