A323327 -id:A323327 - cp's OEIS frontend

A323328 Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.

318, 330, 534, 546, 798, 1122, 1470, 2562, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 106074, 111654, 111690, 176022, 266346, 266382, 266490, 480006, 480330, 674406, 740826, 833814, 834138

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

318 is the least number k whose repeated iteration of the mapping k -> A001615(k) - k yields an unbounded sequence. Since t(m^j * n) = m^j * t(n) if m|n, then if in the sequence a_0 = k, a_1 = t(k), a_2 = t(t(k))... there is a term a_{i1} = m^j * a_0 such that m|k and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded. Since a(13)=9870, after 19 iterations a(32) = 27 * 9870, 27 = 3^3 and 3|9870 then a(n+19) = 27 * a(n) for n >= 13.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.

Cf. A001615, A008892, A323327.

t[1] = 0; t[n_] := (Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]) - 1)*n; NestList[t, 318, 40]

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.

Original entry on oeis.org

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

Amiram Eldar, Jan 11 2019

nonn

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

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.

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

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

A323330 Larger 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.

Original entry on oeis.org

1550, 3100, 4790, 6200, 7750, 9580, 12400, 12922, 15500, 15290, 19160, 20330, 23950, 24800, 25844, 31000, 30580, 38320, 38750, 40660, 47900, 49600, 51688, 62000, 61160, 76640, 77500, 82150, 81320, 76450, 95800, 90454, 99200, 103376, 101650, 119750, 124000, 122320

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The terms are ordered according to the order of their lesser counterparts (A323329).

Amiram Eldar, Table of n, a(n) for n = 1..1000
S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025. See p. 2.

Cf. A001615, A002046, A033845 (Dedekind psi perfect numbers), A323327, A323328, 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,n]], {m, 1, 120000}]; s

A323331 Smallest member of sociable quadruples using Dedekind psi function (A001615).

Original entry on oeis.org

11398670, 22797340, 38369450, 45594680, 56993350, 59334310, 76738900, 91189360, 113986700, 118668620, 153477800, 182378720, 209524210, 227973400, 237337240, 268586150, 284966750, 306955600, 364757440, 419048420, 455946800, 474674480, 537172300, 539867650, 569933500

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

Numbers k whose iterations of k -> A001615(k) - k are cyclic with a period of 4, and in each cyclic quadruple k is the least of the 4 members.

			11398670 is in the sequence since the iterations of k -> A001615(k) - k are cyclic with a period of 4: 11398670, 11475730, 12474350, 14093650, 11398670, ... and 11398670 is the smallest member of the quadruple.

Cf. A001615, A090615, A319902, A319915, A323327, A323328, A323329, A323330.

t[0]=0; t[1]=0; t[n_]:=(Times@@(1+1/Transpose[FactorInteger[n]][[1]])-1)*n;
seq[n_]:=NestList [t, n, 4][[2;; 5]] ; aQ[n_] := Module[ {s=seq[n]}, n==Min[s] && Count[s, n]==1]; s={}; Do[If[aQ[n], AppendTo[s, n]], {n, 1, 10^9}]; s

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A323328 Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.

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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.

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A323330 Larger 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.

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A323331 Smallest member of sociable quadruples using Dedekind psi function (A001615).

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