A357496 -id:A357496 - cp's OEIS frontend

A357495 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.

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

Offset: 1

Amiram Eldar, Oct 01 2022

nonn

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.

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

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

Cf. A162296, A325314, A322609, A357493, A357494, A357496.
Subsequence of A013929.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

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

A357497 Nonsquarefree numbers whose harmonic mean of nonsquarefree divisors in an integer.

Original entry on oeis.org

4, 9, 12, 18, 24, 25, 28, 45, 49, 54, 60, 90, 112, 121, 126, 132, 150, 153, 168, 169, 198, 270, 289, 294, 336, 361, 364, 414, 529, 560, 594, 630, 637, 684, 726, 841, 918, 961, 1014, 1140, 1232, 1305, 1350, 1369, 1512, 1521, 1638, 1680, 1681, 1710, 1734, 1849, 1984

Offset: 1

Amiram Eldar, Oct 01 2022

nonn

Analogous to harmonic numbers (A001599) with nonsquarefree divisors.
The squares of primes (A001248) are terms since they have a single nonsquarefree divisor.
If p is a prime then 6*p^2 is a term.

			12 is a term since its nonsquarefree divisors are 4 and 12 and their harmonic mean is 6 which is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A162296, A322609, A357493, A357494, A357495, A357496.
Subsequence of A013929.
Subsequence: A001248.
Similar sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary), A335387 (tri-unitary).

q[n_] := Length[d = Select[Divisors[n], ! SquareFreeQ[#] &]] > 0 && IntegerQ[HarmonicMean[d]]; Select[Range[2000], q]

cp's OEIS Frontend

A357495 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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