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

Author

Amiram Eldar, Jan 11 2019

Keywords

• nonn

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

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.

Crossrefs

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

Programs

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