A180871 -id:A180871 - cp's OEIS frontend

A007996 Primes that divide at least one term of Sylvester's sequence s = A000058: s(n+1) = s(n)^2 - s(n) + 1, s(0) = 2.

2, 3, 7, 13, 43, 73, 139, 181, 547, 607, 1033, 1171, 1459, 1861, 1987, 2029, 2287, 2437, 4219, 4519, 6469, 7603, 8221, 9829, 12763, 13147, 13291, 13999, 15373, 17881, 17977, 19597, 20161, 20479, 20641, 20857, 20929, 21661, 23689, 23773, 27031

Offset: 1

Bennett Battaile (bennett.battaile(AT)autodesk.com)

nonn

Or, let S_1 = [2] and let S_{n+1} = list formed by sorting the union of S_n together with all prime factors of 1 + Product_i S_n(i) into increasing order; sequence is limit as n -> infinity of S_n.
Prime divisors of the terms of Sylvester's sequence A000058. - Max Alekseyev, Jan 03 2004. Also of A007018. - N. J. A. Sloane, Jan 27 2007
Because all terms of the sequence s(n) are coprime, a prime can divide at most one term. Odoni shows that primes p > 3 in this sequence must satisfy p = 1 (mod 6). - T. D. Noe, Sep 25 2010
See A180871(n) for the index of the first term of A000058 (this is one less than the index of the s-sequence) divisible by a(n). - M. F. Hasler, Apr 24 2014

Max Alekseyev, Table of n, a(n) for n = 1..12046 (first 8181 terms are also given at the Andersen link)
J. K. Andersen, Factorization of Sylvester's sequence
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
R. W. K. Odoni, On the prime divisors of the sequence w_{n+1} = 1+w_1 ... w_n, J. London Math. Soc. 32 (1985), 1-11.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 2006
Eric Weisstein's World of Mathematics, Sylvester's sequence

The missing primes form A096264.
Cf. A014546, A091335, A091336.
Cf. A180871 (k such that a(n) divides A000058(k)).
Cf. A323605 (smallest prime dividing A000058(n)).

n := 1; for p do if isprime(p) then x := 2 mod p; S := {}; while not member(x,S) do if x=0 then a[n] := p; n := n+1; break; fi; S := S union {x}; x := (x^2-x+1) mod p; od; fi; od;

t={}; p=1; While[Length[t]<100, p=NextPrime[p]; s=Mod[2,p]; k=0; modSet={}; While[s>0 && !MemberQ[modSet,s], AppendTo[modSet,s]; k++; s=Mod[s^2-s+1,p]]; If[s==0, AppendTo[t,{p,k}]]]; Transpose[t][[1]] (* T. D. Noe, Sep 25 2010 *)

is(n)=my(k=Mod(2,n)); for(i=1, n, k=(k-1)*k+1; if(k==0, return(isprime(n)))); n==2 \\ Charles R Greathouse IV, Sep 30 2015

More terms from Max Alekseyev, Jan 03 2004
Entry revised by N. J. A. Sloane, Jan 28 2007
Definition corrected (following a remark by Don Reble) by M. F. Hasler, Apr 24 2014

A362250 Primes dividing terms of A231831.

Original entry on oeis.org

3, 5, 7, 11, 19, 23, 89, 101, 137, 157, 211, 373, 659, 877, 881, 1399, 1597, 1627, 1663, 1811, 2029, 2069, 2087, 2153, 2381, 2677, 2939, 3433, 3491, 3511, 3617, 3673, 4111, 4127, 4547, 4721, 5059, 5483, 6529, 6793, 6827, 7757, 8209, 8297, 8677, 9203, 9463, 9811, 10139, 10159, 11321

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Since the terms of A231831 are pairwise coprime, each prime divides at most one term of A231831. Indices of the corresponding terms are listed in A362251, and so a(n) divides A231831(A362251(n)).

Max Alekseyev, Table of n, a(n) for n = 1..16944
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Cf. A000058, A007996, A180871, A231830, A231831, A362251, A362252, A362253.

