A323329 -id:A323329 - cp's OEIS frontend

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.

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

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}]

A385852 Integers x such that there exist two integers 0

Original entry on oeis.org

79170, 150150, 158340, 161070, 232050, 237510, 300300, 316680, 322140, 395850, 450450, 464100, 468930, 474810, 475020, 483210, 554190, 570570, 600600, 622440, 633360, 641550, 644280, 696150, 712530, 750750, 791700, 805350, 937860, 949620, 950040, 963270, 966420

Offset: 1

S. I. Dimitrov, Aug 07 2025

The numbers x, y and z form a psi-amicable triple according to Dimitrov's definition.

			79170 is in the sequence since psi(79170) = psi(80850) = psi(81900) = 241920 = 79170 + 80850 + 81900. Other examples: (161070, 161070, 161700), (7063980, 7112490, 7112490).

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

Cf. A001615, A125490, A323329, A386901, A386933.

A386901 Integers y such that there exist two integers 0

Original entry on oeis.org

80850, 158340, 161070, 161700, 232050, 242550, 316680, 322140, 323400, 404250, 464100, 474810, 475020, 483210, 485100, 485940, 565950, 633360, 641550, 644280, 646800, 662340, 696150, 727650, 791700, 805350, 808500, 963270, 966420, 967890, 970200, 971880

Offset: 1

S. I. Dimitrov, Aug 07 2025

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

			158340 is in the sequence since psi(150150) = psi(158340) = psi(175350) = 483840 = 150150 + 158340 + 175350. Other examples: (232050, 232050, 261660), (7091700, 7098630, 7098630).

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

Cf. A001615, A125491, A323329, A385852, A386933.

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A385852 Integers x such that there exist two integers 0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A386901 Integers y such that there exist two integers 0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica