A053670 -id:A053670 - cp's OEIS frontend

A065091 Odd primes.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Labos Elemer, Nov 12 2001

Rayes et al. prove that the a(n)-th Chebyshev-T polynomial, divided by x, is irreducible over the integers.
Odd primes can be written as a sum of no more than two consecutive positive integers. Powers of 2 do not have a representation as a sum of k consecutive positive integers (other than the trivial n=n, for k=1). See A111774. - Jaap Spies, Jan 04 2007
Intersection of A005408 and A000040. - Reinhard Zumkeller, Oct 14 2008
Primes which are the sum of two consecutive numbers. - Juri-Stepan Gerasimov, Nov 07 2009
The arithmetic mean of divisors of p^3, (1+p)(1+p^2)/4, for odd primes p is an integer. - Ctibor O. Zizka, Oct 20 2009
Primes == -+ 1 (mod 4). - Juri-Stepan Gerasimov, Apr 27 2010
a(n) = A053670(A179675(n)) and a(n) <> A053670(m) for m < A179675(n). - Reinhard Zumkeller, Jul 23 2010
Triads of the form <2*a(n+1), a(n+1), 3*a(n+1)> like <6,3,9>, <10,5,15>, <14,7,21> appear in the EKG sequence A064413, see Theorem (3) there. - Paul Curtz, Feb 13 2011.
Complement of A065090; abs(A151763(a(n))) = 1. - Reinhard Zumkeller, Oct 06 2011
Right edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012
Numbers with two odd divisors. - Omar E. Pol, Mar 24 2012
Odd prime p divides some (2^k + 1) or (2^k - 1), (k>0, minimal, cf. A003558) depending on the parity of A179480((p+1)/2) = r. This is a consequence of the Quasi-order theorem and corollaries, [Hilton and Pederson, pp. 260-264]: 2^k == (-1)^r mod b, b odd; and b divides 2^k - (-1)^r, where p is a subset of b. - Gary W. Adamson, Aug 26 2012
Subset of the arithmetic numbers (A003601). - Wesley Ivan Hurt, Sep 27 2013
Odd primes p satisfy the identity: p = (product(2*cos((2*k+1)*Pi/(2*p)), k=0..(p-3)/2))^2.  This follows from C(2*p, 0) = (-1)^((p-1)/2)*p, n>=2, with the minimal polynomial C(k,x) of rho(k) := 2*cos(Pi/k). See A187360 for C and the W. Lang link on the field Q(rho(n)), eqs. (20) and (37). - Wolfdieter Lang, Oct 23 2013
Numbers m > 1 such that m^2 divides (2m-1)!! + m. - Thomas Ordowski, Nov 28 2014
Numbers m such that m divides 2*(m-3)! + 1. - Thomas Ordowski, Jun 20 2015
Numbers m such that (2m-3)!! == m (mod m^2). - Thomas Ordowski, Jul 24 2016
Odd numbers m such that ((m-3)!!)^2 == +-1 (mod m). - Thomas Ordowski, Jul 27 2016
Primes of the form x^2 - y^2. - Thomas Ordowski, Feb 27 2017
Conjecture: a(n) is the smallest odd number m > prime(n) such that Sum_{k=1..prime(n)-1} k^(m-1) == prime(n)-1 (mod m). This is an extension of the Agoh-Giuga conjecture. - Thomas Ordowski, Aug 01 2018
Numbers k > 1 such that either Phi(k,x) == 1 (mod k) or Phi(k,x) == k (mod k^2) holds, where Phi(k,x) is the k-th cyclotomic polynomial. - Jianing Song, Aug 02 2018

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, ICM Report, 1998.
M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, Computers & Mathematics with Applications, Volume 50, Issues 8-9, October-November 2005, Pages 1231-1240.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A000040, A033270, union of A002144 and A002145.

