A073758 -id:A073758 - cp's OEIS frontend

A073760 Integers m such that A073758(m) = 4.

6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Labos Elemer, Aug 08 2002

Essentially the same as A016825.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016825, A045763, A073758, A073759.

a(n) = 4*n+2; \\ Altug Alkan, Jul 23 2018

a(n) = 4*n + 2. - Max Alekseyev, Mar 03 2007
a(n) = 8*n - a(n-1), with a(1)=6. - Vincenzo Librandi, Aug 08 2010
a(1)=6, a(2)=10; for n>2, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Mar 06 2012

Definition simplified by Michel Marcus, Jul 26 2018

A073762 a(n) = 24*n - 12.

Original entry on oeis.org

12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 852, 876, 900, 924, 948, 972, 996, 1020, 1044, 1068, 1092, 1116, 1140, 1164, 1188, 1212

Offset: 1

Labos Elemer, Aug 08 2002

Previous name: "Smallest unrelated number belonging to a term of this sequence equals 8."
This is also the list of numbers k such that A259748(k)/k = 5/12. - José María Grau Ribas, Jul 12 2015.
Also the total number of line segments creating a stellated octahedron, where the length of each stellated edge equals n-1, and where the octahedron has 12 edges, each fixed at unit length. - Peter M. Chema, Apr 28 2016

			URSet[12] = {8,9,10} so 12 is here.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016825, A045763, A073758, A073759, A073760, A259748.

[24*n-12: n in [1..60]]; // Vincenzo Librandi, Jun 15 2011

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Do[s=Min[unr[n]]; If[Equal[s, 8], Print[n]], {n, 1, 1000}]
Range[12, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)

a(n)=24*n-12 \\ Charles R Greathouse IV, Jun 14 2011

x='x+O('x^100); Vec(12*(1+x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015

Min{URS[m]} = 8, where UNR[m] = Complement[RRS[m], Divisors[m]].
a(n) = 24*n - 12. - Max Alekseyev, Mar 03 2007
a(n) = 12*A005408(n-1). - Danny Rorabaugh, Oct 22 2015
G.f.: 12*x*(1 + x)/(1 - x)^2. - Ilya Gutkovskiy, Apr 28 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/48. - Amiram Eldar, Feb 28 2023
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 12*(exp(x)*(2*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A073759 Largest number < n that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 6, 6, 8, 0, 10, 0, 12, 12, 14, 0, 16, 0, 18, 18, 20, 0, 22, 20, 24, 24, 26, 0, 28, 0, 30, 30, 32, 30, 34, 0, 36, 36, 38, 0, 40, 0, 42, 42, 44, 0, 46, 42, 48, 48, 50, 0, 52, 50, 54, 54, 56, 0, 58, 0, 60, 60, 62, 60, 64, 0, 66, 66, 68, 0, 70, 0, 72, 72, 74, 70, 76

Offset: 1

Labos Elemer, Aug 15 2002

nonn

Largest "unrelated" number to n.
From Michael De Vlieger, Mar 28 2016: (Start)
Primes n have no unrelated numbers m < n since all such numbers are coprime to n.
Unrelated numbers m must be composite since primes must either divide or be coprime to n.
m = 1 is not counted as unrelated as it divides and is coprime to n.
a(4) = 0 since 4 is the smallest composite and unrelated numbers m with respect to n must be composite and smaller than n. All other composite n have at least one unrelated number m.
The test for unrelated numbers m that belong to n is 1 < gcd(m, n) < m. (End)

			n = 20: unrelated set to 20 = {6,8,12,14,15,16,18}, largest is a(20) = 18.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A045763, A073758.

tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Table[Max[Join[{0}, unr[w]]], {w, 1, 78}]
Table[t = Select[r = Range[n - 1], Divisible[n, #] || GCD[n, #] == 1 &]; Max[Join[{0}, Complement[r, t]]], {n, 78}] (* Jayanta Basu, Jul 09 2013 *)
Table[SelectFirst[Range[n - 2, 2, -1], 1 < GCD[#, n] < # &] /. n_ /; MissingQ@ n -> 0, {n, 100}] (* Michael De Vlieger, Mar 28 2016, Version 10.2 *)

a(n) = {forstep(k=n-2, 1, -1, if ((gcd(n,k) != 1) && (n % k), return (k));); 0;} \\ Michel Marcus, Mar 29 2016

Name clarified by Sean A. Irvine, Dec 18 2024

A073763 Least number of unrelated set belonging to these numbers is odd.

Original entry on oeis.org

24, 48, 96, 120, 168, 192, 240, 264, 312, 336, 384, 408, 456, 480, 528, 552, 600, 624, 672, 696, 744, 768, 816, 840, 888, 912, 960, 984, 1032, 1056, 1104, 1128, 1176, 1200, 1248, 1272, 1320, 1344, 1392, 1416, 1464, 1488, 1536, 1560, 1608, 1632, 1680, 1704

Offset: 1

Labos Elemer, Aug 08 2002

nonn

			n=24: UnrelatedSet[24]={9, 10, 14, 15, 16, 18, 20, 21, 22}, Min=9, so 24 is here. In cases of all solutions (<50000) the odd number was always 9. This is not an accident. Primes are either divisors or primes to n. Thus a term here should be a composite odd number from A071904, whose first entry is 9; so next candidates are 15, 21, 25, 27... While 15 and 21 not [yet] found, prime powers 25 and 27 did arise.
Least odd unrelated number to 55440 is 25 and smallest unrelated (i.e. neither divisor, nor in RRS) to 3603600 is 27.
Question: can be a smallest odd unrelated number be other than a true power of odd prime?
Answer: no.  Proof: Suppose A073758(n) = k is odd and not a prime power.  Let k = g*u where g = gcd(n,k) > 1.  Since k does not divide n, u > 1.  Since 2*g < k is not unrelated to n, it must divide n, so n is even.  Let p be a prime factor of u.  Since 2*p is not unrelated to n, p must divide n.  But then p^d < k is unrelated to n, where p^d is the highest power of p dividing k. - _Robert Israel_, Sep 11 2014

Cf. A045763, A073757-A073762.

Cf. A071904.

A073758:= proc(n) local k;
  for k from 2 to n-2 do
    if igcd(k,n) > 1 and n mod k > 1 then return k fi
  od;
  0
end proc:
select(t -> A073758(t)::odd, [$1..1000]); # Robert Israel, Sep 11 2014

tn[x_] := Table[w, {w, 1, x}] di[x_] := Divisors[x] rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]] nd[x_] := Complement[tn[x], di[x]] rs[x_] := Union[rrs[x], di[x]] urs[x_] := Complement[tn[x], rs[x]] Do[s=Min[urs[n]]; If[OddQ[s], Print[{n, s}]], {n, 1, 10000}]
unQ[n_] := OddQ[Min[Complement[r = Range[n - 1], Select[r, Divisible[n, #] || GCD[n, #] == 1 &]]]]; Select[Range[1710], unQ] (* Jayanta Basu, Jul 09 2013 *)

Solutions to Mod[A073758(x), 2]=1.
Conjecture: a(n) = 36*n - 18 - 6*(-1)^n = 24 * A001651(n). - Ralf Stephan, Oct 19 2013
The conjecture is false, first counterexample being a(1541) = 55440. - Robert Israel, Sep 11 2014

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A073760 Integers m such that A073758(m) = 4.

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A073762 a(n) = 24*n - 12.

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A073759 Largest number < n that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.

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A073763 Least number of unrelated set belonging to these numbers is odd.

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