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).

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

Author

Amiram Eldar, Mar 23 2024

Keywords

• nonn
• more

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

• 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.

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, ", ")));}