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.

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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 *)

A260086 Smaller of amicable pair (x, y) as they are listed in A259933.

Original entry on oeis.org

220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 66928, 67095, 63020, 69615, 79750, 100485, 122368, 122265, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319950, 356408, 437456, 469028, 503056, 522405, 600392, 609928, 643336, 624184, 635624, 667964, 726104, 802725, 879712, 898216, 998104, 947835
Offset: 1

Views

Author

Omar E. Pol, Jul 15 2015

Keywords

Comments

Another version of A002025.
First differs from A002025 at a(9).

Links

Crossrefs

Cf. A002025, A063990, A259180, A259933, A259953, A260087.

Formula

a(n) = A259933(2n-1) = A259953(n) - A259933(2n) = A259953(n) - A260087(n).

A260087 Larger of amicable pair (x, y) as they are listed in A259933.

Original entry on oeis.org

284, 1210, 2924, 5564, 6368, 10856, 14595, 18416, 66992, 71145, 76084, 87633, 88730, 124155, 123152, 139815, 153176, 168730, 176336, 180848, 203432, 202444, 365084, 389924, 430402, 399592, 455344, 486178, 514736, 525915, 669688, 686072, 652664, 691256, 712216, 783556, 796696, 863835, 901424, 980984, 1043096, 1125765
Offset: 1

Views

Author

Omar E. Pol, Jul 15 2015

Keywords

Comments

Another version of A002046.
First differs from A002046 at a(9).

Links

Crossrefs

Cf. A002046, A063990, A259180, A259933, A259953, A260086.

Formula

a(n) = A259933(2n) = A259953(n) - A259933(2n-1) = A259953(n) - A260086(n).

A262623 Amicable pairs of odd numbers.

Original entry on oeis.org

12285, 14595, 67095, 71145, 69615, 87633, 100485, 124155, 122265, 139815, 522405, 525915, 802725, 863835, 947835, 1125765, 1175265, 1438983, 1280565, 1340235, 1358595, 1486845, 1798875, 1870245, 4482765, 5120595, 5357625, 5684679, 5730615, 6088905, 6377175, 6680025, 8619765, 9627915, 9071685, 9498555, 9206925, 10791795
Offset: 1

Views

Author

Omar E. Pol, Nov 09 2015

Keywords

Comments

If there are no amicable pairs whose members have distinct parity then this is also the odd terms of A259180.
First differs from A262625 at a(4).

References

  • Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computational Approach, World Scientific, 1996, pages 151 - 153.

Links

Crossrefs

Cf. A002025, A002046, A005408, A063990, A259180, A262622, A262625.

Programs

  • PARI
    lista(nn) = {forstep(n=1, nn, 2, m = sigma(n)-n; if ((m > n) && (n==sigma(m)-m), print1(n, ", ", m, ", ")););} \\ Michel Marcus, Nov 14 2015

A275996 Numbers n whose abundance is 64: sigma(n) - 2n = 64.

Original entry on oeis.org

108, 220, 6808, 8968, 14008, 24448, 66928, 552568, 786208, 1020568, 5303488, 8229568, 10001848, 133685248, 499722448, 2608895488, 4733164768, 7163795488, 13707973408, 14468025568, 16122444736, 27339731968, 34351218688, 34672397728, 35371084288, 69657461248
Offset: 1

Views

Author

Timothy L. Tiffin, Aug 16 2016

Keywords

Comments

Any term x = a(m) of this sequence can be used with any term y of A275997 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable.
The smallest amicable pair is (220, 284) = (a(2), A275997(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
The amicable pair (66928, 66992) = (a(7), A275997(11)) = (A063990(18), A063990(19)), and 66992 - 66928 = 64 is the abundance of 66928 and the deficiency of 66992.

Examples

			a(1) = 108, since sigma(108) - 2*108 = 280 - 216 = 64.

Links

Crossrefs

Subsequence of A005101.
Cf. A002025, A063990, A275997, A088831, A088832, A088833, A141547, A175989, A275701, A066539, A259180.

Programs

  • PARI
    isok(n) = sigma(n) - 2*n == 64; \\ Michel Marcus, Dec 30 2016

Extensions

a(14)-a(15) from Michel Marcus, Dec 30 2016
a(16)-a(21) from Lars Blomberg, Jan 12 2017
Terms a(22) onward from Max Alekseyev, Aug 27 2025

A328064 Amicable pairs with the property that both members have the same number of divisors.

