A167604 -id:A167604 - cp's OEIS frontend

A167605 Positive integers n such that the first n terms of A167604 form a permutation of the first n primes.

1, 2, 3, 16, 18, 19, 26, 27, 34, 35, 36, 38, 39, 40, 48, 51, 52, 53, 54, 55, 63, 73, 75, 76, 77, 78, 83, 93, 94, 98, 99, 106, 113, 114, 121, 122, 123, 128, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 156, 157, 158, 159, 168, 173, 174

Offset: 1

Kok Seng Chua (chuakokseng(AT)hotmail.com), Nov 07 2009

nonn

Is this sequence infinite?

			a(4)=16 since the first 16 terms of A167604 are 2,3,5,11,37,13,7,29,17,19,43,23,47,41,53,31 which is a permutation of the first 16 primes.

A167604, A000945

Edited and extended by Max Alekseyev, Nov 11 2009

A344020 A sequence of prime numbers: a(1)=2, a(n+1) is the least prime dividing Product_{i in S} a(i)^2 + Product_{i not in S} a(i)^2, minimized over all subsets S of {1..n}.

Original entry on oeis.org

2, 5, 29, 17, 41, 13, 37, 53, 61, 97, 101, 73, 89, 109, 149, 137, 113, 173, 181, 157, 229, 197, 241, 257, 233, 193, 277, 269, 349, 317, 337, 293, 281, 313, 353, 373, 389, 409, 421, 397, 457, 461, 401, 433, 521, 509, 449, 541, 569, 557, 701, 593, 613, 653, 641, 617, 577, 661, 673, 709, 677, 601, 761, 733, 757, 769, 773, 797

Offset: 1

Lucas Hoogendijk, May 06 2021

nonn

A variant of Euclid-Mullin (A000945) and Chua's adaptation (A167604).
All a(i) must be unique and, apart from 2, must be congruent to 1 (mod 4) as p only divides Product_{i in S} a(i)^2 + Product_{i not in S} a(i)^2 if -1 is a quadratic residue modulo p.
Whether all primes congruent to 1 (mod 4) occur in this sequence is unknown.
For n > 1, a(n) >= p, where p is the smallest prime p such that p == 1 (mod 4) and a(2)*a(3)*...*a(n-1) is a nonzero square modulo p. Conjecture: a(n) = p. - Jinyuan Wang and Max Alekseyev, Jul 04 2022

			For n=4 we obtain the 4 partitions with their products: 1 + 2^2 * 5^2 * 29^2 = 84101 = 37 * 2273, 2^2 + 5^2 * 29^2 = 21029 = 17*1237, 5^2 + 2^2 * 29^2 = 3389 and 2^2 * 5^2 + 29^2 = 941. The minimum of the primes dividing these is 17, thus a(4)=17.

Max Alekseyev, Table of n, a(n) for n = 1..2500
Andrew R. Booker, A variant of the Euclid-Mullin sequence containing every prime, arXiv preprint arXiv:1605.08929 [math.NT], 2016.
Lucas M. H. Hoogendijk, Prime Generators, UU bachelor thesis, 2020.
Lucas Hoogendijk, Python code used to compute the first 27 terms (all known terms at the time of upload).

Cf. A000945, A167604.

a = {2}; leastPrimeDivisor[n_Integer] := First[Select[FactorInteger[n][[All, 1]], PrimeQ]]; SequenceRange[start_Integer, end_Integer] := Module[{n, subsets, products, minPrime}, While[Length[a] < end, n = Length[a];subsets = Subsets[Range[n]]; products = Table[With[{S = subsets[[i]]}, Times @@ (a[[#]]^2 & /@ S) + Times @@ (a[[#]]^2 & /@ Complement[Range[n], S])], {i, Length[subsets]}]; minPrime = Min[leastPrimeDivisor /@ products]; AppendTo[a, minPrime];]; a[[start ;; end]]]; SequenceRange[1, 15] (* Hilko Koning, Nov 01 2024 *)

{ A344020_list() = my(a, A, m, p, b, q, z); print1(2,", "); a = [2]; A=1; while(1, p=5; while( kronecker(A, p)!=1 || p%4!=1, p=nextprime(p+1) ); b=lift(sqrt(A+O(p))*(1+sqrt(-1+O(p)))); z=znprimroot(p); m = nextprime(random(10^6)); q=lift(prod(i=1, #a, Mod(1+x^znlog(Mod(a[i], p), z, p-1), (1-x^(p-1))*Mod(1,m)) )); if( polcoeff(q, znlog(Mod(b, p), z, p-1), x)==0 && polcoeff(q, znlog(Mod(-b, p), z, p-1), x)==0, error("conjecture failed mod",m) ); a=concat(a, [p]); A*=p; print1(p, ", ") ); } \\ Max Alekseyev, Jul 04 2022

More terms from Max Alekseyev, Jul 03 2022

A258249 Lexicographically smallest sequence of primes such that Legendre symbol (a(1)...a(n-1) / a(n)) = 1 for all n.

Original entry on oeis.org

2, 7, 3, 11, 43, 5, 13, 17, 47, 19, 23, 61, 29, 41, 59, 31, 53, 67, 71, 79, 37, 83, 73, 101, 89, 103, 127, 97, 107, 113, 131, 139, 109, 151, 149, 137, 163, 157, 173, 197, 167, 179, 181, 191, 193, 223, 227, 199, 233, 211, 229, 239, 241, 263, 251, 269, 257, 271, 277, 281, 283, 311, 293, 307, 317, 331

Offset: 1

Max Alekseyev, May 24 2015

nonn

Cf. A167604.

{ P=1; for(n=1,100, p=2; while( kronecker(p,P)!=1, p=nextprime(p+1)); print1(p,", "); P*=p) }

cp's OEIS Frontend

A167605 Positive integers n such that the first n terms of A167604 form a permutation of the first n primes.

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A344020 A sequence of prime numbers: a(1)=2, a(n+1) is the least prime dividing Product_{i in S} a(i)^2 + Product_{i not in S} a(i)^2, minimized over all subsets S of {1..n}.

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A258249 Lexicographically smallest sequence of primes such that Legendre symbol (a(1)...a(n-1) / a(n)) = 1 for all n.

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PARI

cp's OEIS Frontend

A167605 Positive integers n such that the first n terms of A167604 form a permutation of the first n primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A344020 A sequence of prime numbers: a(1)=2, a(n+1) is the least prime dividing Product_{i in S} a(i)^2 + Product_{i not in S} a(i)^2, minimized over all subsets S of {1..n}.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Extensions

A258249 Lexicographically smallest sequence of primes such that Legendre symbol (a(1)*...*a(n-1) / a(n)) = 1 for all n.

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Author

Keywords

Crossrefs

Programs

PARI

A258249 Lexicographically smallest sequence of primes such that Legendre symbol (a(1)...a(n-1) / a(n)) = 1 for all n.