A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 9, 4, 4, 8, 3, 31, 12, 7, 7, 9, 44, 8, 22, 29, 36, 37, 8, 21, 5, 26, 4, 20, 24, 12, 76, 30, 5, 47, 5, 13, 9, 25, 6, 41, 51, 9, 53, 6, 27, 39, 5, 12, 78, 64, 10, 185, 113, 205, 91, 85, 43, 195, 32, 117, 20, 133, 142, 119, 64, 70, 199, 41, 125, 79, 243, 70, 35, 105, 67, 156

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..16944

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362252, A362253.

A362252 Primes dividing terms of A231830.

Original entry on oeis.org

5, 53, 89, 101, 373, 877, 1109, 1181, 1597, 1613, 2029, 2069, 2153, 2213, 2381, 2741, 3617, 4273, 6529, 6737, 7417, 7717, 11321, 12653, 13009, 13309, 16829, 17729, 23581, 23993, 25373, 32569, 33353, 33857, 34841, 35053, 36097, 37201, 38609, 41513, 42461, 48661, 55829, 58369, 59093, 63281

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Since the terms of A231830 are pairwise coprime, each prime divides at most one term of A231830. Indices of the corresponding terms are listed in A362253, and so a(n) divides A231830(A362253(n)).

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362253.

A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

Original entry on oeis.org

1, 4, 7, 2, 19, 25, 30, 38, 45, 4, 26, 33, 27, 46, 10, 59, 102, 38, 84, 37, 22, 77, 80, 37, 240, 57, 45, 240, 173, 38, 41, 100, 88, 44, 114, 39, 63, 24, 14, 121, 177, 12, 155, 270, 65, 109, 44, 391, 54, 22, 96, 320, 194, 347, 182, 226, 143, 290, 105, 135, 29, 198, 113, 302, 572, 53, 692, 168, 366

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362252.

A323605 Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).

Original entry on oeis.org

2, 3, 7, 43, 13, 3263443, 547, 29881, 5295435634831, 181, 2287, 73

Offset: 0

Robert FERREOL, Jan 19 2019

a(n) is also the smallest prime divisor of A007018(n+1) that is not a divisor of A007018(n).
The prime numbers a(n) are all distinct, which proves the infinitude of the prime numbers (Saidak's proof).
a(12) <= 2589377038614498251653. - Daniel Suteu, Jan 20 2019
a(12)..a(50) = [?, 52387, 13999, 17881, 128551, 635263, ?, ?, 352867, 387347773, ?, 74587, ?, ?, 27061, 164299, 20929, 1171, ?, 1679143, ?, ?, 120823, 2408563, 38218903, 333457, 30241, 4219, 1085443, 7603, 1861, ?, 23773, 51769, 1285540933, 429547, ?, 8323570543, ?], where ? denote unknown values > 10^10. - Max Alekseyev, Oct 11 2023

Filip Saidak, A new proof of Euclid's theorem, Amer. Math. Monthly, 113:10 (2006) 937-938.
Wikipedia, Sylvester's sequence: Divisibility and factorizations.

Cf. A007018, A000058, A007996, A091335, A180871.

with(numtheory):
u:=1: P:=NULL: to 9 do P:=P,sort([op(divisors(u+1))])[2]: u:=u*(u+1) od:
P;

f(n)=if(n<1, n>=0, f(n-1)+f(n-1)^2); \\ A007018
a(n)=divisors(f(n)+1)[2]; \\ Michel Marcus, Jan 20 2019

a(n) = A007996(m), where m is the smallest index such that A180871(m) = n. - Max Alekseyev, Oct 11 2023

a(10)-a(11) from Daniel Suteu, Jan 20 2019

cp's OEIS Frontend

A007996 Primes that divide at least one term of Sylvester's sequence s = A000058: s(n+1) = s(n)^2 - s(n) + 1, s(0) = 2.

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A362250 Primes dividing terms of A231831.

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A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

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A362252 Primes dividing terms of A231830.

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A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

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A323605 Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).

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