Original entry on oeis.org

1184, 1210, 2620, 2924, 5020, 5564, 10744, 10856, 66928, 66992, 67095, 71145, 122368, 123152, 171856, 176336, 176272, 180848, 196724, 202444, 437456, 455344, 503056, 514736, 522405, 525915, 1077890, 1099390, 1154450, 1189150, 1280565, 1340235, 1358595, 1486845, 1392368, 1464592, 2082464, 2090656
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 [1184, 1210]. The smaller member has 12 divisors, they are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184. The larger member has 12 divisors, they are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210. The number of divisors of 1184 is equal to the number of divisors of 1210, so the amicable pair [1184, 1210] is in the sequence.

Links

Crossrefs

Subsequence of A259180.
Cf. A000005, A002025, A002046, A328009, A328043, A328063, 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, 10^6}]; 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

A383484 Integers k such that there exists an integer 0

Original entry on oeis.org

3, 33, 69, 77, 133, 161, 235, 267, 287, 321, 385, 699, 715, 1235, 1379, 1437, 1529, 1595, 1653, 1719, 2047, 2233, 2241, 2569, 2727, 2829, 3237, 3269, 3999, 4585, 4683, 4911, 5075, 5163, 5215, 5497, 5667, 5691, 7085, 7089, 7587, 7761, 7797, 7945, 8259, 9159, 9659, 10653
Offset: 1

Views

Author

S. I. Dimitrov, Apr 28 2025

Keywords

Comments

From David A. Corneth, May 04 2025: (Start)
If (t, u) is a divisor pair of sigma(k)^2 then m = (t + u - 2*k)/2, sigma(m) = m + k - t.
Proof:
Since sigma(m)^2 + sigma(k)^2 = (m+k)^2 we have sigma(k)^2 = (m+k)^2 - sigma(m)^2 = (m + k - sigma(m)) * (m + k + sigma(m)) = t * u where t, u | sigma(k)^2.
This gives the system (m + k - sigma(m)) = t and (m + k + sigma(m)) = u. Solving gives
m = (t + u - 2*k)/2, sigma(m) = m + k - t. For every pair (t, u) of divisors of sigma(k)^2 we can test if the given values of m and sigma(m) hold. If at least one of them holds then k is in the sequence. Q. E. D.
Are there any even terms? There are none in the first 1006 terms. (End)

Examples

			(2, 3) is such a pair because sigma^2(2)+sigma^2(3) = 3^2+4^2 = (2+3)^2.
33 is in the sequence. As sigma(33)^2 = 2304 and for the divisor pair (32, 72) we have m = (32 + 72 - 2*33)/2 = 19 and sigma(m) = m + k - 32 = 19 + 33 - 32 = 20 and indeed sigma(19) = 20. - _David A. Corneth_, May 04 2025

Links

Crossrefs

Cf. A063990, A259180, A383239.

Programs

  • PARI
    isok(k) = for (m=1, k, if (sigma(m)^2+sigma(k)^2==(m+k)^2, return(1))); \\ Michel Marcus, Apr 28 2025
  • PARI
    is(n) = {my(sn = sigma(n)^2, d = divisors(sn)); for(i = 1, #d / 2, k = (d[i] + d[#d + 1 - i] - 2*n) / 2; if(denominator(k) == 1, sk = n + k - d[i]; if(k < n && sigma(k) == sk, return(1)))); 0} \\ David A. Corneth, May 04 2025

Extensions

More terms from Michel Marcus, Apr 28 2025

A061469 GCD of the amicable pairs: a(n) = gcd(A002025(n), A002046(n)).

Original entry on oeis.org

4, 2, 4, 4, 8, 8, 105, 16, 92, 16, 135, 819, 10, 45, 585, 16, 8, 10, 16, 16, 8, 44, 4, 4, 14, 8, 16, 98, 16, 585, 8, 8, 8, 8, 8, 4, 8, 105, 16, 8, 135, 8, 10, 50, 10, 5733, 8, 585, 10, 855, 16, 10, 4, 10, 10, 585, 32, 10, 8, 16, 14, 8, 16, 4, 8, 310, 8, 16, 8, 10, 50, 855, 14, 8, 10
Offset: 1

Views

Author

Jason Earls, Jun 11 2001

Keywords

Comments

The pairs are ordered by the smaller number. - David Wasserman, Jun 18 2002

Examples

			a(1) = gcd(220, 284) = 4;
a(2) = gcd(1184, 1210) = 2.

Links

Crossrefs

Cf. A002025, A002046, A259180 (amicable pairs).

Programs

  • Mathematica
    With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Flatten@ Map[GCD @@ # &, Apply[Join, Map[Function[n, Select[Subsets[Lookup[s, n], {2}], Total@ # == n &]], Sort@ Select[Keys@ s, Length@ Lookup[s, #] > 1 &]]]]] (* Michael De Vlieger, Oct 22 2017 *)

Extensions

More terms from David Wasserman, Jun 18 2002
a(22) corrected by T. D. Noe, Aug 15 2010
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