A087966 -id:A087966 - cp's OEIS frontend

A364309 Numbers k such that k, k+1 and k+2 have exactly 4 distinct prime factors.

37960, 44484, 45694, 50140, 51428, 55130, 55384, 61334, 63364, 64294, 67164, 68264, 68474, 70004, 70090, 71708, 72708, 76152, 80444, 81548, 81718, 82040, 84434, 85490, 86240, 90363, 95380, 97382, 98020, 99084, 99384, 99428, 99788, 100164, 100490, 100594, 102254, 102542, 104804, 105994, 108204

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			37960 = 2^3*5*13*73, 37961 = 7*11*17*29, and 37962 = 2*3^3*19*37 each have 4 distinct prime factors, so 37960 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Subsequence of A006073 and of A140078.
A176167 is a subsequence.
Cf. A364307 (2 factors), A364308 (3 factors), A364266 (5 factors), A364265 (6 factors), A001221, A087966, A168628.

q[n_] := q[n] = PrimeNu[n] == 4; Select[Range[10^5], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A087966(3).
a(n)+1 = A168628(n).
{k: A001221(k) = A001221(k+1) = A001221(k+2) = 4}.

A087967 Integers m such that gcd(2^m-1, m^2) is not a square number.

Original entry on oeis.org

18, 36, 72, 90, 100, 126, 144, 147, 162, 180, 198, 200, 234, 252, 288, 294, 300, 306, 324, 342, 360, 396, 400, 414, 441, 450, 468, 504, 522, 558, 576, 588, 600, 612, 630, 648, 666, 684, 700, 720, 735, 738, 774, 792, 800, 810, 828, 846, 882, 900, 936, 954

Offset: 1

Labos Elemer, Sep 22 2003

nonn

			For m = 18: gcd(262143, 324) = 27 is not a square number, so 18 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A087966, A087969.

Select[Range[1000], !IntegerQ[Sqrt[GCD[2^#-1, #^2]]] &]

isok(m) = !issquare(gcd(2^m-1, m^2)); \\ Michel Marcus, Aug 27 2019

A087968 a(n) = gcd(1 + 2^n, n^2).

Original entry on oeis.org

1, 1, 9, 1, 1, 1, 1, 1, 27, 25, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 81, 1, 1, 25, 1, 1, 9, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 125, 9, 1, 1, 1, 121, 1, 9, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 289, 9, 25, 1, 1, 1, 1, 9, 1, 1, 169, 1, 1, 243, 1, 1, 1, 1, 1, 9, 1, 1, 25, 1, 1, 9, 1, 1, 1

Offset: 1

Labos Elemer, Sep 22 2003

nonn

			a(9) = gcd(2^9 + 1,9^2) = gcd(513,81) = 27.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A087966, A087969.

List([1..100],n->Gcd(2^n+1,n^2)); # Muniru A Asiru, Sep 22 2018

seq(gcd(2^n+1,n^2),n=1..100); # Muniru A Asiru, Sep 22 2018

A087968(n) = gcd((2^n)+1, n^2); \\ Antti Karttunen, Sep 22 2018

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A364309 Numbers k such that k, k+1 and k+2 have exactly 4 distinct prime factors.

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A087967 Integers m such that gcd(2^m-1, m^2) is not a square number.

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A087968 a(n) = gcd(1 + 2^n, n^2).

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