A000946 -id:A000946 - cp's OEIS frontend

A102926 Smallest prime factor in product of previous terms +1 or -1.

2, 3, 5, 29, 11, 7, 13, 37, 17, 79, 23, 4129, 193, 2593, 101, 19, 39163, 577, 26431, 131, 308798542881428667318174028327605372989, 103, 163, 179, 293, 127, 6287, 683437, 31, 89, 13590243019242466336587034391, 113, 2207, 59, 109, 223, 2351

Offset: 1

Marc LeBrun, Jan 19 2005

nonn

A variant of the Euclid-Mullin construction.

This sequence is listed on the OEIS wiki page "OEIS sequences needing factors" and on the corresponding thread on mersenneforum.org. - M. F. Hasler, Mar 21 2013

			a(5)=11 because 2*3*5*29=870, 869=11*79, 871=13*67.
a(31) = 13590243019242466336587034391 because this is the least prime factor of A102927(30)+1. The least prime factor of A102927(30)-1 is 44989026625856465412069667987. Remarkably, both are 29-digit numbers. - _David Wasserman_, Apr 15 2008

Donovan Johnson, Table of n, a(n) for n = 1..111

Cf. A000945, A000946, A005265, A102927.

spf[{p_,a_}]:=With[{f=FactorInteger[p^2-1][[1,1]]},{p*f,f}]; NestList[ spf,{2,2},36][[All,2]] (* Harvey P. Dale, May 05 2018 *)

a(n) = least prime factor of b(n)^2-1, where b(n) = product a(k), 0A102927.

More terms from Don Reble, Jan 23 2005, corrected Sep 26 2006

Further terms from David Wasserman, Apr 15 2008

A217759 Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.

Original entry on oeis.org

3, 7, 43, 19, 6863, 883, 23, 191, 2927, 205677423255820459, 11, 163, 227, 9127, 59, 31, 71, 131627, 2101324929412613521964366263134760336303, 127, 1302443, 4065403, 107, 2591, 21487, 223, 12823, 167, 53720906651, 5452254637117019, 39827899, 11719, 131

Offset: 1

Daran Gill, Mar 23 2013

nonn

Contrast A057207, where all the factors are congruent to 1 (mod 4), here only one is guaranteed to be congruent to 3 (mod 4).

			a(10) is 205677423255820459 because it is the only prime factor congruent to 3 (mod 4) of 4*(3*7*43*19*6863*883*23*191*2927)^2-1 = 5*13*2088217*256085119729*205677423255820459. The four smaller factors are all congruent to 1 (mod 4).

Dirichlet,P.G.L (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 13.

Daran Gill, Table of n, a(n) for n = 1..51
Mersenne Forum, Two new sequences

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002145, A057204-A057208.

A240803 a(n) = 2 + product of first n odd primes.

Original entry on oeis.org

5, 17, 107, 1157, 15017, 255257, 4849847, 111546437, 3234846617, 100280245067, 3710369067407, 152125131763607, 6541380665835017, 307444891294245707, 16294579238595022367, 961380175077106319537, 58644190679703485491637, 3929160775540133527939547, 278970415063349480483707697

Offset: 1

N. J. A. Sloane, Apr 14 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.

Cf. A000945, A000946, A005265, A005266, A240804.

[2+&*[NthPrime(i+1): i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Apr 15 2014

Table[Product[Prime[k + 1], {k, 1, n}] + 2, {n, 1, 30}] (* Vincenzo Librandi, Apr 15 2014 *)

A240804 a(n) = -2 + product of first n odd primes.

Original entry on oeis.org

1, 13, 103, 1153, 15013, 255253, 4849843, 111546433, 3234846613, 100280245063, 3710369067403, 152125131763603, 6541380665835013, 307444891294245703, 16294579238595022363, 961380175077106319533, 58644190679703485491633, 3929160775540133527939543, 278970415063349480483707693

Offset: 1

N. J. A. Sloane, Apr 14 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.

Cf. A000945, A000946, A005265, A005266, A240803.

