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.

A328043 Number of divisors of the amicable pairs.

Original entry on oeis.org

12, 6, 12, 12, 12, 12, 12, 12, 16, 12, 16, 16, 32, 16, 20, 10, 24, 12, 20, 20, 32, 32, 48, 24, 32, 16, 48, 24, 48, 24, 20, 20, 24, 16, 32, 16, 20, 20, 20, 20, 32, 16, 24, 24, 36, 12, 24, 12, 48, 16, 32, 16, 20, 20, 18, 36, 20, 20, 48, 48, 32, 16, 32, 16, 32, 16, 32, 16, 32, 16, 48, 24, 32, 16, 48, 32, 24, 20
Offset: 1

Views

Author

Omar E. Pol, Oct 02 2019

Keywords

Examples

			Consider the first amicable pair [220, 284]. The smaller member has 12 divisors, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220. The larger member has 6 divisors, they are 1, 2, 4, 71, 142, 284. So a(1) = 12 and a(2) = 6.

Links

Crossrefs

Row lengths of A328009.
Cf. A000005, A259180, A328063, A328064, A328065.

Programs

  • Mathematica
    With[{s = Array[{#, DivisorSigma[1, #] - #} &, 10^5]}, DivisorSigma[0, #] &@ Flatten@ DeleteDuplicates[Sort /@ Select[Reverse /@ s, And[! FreeQ[s, #], UnsameQ @@ #] &]]] (* Michael De Vlieger, Oct 08 2019 *)

Formula

a(n) = A000005(A259180(n)).

A328063 Amicable pairs with the property that the number of divisors of the smaller member is greater than the number of divisors of the larger member.

Original entry on oeis.org

220, 284, 6232, 6368, 12285, 14595, 17296, 18416, 63020, 76084, 69615, 87633, 79750, 88730, 100485, 124155, 122265, 139815, 141664, 153176, 142310, 168730, 185368, 203432, 280540, 365084, 308620, 389924, 319550, 430402, 356408, 399592, 600392, 669688, 609928, 686072, 624184, 691256
Offset: 1

Views

Author

Omar E. Pol, Oct 03 2019

Keywords

Comments

Amicable pairs(x,y) such that d(x) > d(y), where d(n) is the number of divisors of n.

Examples

			Consider the amicable pair [220, 284]. The smaller member has 12 divisors, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220. The larger member has 6 divisors, they are 1, 2, 4, 71, 142, 284. The number of divisors of 220 is greater than the number of divisors of 284, so the amicable pair [220, 284] is in the sequence.

Links

Crossrefs

Subsequence of A259180.
Cf. A000005, A002025, A002046, A328009, A328043, A328064, A328065, A328255.

Programs

  • Mathematica
    seq = {}; s[n_] := DivisorSigma[1, n] - n; Do[m = s[n]; If[m > n && s[m] == n && DivisorSigma[0, n] > DivisorSigma[0, m], seq = Join[seq, {n, m}]], {n, 1, 7*10^5}]; seq (* Amiram Eldar, Oct 11 2019 *)

A328065 Amicable pairs with the property that the number of divisors of the smaller member is twice the number of divisors of the larger member.

Original entry on oeis.org

220, 284, 12285, 14595, 17296, 18416, 63020, 76084, 69615, 87633, 79750, 88730, 100485, 124155, 122265, 139815, 142310, 168730, 185368, 203432, 308620, 389924, 356408, 399592, 600392, 669688, 609928, 686072, 624184, 691256, 635624, 712216, 643336, 652664, 667964, 783556, 726104, 796696, 898216, 980984
Offset: 1

Views

Author

Omar E. Pol, Oct 03 2019

Keywords

Comments

Amicable pairs(x,y) such that d(x) = 2*d(y), where d(n) is the number of divisors of n.

