A323330 -id:A323330 - 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

A386933 Integers z such that there exist two integers 0

Original entry on oeis.org

81900, 161700, 163800, 175350, 245700, 261660, 323400, 327600, 350700, 409500, 485100, 490770, 491400, 499380, 523320, 526050, 526260, 573300, 646800, 647010, 655200, 671370, 701400, 702450, 737100, 784980, 808500, 819000, 876750, 970200, 971880, 981540, 982800, 990150, 998760

Offset: 1

S. I. Dimitrov, Aug 09 2025

The numbers x, y and z form a psi-amicable triple.

			163800 is in the sequence since psi(158340) = psi(161700) = psi(163800) = 564480 = 158340 + 161700 + 163800. Other examples: (322140, 322140, 323400), (14127960, 14224980, 14224980).

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

Cf. A001615, A125492, A323330, A385852, A386901.

A332329 Decimal expansion of the next-to-least positive zero of the 4th Maclaurin polynomial of cos x.

Original entry on oeis.org

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

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 x^(2n)/(2n)!.

Let z(n) be the next-to-least positive zero of p(2n,x) if there is such a zero. The limit of z(n) is 3 Pi/2 = 4.7123889..., as in A197723.

			Next-to-least positive zero = 3.0763780026417030969660258639367224

Cf. A197723, A323326, A323330, A323331.

z = 150; p[n_, x_] := Normal[Series[Cos[x], {x, 0, n}]]
t = x /. NSolve[p[4, x] == 0, x, z][[4]]
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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A386933 Integers z such that there exist two integers 0

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A332329 Decimal expansion of the next-to-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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