A259039 -id:A259039 - cp's OEIS frontend

A259038 Smaller of a non-unitary amicable pair.

48, 192, 252, 448, 768, 1792, 3240, 7936, 11616, 11808, 49152, 114688, 507904, 786432, 1835008, 2080768, 3145728, 7340032, 8126464, 32505856, 33292288, 133169152, 2147221504, 8588886016

Offset: 1

Mauro Fiorentini, Jun 17 2015

This sequence is strictly increasing (and A259039, which contains the larger number in each pair, is sorted by this sequence).

Cf. A034448, A259037, A259039.

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

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

Amiram Eldar, Oct 13 2021

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

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

Cf. A327633, A348271, A348343.

Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709.

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

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

Mauro Fiorentini, Jun 17 2015

nonn

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

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..48
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Subsequence of A013929.

Cf. A000043, A034448, A048146, A259038, A259039.

A048146(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1)
is(n)=my(k=A048146(n)); k>1 && k!=n && A048146(k)==n \\ Charles R Greathouse IV, Jun 17 2015

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

Amiram Eldar, Oct 01 2022

nonn

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

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

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

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

Amiram Eldar, Mar 23 2024

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

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

Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371419.

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

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

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

Amiram Eldar, Oct 25 2021

nonn

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

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

Cf. A160135, A348601, A348602.

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

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

cp's OEIS Frontend

A259038 Smaller of a non-unitary amicable pair.

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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).

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A259037 Non-unitary amicable numbers.

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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.

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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).

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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).

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