A323328 -id:A323328 - 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.

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

A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

Original entry on oeis.org

222, 228, 280, 360, 585, 546, 672, 1008, 1612, 1568, 1197, 1040, 1302, 1536, 2046, 2304, 949, 518, 456, 600, 930, 1152, 1105, 756, 1120, 1512, 2400, 3906, 4992, 7140, 12096, 20320, 24192, 40800, 70308, 108416, 135660, 241920, 490560, 902208, 1235456, 1309440, 2354688

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

222 is the least number k for which the repeated iterations of the mapping k -> A371418(k) seem to generate an unbounded sequence.

			a(1) = 222 by definition.
a(2) = A371418(a(1)) = A371418(222) = 228.
a(3) = A371418(a(2)) = A371418(228) = 280.

Amiram Eldar, Table of n, a(n) for n = 1..1389
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371421, A371422.

Similar sequences: A008892, A323328, A361421.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]]; NestList[f, 222, 60]

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m = 222); for(n = 1, nmax, print1(m, ", "); m = f(m));}

A332326 Decimal expansion of the least positive zero of the 4th Maclaurin polynomial of cos x.

Original entry on oeis.org

1, 5, 9, 2, 4, 5, 0, 4, 3, 4, 0, 3, 6, 2, 5, 1, 3, 8, 1, 6, 6, 8, 9, 9, 8, 6, 7, 0, 4, 8, 4, 0, 0, 1, 9, 6, 9, 6, 5, 9, 5, 5, 0, 5, 6, 2, 7, 0, 7, 2, 2, 1, 7, 1, 8, 2, 1, 7, 6, 8, 6, 4, 5, 5, 1, 7, 7, 5, 6, 6, 8, 0, 7, 6, 2, 1, 2, 2, 5, 3, 4, 1, 3, 2, 9, 1

Offset: 1

Clark Kimberling, Feb 11 2020

The Maclaurin polynomial p(2n,x) of cos x is 1 - x^2/2! + x^4/4! + ... + (-1)^n ^(2n)/(2n)!.

Let z(n) be the least positive zero of p(2n,x). The limit of z(n) is Pi/2 = 1.570796326..., as in A019669.

			Least positive zero = 1.592450434036251381668998670484001969...

Cf. A019669, A332325, A332327, A323328, A323329.

z = 150; p[n_, x_] := Normal[Series[Cos[x], {x, 0, n}]]
t = x /. NSolve[p[4, x] == 0, x, z][[3]]
u = RealDigits[t][[1]]
Plot[Evaluate[p[4, x]], {x, -1, 4}]

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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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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A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

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A332326 Decimal expansion of the least positive zero of the 4th Maclaurin polynomial of cos x.

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

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