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 1-6 of 6 results.

A259039 Larger of a non-unitary amicable pair.

Original entry on oeis.org

56, 248, 328, 496, 1016, 2032, 6462, 8128, 17412, 20538, 65528, 131056, 524224, 1048568, 2097136, 2096896, 4194296, 8388592, 8388544, 33554368, 33554176, 134217472, 2147467264, 8589918208
Offset: 1

Views

Author

Mauro Fiorentini, Jun 17 2015

Keywords

Comments

The elements 2097136, 8388592, etc. are intentionally out of numerical order so that a(n) and A259038(n) form an amicable pair.

Crossrefs

Cf. A034448, A259037, A259038.

Extensions

a(23)-a(24) added by Amiram Eldar, Sep 27 2018 from the b-file at A259037.

A348343 Smaller member of a noninfinitary amicable pair: numbers (k, m) such that nisigma(k) = m and nisigma(m) = k, where nisigma(k) is the sum of the noninfinitary divisors of k (A348271).

Original entry on oeis.org

336, 1792, 5376, 6096, 21504, 32004, 97536, 34062336, 64512000, 118008576, 30064771072
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

The larger counterparts are in A348344.

Examples

			336 is a term since A348271(336) = 448 and A348271(448) = 336.

Crossrefs

Cf. A327633, A348271, A348344.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, n]], {n,1,10^4}]; seq

A259037 Non-unitary amicable numbers.

Original entry on oeis.org

48, 56, 192, 248, 252, 328, 448, 496, 768, 1016, 1792, 2032, 3240, 6462, 7936, 8128, 11616, 11808, 17412, 20538, 49152, 65528, 114688, 131056, 507904, 524224, 786432, 1048568, 1835008, 2080768, 2096896, 2097136, 3145728, 4194296, 7340032, 8126464, 8388544, 8388592, 32505856, 33292288, 33554176, 33554368, 133169152, 134217472
Offset: 1

Views

Author

Mauro Fiorentini, Jun 17 2015

Keywords

Comments

A pair of integers x and y is called non-unitary amicable if the sum of the non-unitary divisors of either one is equal to the other. Union of A259038 and A259039.
The sequence lists the non-unitary amicable numbers in increasing order. Note that the pairs x, y are not always adjacent to each other in the list. See also A259038 for the x's, A259039 for the y's. The first time a pair is not adjacent is x = 11616, y = 17412 which correspond to a(17) and a(19), respectively.
No other pair below 10^9.
Ligh & Wall showed that if p and q are different Mersenne exponents (A000043) (i.e., 2^p - 1 and 2^q - 1 are Mersenne primes), then 2^(p+1) * (2^q-1) and 2^(q+1) * (2^p-1) is a nonunitary amicable pair. They also found the pairs (252, 328), (3240, 6462), (11616, 17412), (11808, 20538), which are all the known pairs that are not based on Mersenne primes. - Amiram Eldar, Sep 27 2018

Examples

			48 and 56 are in the sequence, as sigma(48)-usigma(48) = 56 and sigma(56)-usigma(56) = 48.

Links

Crossrefs

Subsequence of A013929.
Cf. A000043, A034448, A048146, A259038, A259039.

Programs

A357495 Lesser of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

880, 10480, 20080, 24928, 42976, 69184, 110565, 252080, 267712, 489472, 566656, 569240, 603855, 626535, 631708, 687424, 705088, 741472, 786896, 904365, 1100385, 1234480, 1280790, 1425632, 1749824, 1993750, 2012224, 2401568, 2439712, 2496736, 2542496, 2573344, 2671856
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The larger counterparts are in A357496.
Both members of each pair are necessarily nonsquarefree numbers.

Examples

			880 is a term since s(880) = 1136 and s(1136) = 880.

Links

Crossrefs

Cf. A162296, A325314, A322609, A357493, A357494, A357496.
Subsequence of A013929.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 2, 3*10^6}]; seq

A371419 Lesser member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

Original entry on oeis.org

12, 48, 112, 160, 192, 448, 1984, 12288, 28672, 126976, 196608, 458752, 520192, 786432, 1835008, 2031616, 8126464, 8323072, 33292288, 536805376, 2147221504, 3221225472, 7516192768, 33285996544, 34359476224, 136365211648
Offset: 1

Views

Author

Amiram Eldar, Mar 23 2024

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with the largest aliquot divisor of the sum of divisors (A371418) instead of the sum of aliquot divisors (A001065).
Carmichael (1921) proposed this function (A371418) for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> A371418(x). The chains of cycle 2 are analogous to amicable numbers.
Carmichael noted that if q < p are two different Mersenne exponents (A000043), then 2^(p-1)*(2^q-1) and 2^(q-1)*(2^p-1) are an amicable pair. With the 51 Mersenne exponents that are currently known it is possible to calculate 51 * 50 / 2 = 1275 amicable pairs. (160, 189) is a pair that is not of this "Mersenne form". Are there any other pairs like it? There are no other such pairs with lesser member below a(26).
a(27) <= 8795019280384.
The greater counterparts are in A371420.

Examples

			12 is a term since A371418(12) = 14 > 12, and A371418(14) = 12.

Links

Crossrefs

Cf. A000043, A001065, A002046, A337874, A337876, A371418, A371420.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

Programs

  • Mathematica
    r[n_] := n/FactorInteger[n][[1, 1]]; s[n_] := r[DivisorSigma[1, n]]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 10^6}]; seq
  • PARI
    f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m); for(n = 1, nmax, m = f(n); if(m > n && f(m) == n, print1(n, ", ")));}

A348602 Smaller member of a nonexponential amicable pair: numbers (k, m) such that nesigma(k) = m and nesigma(m) = k, where nesigma(k) is the sum of the nonexponential divisors of k (A160135).

Original entry on oeis.org

198, 18180, 142310, 1077890, 1156870, 1511930, 1669910, 2236570, 2728726, 3776580, 4246130, 4532710, 5123090, 5385310, 6993610, 7288930, 8619765, 8754130, 8826070, 9478910, 10254970, 14426230, 17041010, 17257695, 21448630, 30724694, 34256222, 35361326, 37784810
Offset: 1

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

The larger counterparts are in A348603.

Examples

			198 is a term since A160135(198) = 204 and A160135(204) = 198.

Crossrefs

Cf. A160135, A348601, A348603.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 1.7*10^6}]; seq
Showing 1-6 of 6 results.

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