[-2+&*[NthPrime(i+1): i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Apr 15 2014

Table[Product[Prime[k + 1], {k, 1, n}] - 2, {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)
Rest[FoldList[Times,1,Prime[Range[2,20]]]]-2 (* Harvey P. Dale, Mar 17 2015 *)

a(n) = A070826(n+1)-2. - R. J. Mathar, May 03 2017

A359504 a(n) is calculated by considering in ascending order all products P of zero or more terms from {a(1..n-1)} until finding one where P+1 has a prime factor not in {a(1..n-1)}, in which case a(n) is the smallest such prime factor.

Original entry on oeis.org

2, 3, 7, 5, 11, 23, 31, 17, 43, 47, 13, 67, 71, 79, 29, 59, 19, 103, 53, 107, 37, 131, 139, 73, 83, 167, 89, 179, 61, 41, 191, 101, 211, 109, 223, 239, 127, 263, 137, 283, 97, 151, 311, 331, 173, 347, 359, 367, 383, 193, 197, 419, 431, 439, 443, 149, 113, 227, 463

Offset: 1

Joel Brennan, Jan 03 2023

nonn

A new prime is always found since at worst P can be the product of all primes {a(1..n-1)} and per Euclid's proof of the infinitude of primes, P+1 then certainly has a prime factor not among a(1..n-1).
Taking products P in ascending order generally results in smaller quantities to consider than always taking the product of all primes as done in A000945, the Euclid-Mullin sequence.
Conjecture: P+1 has at most one prime factor not already in the sequence, so the requirement of taking "the smallest such" is unnecessary.

			For n=1, the sole product P is the empty product P=1, and P+1 = 2 is itself prime so a(1) = 2.
For n=3, the primes so far are 2,3 but products P=2 or P=3 have P+1 = 3 or 4 which have no new prime factor. Product P = 2*3 = 6 has P+1 = 7 which is a new prime so a(3) = 7.
For n=4, the smallest product P with a new prime in P+1 is P = 2*7 = 14 for which P+1 = 15 and a(4) = 5 is its smallest new prime factor.

Joel Brennan, Table of n, a(n) for n = 1..5000

Cf. A000945, A000946.

More terms from Kevin Ryde, Jan 10 2023

A051310 Euclid-Mullin sequence (A000945) with initial value a(1)=13 instead of a(1)=2.

Original entry on oeis.org

13, 2, 3, 79, 6163, 7, 1601, 11, 137, 5, 199, 151, 263, 983, 31, 83, 30187890723499, 23847817657, 37, 67, 9661, 251, 73, 1214623152057970133, 24597089626521443731307390760915220105471840174452030562332559181845834101711082531

Offset: 1

Labos Elemer

nonn

Robert Price, Table of n, a(n) for n = 1..36
A. R. Booker, S. A. Irvine, The Euclid-Mullin graph, arXiv preprint arXiv:1508.03039 [math.NT], 2015-2016.

Cf. A000945, A000946, A005265, A005266.

a[1]=13; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

spf(n)=factor(n)[1,1]
first(m)=my(v=vector(m)); v[1]=13; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 22 2015

A051311 Euclid-Mullin sequence (A000945) with initial value a(1)=17 instead of a(1)=2.

Original entry on oeis.org

17, 2, 5, 3, 7, 3571, 31, 395202571, 13, 29, 137, 23, 97, 1896893, 34138453466895150823580146142491, 4639, 61, 181, 43, 19, 11, 59, 25292522503108044617, 4909, 18305191, 467

Offset: 1

Labos Elemer

nonn

Robert Price, Table of n, a(n) for n = 1..30

Cf. A000945, A000946, A005265, A005266.

a[1]=17; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

spf(n)=factor(n)[1,1]
first(m)=my(v=vector(m)); v[1]=17; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 22 2015

A051313 Euclid-Mullin sequence (A000945) with initial value a(1)=23 instead of a(1)=2.