Examples

			Consider the amicable pair [220, 284]. The smaller member has 12 divisors, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220. The larger member has 6 divisors, they are 1, 2, 4, 71, 142, 284. The number of divisors of 220 is twice the number of divisors of 284, so the amicable pair [220, 284] is in the sequence.

Links

Crossrefs

Subsequence of A259180 and of A328063.
Cf. A000005, A002025, A002046, A062011, A328009, A328043, A328064, A328255.

Programs

  • Mathematica
    seq = {}; s[n_] := DivisorSigma[1, n] - n; Do[m = s[n]; If[m > n && s[m] == n && DivisorSigma[0, n] == 2 * DivisorSigma[0, m], seq = Join[seq, {n, m}]], {n, 1, 10^6}]; seq (* Amiram Eldar, Oct 11 2019 *)

A328255 Amicable pairs with the property that the number of divisors of the smaller member is smaller than the number of divisors of the larger member.

Original entry on oeis.org

469028, 486178, 1511930, 1598470, 4246130, 4488910, 5232010, 5799542, 10533296, 10949704, 11693290, 12361622, 20308995, 20955645, 37784810, 39944086, 46991890, 48471470, 48641584, 48852176, 80422335, 82977345, 96304845, 96747315, 103034776, 105016424, 115749344, 116983744, 118458830, 131819506
Offset: 1

Views

Author

Omar E. Pol, Oct 09 2019

Keywords

Comments

Amicable pairs(x,y) such that d(x) < d(y), where d(n) is the number of divisors of n.

Examples

			Consider the amicable pair [469028, 486178]. The smaller member has 18 divisors and the larger member has 36 divisors. 18 is smaller than 36, so the amicable pair [469028, 486178] is in the sequence.

Links

Crossrefs

Subsequence of A259180.
Cf. A000005, A002025, A002046, A328009, A328043, A328063, A328064, A328065.

Extensions

More terms from Amiram Eldar, Oct 09 2019

A328009 Irregular array read by rows in which row n lists the divisors of the n-th term of the sequence of amicable pairs (A259180).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220, 1, 2, 4, 71, 142, 284, 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184, 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210, 1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310, 2620, 1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462, 2924, 1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255
Offset: 1

Views

Author

Omar E. Pol, Oct 01 2019

Keywords

Comments

Row sums give a sequence formed by the terms of A180164 repeated as follows: 504, 504, 2394, 2394, 5544, 5544, ...

Examples

			Array begins:
1, 2, 4,  5,  10,  11,  20,  22,   44,  55,   110,  220;
1, 2, 4, 71, 142, 284;
1, 2, 4,  8,  16,  32,  37,  74,  148,  296,  592, 1184;
1, 2, 5, 10,  11,  22,  55, 110,  121,  242,  605, 1210;
1, 2, 4,  5,  10,  20, 131, 262,  524,  655, 1310, 2620;
1, 2, 4, 17,  34,  43,  68,  86,  172,  731, 1462, 2924;
1, 2, 4,  5,  10,  20, 251, 502, 1004, 1255, 2510, 5020;
1, 2, 4, 13,  26,  52, 107, 214,  428, 1391, 2782, 5564,
1, 2, 4,  8,  19,  38,  41,  76,   82,  152,  164,  328, 779, 1558, 3116, 6232;
1, 2, 4,  8,  16,  32, 199, 398,  796, 1592, 3184, 6368;
...

Crossrefs

Right border gives A259180 (amicable pairs).
The length of row n is A328043(n).
Column 1 gives A000012.
Cf. A002025, A002046, A018339, A018373, A027750, A180164, A328063, A328064, A328065.

Programs

  • Mathematica
    With[{s = Array[{#, DivisorSigma[1, #] - #} &, 6000]}, Flatten@ Divisors@ DeleteDuplicates[Sort /@ Select[Reverse /@ s, And[! FreeQ[s, #], UnsameQ @@ #] &]]] (* Michael De Vlieger, Oct 08 2019 *)
Showing 1-5 of 5 results.

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