cp's OEIS Frontend

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.

%I A323329 #38 Aug 09 2025 10:10:18
%S A323329 1330,2660,3850,5320,6650,7700,10640,11270,13300,14950,15400,18550,
%T A323329 19250,21280,22540,26600,29900,30800,33250,37100,38500,42560,45080,
%U A323329 53200,59800,61600,66500,73370,74200,74750,77000,78890,85120,90160,92750,96250,106400,119600
%N 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.
%C A323329 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.
%C A323329 The asymptotic density of the terms relative to the positive integers is zero. See Dimitrov link. - _S. I. Dimitrov_, Aug 06 2025
%H A323329 Amiram Eldar, <a href="/A323329/b323329.txt">Table of n, a(n) for n = 1..1000</a>
%H A323329 Kevin Brown and Charles Vanden Eynden, <a href="https://www.jstor.org/stable/2974888">Pseudo-aliquot Sequences, Solution to Problem 10323</a>, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
%H A323329 S. I. Dimitrov, <a href="https://arxiv.org/abs/2508.02318">On psi-amicable numbers and their generalizations</a>, arXiv:2508.02318 [math.NT], 2025. See p. 2.
%H A323329 David E. Penney and Carl Pomerance, <a href="https://www.jstor.org/stable/10.2307/2323896">Problem 10323</a>, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.
%t A323329 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
%Y A323329 Cf. A001615, A002025, A033845 (Dedekind psi perfect numbers), A323327, A323328, A323330.
%K A323329 nonn
%O A323329 1,1
%A A323329 _Amiram Eldar_, Jan 11 2019