A053670 -id:A053670 - cp's OEIS frontend

A190911 Least number coprime to n and n+3.

3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 11, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 11, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3

Offset: 1

Jonathan Vos Post, May 26 2011

This is to n+3 as A053670 is to n+1. This is the 3rd row of the array A(k,n) = Least number coprime to n and n+k where A053670 is the 1st row. First n for which a(n) = (3, 5, 7, 11, 13, 17, 19, ...) are A191112 = (1, 3, 12, 42, 165, 3000, 2142, ...).

			a(12) = 7 because the prime factors of 12 and 12+3 = 15 are {2, 3, 5} and the next available prime is 7.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A053669-A053674, A191112.

A190911 := proc(n) local k: for k from 3 by 2 do if(gcd(k,n)=1 and gcd(k,n+3)=1)then return k: fi: od: end: seq(A190911(n),n=1..100); # Nathaniel Johnston, May 26 2011

a(15) corrected by Nathaniel Johnston, May 26 2011

A135376 a(n) is the smallest prime that does not divide n(n+1)/2.

Original entry on oeis.org

2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2, 7, 3, 2, 2, 3, 11, 2, 2, 5, 7, 2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 11, 5, 2, 2, 7, 3, 2, 2, 3, 7, 2, 2, 5, 5, 2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 7, 7, 2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2, 7, 3, 2, 2, 3, 7, 2, 2, 5, 11, 2, 2, 5, 3, 2, 2, 3, 5, 2, 2, 7, 5, 2, 2, 7, 3, 2, 2, 3

Offset: 1

Leroy Quet, Dec 09 2007

nonn

The sums of the first 10^k terms, for k = 1, 2, ..., are 28, 354, 3596, 36026, 360402, 3604134, 36041392, 360413970, 3604140072, 36041400856, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3.604140... . - Amiram Eldar, Sep 10 2022

			The 11th triangular number is 66 = 2*3*11. 5 is the smallest prime that is coprime to 66, so a(11) = 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A053669, A053670.

A135376 := proc(n) local T,p ; T := n*(n+1)/2 ; p := 2 ; while T mod p = 0 do p := nextprime(p) ; od: RETURN(p) ; end: seq(A135376(n),n=1..120) ; # R. J. Mathar, Dec 11 2007

a = {}; For[n = 1, n < 80, n++, j = 1; While[Mod[n*(n + 1)/2, Prime[j]] == 0, j++ ]; AppendTo[a, Prime[j]]]; a (* Stefan Steinerberger, Dec 10 2007 *)
sp[n_]:=Module[{p=2},While[Mod[n,p]==0,p=NextPrime[p]];p]; sp[#]&/@ Accumulate[ Range[110]] (* Harvey P. Dale, Jul 26 2018 *)

a(4n+1) = a(4n+2) = 2 for all nonnegative integers n.
a(n) = A053670(n) for all n congruent to 0 or 3 (mod 4).
a(n) = A053669(A000217(n)). - R. J. Mathar, Dec 11 2007

More terms from Stefan Steinerberger and R. J. Mathar, Dec 10 2007

A089091 a(n) is the smallest composite number coprime to n and n+1.

Original entry on oeis.org

9, 25, 25, 9, 49, 25, 9, 25, 49, 9, 25, 25, 9, 121, 49, 9, 25, 25, 9, 121, 25, 9, 25, 49, 9, 25, 25, 9, 49, 49, 9, 25, 25, 9, 121, 25, 9, 25, 49, 9, 25, 25, 9, 49, 49, 9, 25, 25, 9, 49, 25, 9, 25, 49, 9, 25, 25, 9, 49, 49, 9, 25, 25, 9, 49, 25, 9, 25, 121, 9, 25, 25, 9, 49, 49, 9, 25, 25

Offset: 1

Labos Elemer, Nov 26 2003

Cf. A007978, A053669, A053670, A053671, A053672, A053673, A053674.

Cf. A090092, A090093.

m=0;Table[fla=1;Do[s=GCD[n, k]; s1=GCD[n, k+1];s2=GCD[n, k+2];s3=GCD[n, k+3]; If[Equal[s, 1]&&Equal[s1, 1]&&!PrimeQ[n]&&!Equal[n, 1]&& Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 1000}], {k, 1, 256}]

from math import gcd
def a(n):
    k, m = 3, n*(n+1)
    while gcd(k, m) != 1: k += 2
    return k*k
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Sep 25 2021

a(n) = A053670(n)^2.

Offset corrected by Mohammed Yaseen, Aug 15 2023

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A190911 Least number coprime to n and n+3.

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A135376 a(n) is the smallest prime that does not divide n(n+1)/2.

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A089091 a(n) is the smallest composite number coprime to n and n+1.

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Author

Keywords

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Mathematica

Python

Formula

Extensions