Cf. A230953 (boustrophedon transform).

a065091 n = a065091_list !! (n-1)
a065091_list = tail a000040_list  -- Reinhard Zumkeller, Jan 30 2012

[NthPrime(n): n in [2..100]]; // Vincenzo Librandi, Jun 21 2015

A065091 := proc(n) RETURN(ithprime(n+1)) end:

Prime[Range[2, 33]] (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)

forprime(p=3, 200, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

from sympy import prime
def A065091(n): return prime(n+1) # Chai Wah Wu, Jul 13 2024

def A065091_list(limit):  # after Minác's formula
    f = 3; P = [f]
    for n in range(3, limit, 2):
        if (f+1)>n*(f//n)+1: P.append(n)
        f = f*n
    return P
A065091_list(100)  # Peter Luschny, Oct 17 2013

a(n) = A000040(n+1). - M. F. Hasler, Oct 26 2013

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 05 2002

Edited (moved contributions from A000040 to here) by M. F. Hasler, Oct 26 2013

A053669 Smallest prime not dividing n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Feb 15 2000

Smallest prime coprime to n.
Smallest k >= 2 coprime to n.
a(#(p-1)) = a(A034386(p-1)) = p is the first appearance of prime p in sequence.
a(A005408(n)) = 2; for n > 2: a(n) = A112484(n,1). - Reinhard Zumkeller, Sep 23 2011
Average value is 2.920050977316134... = A249270. - Charles R Greathouse IV, Nov 02 2013
Differs from A236454, "smallest number not dividing n^2", for the first time at n=210, where a(210)=11 while A236454(210)=8. A235921 lists all n for which a(n) differs from A236454. - Antti Karttunen, Jan 26 2014
For k >= 0, a(A002110(k)) is the first occurrence of p = prime(k+1). Thereafter p occurs whenever A007947(n) = A002110(k). Thus every prime appears in this sequence infinitely many times. - David James Sycamore, Dec 04 2024

			a(60) = 7, since all primes smaller than 7 divide 60 but 7 does not.
a(90) = a(120) = a(150) = a(180) = 7 because 90,120,150,180 all have same squarefree kernel = 30 = A002110(3), and 7 is the smallest prime which does not divide 30. - _David James Sycamore_, Dec 04 2024

T. D. Noe, Table of n, a(n) for n = 1..10000
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73; ResearchGate link.
James Grime and Brady Haran, 2.920050977316, Numberphile video (2020)
Igor Rivin, Geodesics with one self-intersection, and other stories, Advances in Mathematics, Vol. 231, No. 5 (2012), pp. 2391-2412; arXiv preprint, arXiv:0901.2543 [math.GT], 2009-2011.

One more than A071222(n-1).
Cf. A000040, A020639, A053670, A053671, A053672, A053673, A053674, A055874, A079578, A087560, A096014, A235921, A236454, A249270, A257993 (the indices of these primes), A276086, A324895, A353526, A353528, A353529, A358754, A358755, A370125.
Cf. A002110, A007947.

a053669 n = head $ dropWhile ((== 0) . (mod n)) a000040_list
-- Reinhard Zumkeller, Nov 11 2012

f:= proc(n) local p;
p:= 2;
while n mod p = 0 do p:= nextprime(p) od:
p
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2016

Table[k := 1; While[Not[GCD[n, Prime[k]] == 1], k++ ]; Prime[k], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
With[{prs=Prime[Range[10]]},Flatten[Table[Select[prs,!Divisible[ n,#]&,1],{n,110}]]] (* Harvey P. Dale, May 03 2012 *)

a(n)=forprime(p=2,,if(n%p,return(p))) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import nextprime
def a(n):
    p = 2
    while True:
        if n%p: return p
        else: p=nextprime(p) # Indranil Ghosh, May 12 2017

# using standard library functions only
import math
def a(n):
    k = 2
    while math.gcd(n,k) > 1: k += 1
    return k # Ely Golden, Nov 26 2020

(define (A053669 n) (let loop ((i 1)) (cond ((zero? (modulo n (A000040 i))) (loop (+ i 1))) (else (A000040 i))))) ;; Antti Karttunen, Jan 26 2014

a(n) = A071222(n-1)+1. [Because the right hand side computes the smallest k >= 2 such that gcd(n,k) = gcd(n-1,k-1) which is equal to the smallest k >= 2 coprime to n] - Antti Karttunen, Jan 26 2014
a(n) = 1 + Sum_{k=1..n}(floor((n^k)/k!)-floor(((n^k)-1)/k!)) = 2 + Sum_{k=1..n} A001223(k)*( floor(n/A002110(k))-floor((n-1)/A002110(k)) ). - Anthony Browne, May 11 2016
a(n!) = A151800(n). - Anthony Browne, May 11 2016
a(2k+1) = 2. - Bernard Schott, Jun 03 2019
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A249270. - Amiram Eldar, Oct 29 2020
a(n) = A000040(A257993(n)) = A020639(A276086(n)) = A276086(n) / A324895(n). - Antti Karttunen, Apr 24 2022
a(n) << log n. For every e > 0, there is some N such that for all n > N, a(n) < (1 + e)*log n. - Charles R Greathouse IV, Dec 03 2022
A007947(n) = A002110(k) ==> a(n) = prime(k+1). - David James Sycamore, Dec 04 2024

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000 and James Sellers, Feb 22 2000

Entry revised by David W. Wilson, Nov 25 2006

A280302 Smallest k such that (n+1)^k - n^k is divisible by a square > 1.

