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 1-10 of 10 results.

A292980 Smaller of bi-unitary amicable pair.

Original entry on oeis.org

114, 594, 1140, 3608, 4698, 5940, 6232, 7704, 9520, 10744, 12285, 13500, 41360, 44772, 46980, 60858, 62100, 67095, 67158, 73360, 79650, 79750, 105976, 118500, 141664, 142310, 177750, 185368, 193392, 217840, 241024, 298188, 308220, 308992, 356408, 399200
Offset: 1

Views

Author

Amiram Eldar, Sep 27 2017

Keywords

Comments

Analogous to amicable numbers with bi-unitary sigma (A188999) instead of sigma (A000203).
Hagis found all the bi-unitary amicable pairs with smaller members below 10^6.
The larger members are in A292981.

Examples

			3608 is in the sequence since A188999(3608) - 3608 = 3952 and A188999(3952) - 3952 = 3608.

Links

Crossrefs

Cf. A002025, A188999, A292981.

Programs

  • Mathematica
    fun[p_,e_]:=If[Mod[e,2]==1,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)];
bsigma[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Do[s = bsigma[n]; If[s > 2 n && bsigma[s - n] == s, Print[n]],{n,1,10000}] (* Amiram Eldar, Sep 29 2018 *)

A324709 Larger of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

Original entry on oeis.org

126, 846, 1260, 8460, 11760, 10856, 14595, 17700, 49308, 83142, 62700, 71145, 73962, 83904, 107550, 88730, 131100, 168730, 149952, 196650, 203432, 306612, 365700, 399592, 419256, 548550, 721962, 669688, 831420, 686072, 691256, 712216, 652664, 661824, 827700
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2019

Keywords

Comments

The terms are ordered according to their lesser counterparts (A324708).

Examples

			126 is in the sequence since it is the larger of the amicable pair (114, 126): tsigma(114) = tsigma(126) = 114 + 126.

Links

Crossrefs

Cf. A002046, A002953, A126170, A292981, A322542, A324706, A324707, A324708.

Programs

  • Mathematica
    f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; tsigma[1]=1; tsigma[n_]:= Times @@ f @@@ FactorInteger[n]; s[n_] := tsigma[n] - n; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, m]] ,{n,1,700000}]; seq

A348344 Larger 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

448, 2032, 8128, 7168, 24384, 41984, 130048, 41940480, 102222432, 221316608, 34359738352
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

The terms are ordered according to their smaller counterparts (A348343).

Examples

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

Crossrefs

Cf. A327633, A348271, A348343.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709.

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, m]], {n,1,10^4}]; seq

A357496 Greater 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

1136, 11696, 22256, 25472, 43424, 73664, 131355, 304336, 267968, 492608, 612704, 674920, 640305, 788697, 691292, 705344, 723392, 813728, 809776, 1117395, 1258335, 1559696, 1518570, 1598368, 1821376, 2218250, 2058944, 2678752, 2744288, 2765024, 2848864, 2610656, 3134224
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The terms are ordered according to their lesser counterparts (A357495).
Both members of each pair are necessarily nonsquarefree numbers.

Examples

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

Links

Crossrefs

Cf. A162296, A325314, A322609, A357493, A357494, A357495.
Subsequence of A013929.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

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, m]], {n, 2, 3*10^6}]; seq

A306868 Larger of augmented bi-unitary amicable pair.

Original entry on oeis.org

871585, 1388145, 1483785, 2479065, 2580105, 4895241, 3830625, 7336455, 9100905, 10350345, 16933105, 9843526, 16367481, 24829945, 15706090, 18653745, 27866241, 21080865, 15439545, 23872185, 24401601, 32263905, 53763535, 63075321, 41337555, 60923577, 90245793
Offset: 1

Views

Author

Amiram Eldar, Mar 14 2019

Keywords

Comments

A pair m < n is an augmented bi-unitary amicable pair if bsigma(m) = bsigma(n) = m + n - 1, where bsigma(n) is the sum of bi-unitary divisors of n (A188999).
The terms are ordered according to their lesser counterparts (A306867).

Examples

			871585 is in the sequence since it is the larger of the amicable pair (434784, 871585): bsigma(434784) = bsigma(871585) = 1306368 = 434784 + 871585 - 1.

Links

Crossrefs

Cf. A015630, A126175, A188999, A292981, A306867.

Programs

  • Mathematica
    fun[p_, e_]:=If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); f[n_] := bsigma[n] - n + 1; s={}; Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, m]], {n, 1, 10^7}]; s

A306871 Larger of reduced bi-unitary amicable pair.

