A323329 Lesser of amicable pair m < n defined by t(n) = m and t(m) = n where t(n) = psi(n) - n and psi(n) = A001615(n) is the Dedekind psi function.

1330, 2660, 3850, 5320, 6650, 7700, 10640, 11270, 13300, 14950, 15400, 18550, 19250, 21280, 22540, 26600, 29900, 30800, 33250, 37100, 38500, 42560, 45080, 53200, 59800, 61600, 66500, 73370, 74200, 74750, 77000, 78890, 85120, 90160, 92750, 96250, 106400, 119600

Offset

1,1

Comments

t(n) = psi(n) - n is the sum of aliquot divisors of n, d, such that n/d is squarefree. Penney & Pomerance proposed a problem to show that the "pseudo-aliquot" sequence related to this function is unbounded. This sequence lists number with pseudo-aliquot sequence of cycle 2. The sequence that is analogous to perfect numbers is A033845.

The asymptotic density of the terms relative to the positive integers is zero. See Dimitrov link. - S. I. Dimitrov, Aug 06 2025

All terms are psi-abundant (A387893). Each term in the sequence has the same parity as its larger counterpart in A323330, which follows directly from psi(m) = psi(n) = m + n and the fact that psi(k) is even for k=>3. - S. I. Dimitrov, Sep 16 2025

Links

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

Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025. See p. 2.

David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.

Mathematica

psi[n_] := n*Times@@(1+1/Transpose[FactorInteger[n]][[1]]); t[n_]:= psi[n] - n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s, m]], {m, 1, 120000}]; s

Crossrefs

Cf. A001615, A002025, A033845 (Dedekind psi perfect numbers), A323327, A323328, A323330, A387641, A387643, A387885.

Sequence in context: A250873 A205091 A038854 * A353027 A182371 A252258

Adjacent sequences: A323326 A323327 A323328 * A323330 A323331 A323332

Keyword

nonn

Author

Amiram Eldar, Jan 11 2019

Status

approved