Original entry on oeis.org

6, 10, 4, 2, 21, 20, 3, 20, 33, 6, 20, 2, 2, 5, 21, 6, 10, 6, 6, 4, 4, 2, 7, 2, 6, 3, 10, 4, 18, 6, 2, 10, 20, 6, 57, 17, 2, 14, 42, 2, 10, 10, 6, 39, 14, 4, 10, 20, 2, 21, 20, 6, 4, 21, 6, 20, 10, 2, 5, 2, 5, 2, 20, 6, 42, 14, 2, 6, 55, 6, 3, 7, 2, 42, 3, 2

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2016

nonn

a(209) > 70.

a(n) <= p^2 - p, where p = A053670(n). - Jinyuan Wang, May 15 2020

			a(1) = 6 is because (1+1)^6 - 1^6 = 63 is divisible by 9 = 3^2.

Lars Blomberg, Table of n, a(n) for n = 1..208

Cf. A049094, A053670, A237043, A280203, A280208, A280209, A280296, A282176, A282177, A280307.

Cf. A289629, A280547, A289985.

a(n) = {my(k = 1); while (issquarefree((n+1)^k - n^k), k++); k;} \\ Michel Marcus, Jan 14 2017

More terms from Lars Blomberg, Jan 10 2017

A090093 a(n) is the smallest composite number coprime to n, n+1, n+2, n+3.

Original entry on oeis.org

25, 49, 49, 121, 121, 25, 121, 49, 49, 49, 25, 121, 121, 121, 49, 25, 49, 121, 169, 169, 25, 49, 49, 49, 121, 25, 121, 121, 49, 49, 25, 169, 169, 121, 121, 25, 49, 49, 121, 121, 25, 169, 49, 49, 49, 25, 121, 121, 121, 49, 25, 49, 169, 169, 169, 25, 49, 49, 49, 121, 25

Offset: 1

Labos Elemer, Nov 26 2003

nonn

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

Cf. A090092, A090094, A090095.

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] &&Equal[s2, 1]&&Equal[s3, 1]&&!PrimeQ[n]&&!Equal[n, 1] &&Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 1000}], {k, 1, 256}]
With[{compnos=Complement[Range[2,500],Prime[Range[PrimePi[500]]]]}, Table[ First[Select[compnos,And@@CoprimeQ[#,n+Range[0,3]]&]],{n,70}]] (* Harvey P. Dale, Aug 10 2011 *)

a(n) = A053672(n)^2.

A118478 a(n) is the smallest m such that m*(m+1) is divisible by the first n prime numbers.

Original entry on oeis.org

1, 2, 5, 14, 209, 714, 714, 62985, 367080, 728364, 64822394, 1306238010, 11182598504, 715041747420, 51913478860880, 454746157008780, 9314160363311804, 261062105979210899, 261062105979210899, 696537082207206753590, 54097844397380813592485, 286495021083846822067820

Offset: 1

Giovanni Resta, May 05 2006

nonn

a(n)*(a(n)+1)/(product of first n primes) = 1, 1, 1, 1, 19, 17, 1, 409, 604, 82, 20951, 229931, 411012, 39080794, 4382914408, ... - Robert G. Wilson v, May 13 2006 [This is now A215021. - N. J. A. Sloane, Aug 02 2012]

			a(8) = 62985 since 62985*62986 = 2*3*5*7*11*13*17*19*409, i.e., it is divisible by the first 8 prime numbers (2,3,..,19).