Original entry on oeis.org

2295, 20735, 75495, 1148735, 817479, 774375, 1902215, 1341495, 1348935, 2226014, 2421704, 3123735, 3010215, 5644415, 3894344, 4282215, 4994055, 7509159, 12251679, 10106504, 12900734, 20444319, 24519159, 28206815, 31356314, 33362175, 41950359, 36129375, 43321095
Offset: 1

Views

Author

Amiram Eldar, Mar 14 2019

Keywords

Comments

A pair m < n are a reduced bi-unitary amicable pair if bsigma(m) = bsigma(n) = m + n + 1, where bsigma(n) is the sum of bi-unitary divisors of n (A188999).
The terms are ordered according to their lesser counterparts (A306870).

Examples

			2295 is in the sequence since it is the larger of the amicable pair (2024, 2295): bsigma(2024) = bsigma(2295) = 4320 = 2024 + 2295 + 1.

Links

Crossrefs

Cf. A003503, A126173, A188999, A292981, A306870.

Programs

  • Mathematica
    fun[p_, e_]:=If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); f[n_] := bsigma[n] - n - 1; s={}; Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, m]], {n, 1, 10^7}]; s

A319915 Smallest member of bi-unitary sociable quadruples.

Original entry on oeis.org

162, 1026, 1620, 10098, 10260, 41800, 51282, 100980, 107920, 512820, 1479006, 4612720, 4938136, 14790060, 14800240, 23168840, 28158165, 32440716, 55204500, 81128632, 84392560, 88886448, 209524210, 283604220, 325903500, 498215416, 572062304, 881697520
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2018

Keywords

Comments

The bi-unitary version of A090615.

Examples

			162 is in the sequence since the iterations of the sum of bi-unitary proper divisors function (A188999(n) - n) are cyclic with a period of 4: 162, 174, 186, 198, 162, ... and 162 is the smallest member of the quadruple.

Crossrefs

Cf. A090615, A188999, A292980, A292981, A319902.

Programs

  • Mathematica
    fun[p_, e_]:=If[Mod[e, 2]==1, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)];
bs[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]-n;seq[n_]:=NestList [bs, n,4][[2;;5]] ;aQ[n_] := Module[ {s=seq[n]}, n==Min[s] && Count[s,n]==1]; Do[If[aQ[n],Print[n]],{n,1,10^9}]
  • PARI
    fn(n) = {if (n==1, 1, f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f) - n;);}
isok(n) = my(v = vector(5)); v[1] = n; for(k=2, 5, v[k] = fn(v[k-1])); (v[5] == n) && (vecmin(v) == n) && (#vecsort(v,,8)==4); \\ Michel Marcus, Oct 02 2018
  • PARI
    is(n) = my(c = n); for(i = 1, 3, c = fn(c); if(c <= n, return(0))); c = fn(c); c == n \\ uses Michel Marcus' fn David A. Corneth, Oct 02 2018

A333930 Larger of recursive amicable numbers pair: numbers m < k such that m = s(k) and k = s(m), where s(k) = A333926(k) - k is the sum of proper recursive divisors of k.

Original entry on oeis.org

284, 378, 2924, 4584, 5564, 16632, 16728, 28752, 30912, 53692, 76084, 69552, 87633, 124155, 139815, 179118, 168730, 225096, 202444, 256338, 245904, 266568, 365084, 389924, 320016, 430402, 391656, 353616, 387720, 393528, 486178, 525915, 555216, 642720, 814698, 682896
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2020

Keywords

Comments

The terms are ordered according to their lesser counterparts (A333929).

Examples

			284 is a terms since A333926(284) - 284 = 220 and A333926(220) - 220 = 284.

Links

Crossrefs

Cf. A333926, A333927, A333929.
Analogous sequences: A002046, A002953 (unitary), A126166 (exponential), A126170 (infinitary), A292981 (bi-unitary).

Programs

  • Mathematica
    recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); s[n_] := recDivSum[n] - n; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 1, 10^5}]; seq

A371420 Greater 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

14, 62, 124, 189, 254, 508, 2032, 16382, 32764, 131056, 262142, 524284, 524224, 1048574, 2097148, 2097136, 8388592, 8388544, 33554368, 536866816, 2147479552, 4294967294, 8589934588, 34359738352, 34359672832, 137438953408
Offset: 1

Views

Author

Amiram Eldar, Mar 23 2024

Keywords

Comments

The terms are ordered according to their lesser counterparts (A371419).

Examples

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

Links

Crossrefs

Cf. A371418, A371419.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

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, m]], {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(m, ", ")));}

A348603 Larger 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

204, 19332, 168730, 1099390, 1292570, 1598470, 2062570, 2429030, 3077354, 3903012, 4488910, 6135962, 5504110, 5812130, 7158710, 8221598, 9627915, 10893230, 10043690, 11049730, 10273670, 18087818, 19150222, 17578785, 23030090, 32174506, 35997346, 40117714, 39944086
Offset: 1

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

The terms are ordered according to their smaller counterparts (A348602).

Examples

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

Crossrefs

Cf. A160135, A348601, A348602.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

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, m]], {n, 1, 1.7*10^6}]; seq
Showing 1-10 of 10 results.

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