Search: a259038 -id:a259038
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A259039
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Larger of a non-unitary amicable pair.
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+10
6
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56, 248, 328, 496, 1016, 2032, 6462, 8128, 17412, 20538, 65528, 131056, 524224, 1048568, 2097136, 2096896, 4194296, 8388592, 8388544, 33554368, 33554176, 134217472, 2147467264, 8589918208
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OFFSET
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1,1
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COMMENTS
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The elements 2097136, 8388592, etc. are intentionally out of numerical order so that a(n) and A259038(n) form an amicable pair.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A348343
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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).
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+10
4
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336, 1792, 5376, 6096, 21504, 32004, 97536, 34062336, 64512000, 118008576, 30064771072
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OFFSET
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1,1
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COMMENTS
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The larger counterparts are in A348344.
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LINKS
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EXAMPLE
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A259037
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Non-unitary amicable numbers.
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+10
3
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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48 and 56 are in the sequence, as sigma(48)-usigma(48) = 56 and sigma(56)-usigma(56) = 48.
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PROG
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(PARI) A048146(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A357495
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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.
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+10
3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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880 is a term since s(880) = 1136 and s(1136) = 880.
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A371419
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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).
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+10
2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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MATHEMATICA
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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
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PROG
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(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, ", "))); }
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A348602
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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).
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+10
1
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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
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OFFSET
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1,1
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COMMENTS
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The larger counterparts are in A348603.
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LINKS
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EXAMPLE
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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