Jinyuan Wang, Table of n, a(n) for n = 1..30 (terms 1..25 from Robert Gerbicz)

Cf. A002110, A002378, A193314, A215021, A053670, A059958.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a118478 n = (+ 1) . fromJust $ elemIndex 0 $
            map (flip mod (a002110 n)) $ tail a002378_list
-- Reinhard Zumkeller, Jun 14 2015
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import sieve, prime, primorial
from sympy.ntheory.modular import crt
def A118478(n): return 1 if n == 1 else int(min(min(crt((m, (k:=primorial(n))//m), (0, -1))[0], crt((k//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, n//2+1) for d in combinations(sieve.primerange(prime(n)+1), l)))) # Chai Wah Wu, May 31 2022

f[n_] := Block[{k = 1, p = Times @@ Prime@Range@n}, While[ !IntegerQ@ Sqrt[4k*p + 1], k++ ]; Floor@ Sqrt[k*p]]; Array[f, 15] (* Robert G. Wilson v, May 13 2006 *)

P=primes(25);T=1;for(n=1,25,T*=P[n];m=T;for(k=2^(n-1),2^n-1,u=binary(k); a=1;for(i=1,n,if(u[i],a*=P[i]));b=T/a;w=bezout(a,b);if(w[1]<=0,w[1]+=b); c=a*w[1]-1;m=min(m,c);w[1]=b-w[1];if(w[1]<=0,w[1]+=b);c=a*w[1];m=min(m,c)); print1(m,",")) \\ Robert Gerbicz, Aug 24 2006

a(n) = Min_{m | m*(m+1) is divisible by A002110(n)}.

More terms from Robert Gerbicz, Aug 24 2006

A090092 a(n) is the smallest composite number coprime to n, n+1 and n+2.

Original entry on oeis.org

25, 25, 49, 49, 121, 25, 25, 49, 49, 49, 25, 25, 121, 121, 49, 25, 25, 49, 121, 169, 25, 25, 49, 49, 49, 25, 25, 121, 49, 49, 25, 25, 169, 121, 121, 25, 25, 49, 49, 121, 25, 25, 49, 49, 49, 25, 25, 121, 121, 49, 25, 25, 49, 169, 169, 25, 25, 49, 49, 49, 25, 25, 121, 49, 49, 25

Offset: 1

Labos Elemer, Nov 26 2003

nonn

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

Cf. A090093, A090094, A090095.

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]&&Equal[s2, 1] &&!PrimeQ[n]&&!Equal[n, 1] &&Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 1000}], {k, 1, 256}]
nn=200;With[{c=Complement[Range[2,nn],Prime[Range[PrimePi[nn]]]]}, Flatten[ Table[ Select[c,And@@Thread[CoprimeQ[{n,n+1,n+2},#]]&,1],{n,nn}]]](* Harvey P. Dale, Sep 28 2013 *)

a(n) = A053671(n)^2.

A090095 a(n) is the smallest composite number coprime to n, n+1, n+2, n+3, n+4 and n+5.

Original entry on oeis.org

49, 121, 121, 121, 121, 169, 169, 49, 289, 289, 289, 121, 121, 121, 49, 121, 169, 169, 169, 169, 289, 49, 121, 121, 121, 121, 121, 169, 49, 169, 169, 169, 169, 121, 121, 49, 121, 121, 289, 169, 169, 169, 49, 169, 121, 121, 121, 121, 121, 49, 361, 289, 169, 169

Offset: 1

Labos Elemer, Nov 26 2003

nonn

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

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

Cf. A090092, A090093, A090094.

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];s4=GCD[n, k+4];s5=GCD[n, k+5]; If[Equal[s, 1]&&Equal[s1, 1]&&Equal[s2, 1]&&Equal[s3, 1]&& Equal[s4, 1]&&Equal[s5, 1]&&!PrimeQ[n]&&!Equal[n, 1]&&Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 1000}], {k, 1, 256}]
scn[n_]:=Module[{k=4,c=n+Range[0,5]},While[PrimeQ[k]||!AllTrue[c,CoprimeQ[ k,#]&],k++];k]; Array[scn,60] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 30 2020 *)

a(n) = A053674(n)^2.

A089090 a(n) is the smallest composite number coprime to n.

Original entry on oeis.org

4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4

Offset: 1

Labos Elemer, Nov 26 2003

If n is the n-th primorial, then a(n) = prime(n+1)^2.

			n=30: below 30 coprimes to 30 phi(30)=8 numbers are relevant but each 1 or primes; so a(8)>30; the first suitable number is a(30)=49.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A053669, A007978, A053670-A053674, A090091-A090093.

m=0;Table[fla=1;Do[s=GCD[n, k];If[Equal[s, 1]&&!PrimeQ[n]&&!Equal[n, 1]&& Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 130}], {k, 1, 256}]

A089090(n) = forprime(p=2, , if(n%p, return(p*p))); \\ Antti Karttunen, Dec 19 2018

a(n) = A053669(n)^2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} ((p^2*(p-1)/Product_{q prime <= p} q)) = 10.3344588090... . - Amiram Eldar, Jul 25 2022

Offset corrected by Antti Karttunen, Dec 19 2018

A090094 a(n) is the smallest composite number coprime to n, n+1, n+2, n+3 and n+4.

