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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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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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