Original entry on oeis.org

23, 2, 47, 3, 13, 84319, 7109609443, 463, 23403050994721829453179, 7, 5, 57367, 239, 40237, 10575444619218059847586376042094152838881224222904607376771, 31333, 742759, 9444637217

Offset: 1

Labos Elemer

nonn

Robert Price, Table of n, a(n) for n = 1..37

Cf. A000945, A000946, A005265, A005266.

a[1]=23; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

spf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m)); v[1]=23; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Nov 22 2015

a(18)-a(37) from Robert Price, Jul 17 2015

A051314 Euclid-Mullin sequence (A000945) with initial value a(1)=29 instead of a(1)=2.

Original entry on oeis.org

29, 2, 59, 3, 10267, 7, 5, 3689035771, 19, 396029, 489851, 2971, 179, 13, 4441009, 419, 79, 53, 3109, 538004633, 138285071, 241, 263, 443, 11, 17, 951837583454247922680798591029699, 739, 43, 181, 131, 3257, 31, 2237

Offset: 1

Labos Elemer

nonn

Tyler Busby, Table of n, a(n) for n = 1..56 (terms 1..46 from Robert Price)

Cf. A000945, A000946, A005265, A005266.

a[1]=29; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

spf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m)); v[1]=29; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Nov 21 2015

A051315 Euclid-Mullin sequence (A000945) with initial value a(1)=31 instead of a(1)=2.

Original entry on oeis.org

31, 2, 3, 11, 23, 47059, 7, 5, 89, 19, 2287, 233, 17, 647, 1607, 12637, 103, 13, 163, 4980301, 521, 83, 16561, 540233, 443516695049428313, 109, 37, 1811, 53, 487, 548519020982014152563328120144563684918808813765009178152503015356294212417026402782591

Offset: 1

Labos Elemer

Tyler Busby, Table of n, a(n) for n = 1..37 (terms 1..36 from Robert Price)

Cf. A000945, A000946, A005265, A005266.

spf:= proc(n) local F;
  F:= select(type, map(t -> t[1], ifactors(n,easy)[2]), integer);
   if F <> [] then min(F)
   else min(numtheory:-factorset(n))
   fi
end proc:
a[1]:= 31:
for i from 2 to 31 do
  a[i]:= spf(1 + mul(a[j],j=1..i-1))
od:
seq(a[i],i=1..31); # Robert Israel, Nov 25 2015

a[1]=31; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]

gpf(n)=my(f=factor(n)[, 1]); f[#f];
first(m)=my(v=vector(m)); v[1]=31; for(i=2, m, v[i]=gpf(1+prod(j=1, i-1, v[j]))); v \\ Anders Hellström, Nov 25 2015

cp's OEIS Frontend

A102926 Smallest prime factor in product of previous terms +1 or -1.

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A217759 Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.

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A240803 a(n) = 2 + product of first n odd primes.

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Magma

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A240804 a(n) = -2 + product of first n odd primes.

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Magma

Mathematica

Formula

A359504 a(n) is calculated by considering in ascending order all products P of zero or more terms from {a(1..n-1)} until finding one where P+1 has a prime factor not in {a(1..n-1)}, in which case a(n) is the smallest such prime factor.

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A051310 Euclid-Mullin sequence (A000945) with initial value a(1)=13 instead of a(1)=2.

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PARI

A051311 Euclid-Mullin sequence (A000945) with initial value a(1)=17 instead of a(1)=2.

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Mathematica

PARI

A051313 Euclid-Mullin sequence (A000945) with initial value a(1)=23 instead of a(1)=2.

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PARI

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A051314 Euclid-Mullin sequence (A000945) with initial value a(1)=29 instead of a(1)=2.

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