Original entry on oeis.org

49, 49, 121, 121, 121, 121, 169, 49, 49, 289, 289, 121, 121, 121, 49, 49, 121, 169, 169, 169, 169, 49, 49, 121, 121, 121, 121, 121, 49, 49, 169, 169, 169, 121, 121, 49, 49, 121, 121, 169, 169, 169, 49, 49, 121, 121, 121, 121, 121, 49, 49, 289, 169, 169, 169, 121, 49

Offset: 1

Labos Elemer, Nov 26 2003

nonn

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

Cf. A090092, A090093, A090095.

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];s4=GCD[n, k+4];s5=GCD[n, k+5]; If[Equal[s, 1]&&Equal[s1, 1]&&Equal[s2, 1]&&Equal[s3, 1]&& Equal[s4, 1]&&!PrimeQ[n]&&!Equal[n, 1]&&Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 1000}], {k, 1, 256}]
scn[n_]:=Module[{cn=4},While[!CompositeQ[cn]||!And@@CoprimeQ[ Range[ n,n+4], cn], cn++]; cn]; Array[scn,60] (* Harvey P. Dale, Aug 12 2014 *)

a(n) = A053673(n)^2.

A179675 Smallest m such that the n-th odd prime is the smallest number coprime to m and m+1.

Original entry on oeis.org

1, 2, 5, 14, 209, 1364, 714, 62985, 367080, 728364, 64822394, 1306238010, 11182598504, 715041747420, 51913478860880, 454746157008780, 9314160363311804, 560934821217378530, 261062105979210899, 696537082207206753590, 54097844397380813592485, 286495021083846822067820, 80126789479717708423427654

Offset: 1

Reinhard Zumkeller, Jul 23 2010; corrected Jul 27 2010

nonn

A053670(a(n)) = A065091(n) and A053670(m) <> A065091(n) for m < a(n).

			n=4: a(4) = 14, A065091(4) = A053670(14) = 11,
a(4) = 14 = 2*7, a(4)+1 = 15 = 3*5;
n=5: a(5) = 209, A065091(5) = A053670(209) = 13,
a(5) = 209 = 11*19, a(5)+1 = 210 = 2*3*5*7;
n=6: a(6) = 1364, A065091(6) = A053670(1364) = 17,
a(6) = 1364 = 2*2*11*31, a(6)+1 = 1365 = 3*5*7*13.
a(6) is not 714 even though 714 * 715 is divisible by the first 6 primes. It is disqualified as it is also a multiple of prime(6+1) = 17. - _David A. Corneth_, Aug 16 2023

David A. Corneth, Table of n, a(n) for n = 1..42

Cf. A053670, A065091.

With[{s = Table[k = 1; While[! CoprimeQ[n, n + 1, Set[p, Prime@ k]], k++]; p, {n, 10^6}]}, Flatten@ Table[FirstPosition[s, Prime@ n], {n, 2, PrimePi@ Max@ s}]] (* Michael De Vlieger, Aug 01 2017 *)

a(n) = {my(res = oo, pr = primes(n+1), mp = pr[#pr], mpm1 = mp - 1); v = vector(n, i, [-1, 0]); forvec(x = v, c = lift(chinese(vector(n, i, Mod(x[i], pr[i])))); if(0 < c && c < res, if(c % mp != 0 && c % mp != mpm1, res = c))); res} \\ David A. Corneth, Aug 16 2023

a(11)-a(19) from Donovan Johnson, Jul 30 2010

More terms from David A. Corneth, Aug 16 2023

cp's OEIS Frontend

A065091 Odd primes.

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A053669 Smallest prime not dividing n.

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A280302 Smallest k such that (n+1)^k - n^k is divisible by a square > 1.

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A090093 a(n) is the smallest composite number coprime to n, n+1, n+2, n+3.

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A118478 a(n) is the smallest m such that m*(m+1) is divisible by the first n prime numbers.

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A090092 a(n) is the smallest composite number coprime to n, n+1 and n+2.

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A090095 a(n) is the smallest composite number coprime to n, n+1, n+2, n+3, n+4 and n+5.

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