A006862 -id:A006862 - cp's OEIS frontend

A100465 Let p(1)=2, p(2)=3, p(3)=5, ... denote the primes and let E(n) = 1 + p(1) * p(2) * ... * p(n). Sequence gives primes p such that p(n+2) | E(n).

7, 271, 307, 673

Offset: 1

Lévai Gábor (gablevai(AT)vipmail.hu), Nov 23 2004

No other terms for p < 80000000.

			7 is a term of the sequence, because it is the 4th prime and divides E(2)=2*3+1=7 trivially. - _Martin Ehrenstein_, Feb 05 2021

See A066735 for further information.

Cf. A006862 Euclid numbers.

a(2) corrected by Martin Ehrenstein, Feb 05 2021

A119988 Divisors of 69371610.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 59, 66, 70, 77, 105, 110, 118, 154, 165, 177, 210, 231, 295, 330, 354, 385, 413, 462, 509, 590, 649, 770, 826, 885, 1018, 1155, 1239, 1298, 1527, 1770, 1947, 2065, 2310, 2478, 2545, 3054, 3245, 3563

Offset: 1

Reinhard Zumkeller, Jun 18 2006

69371610 = (2*3*5*7*11)*(59*509) = 11# * (13# + 1) = A002110(5)*A006862(6);

a(A000005(69371610)) = a(128) = 69371610.

R. Zumkeller, Table of n, a(n) for n = 1..128
Index entries for sequences related to divisors of numbers

Divisors[69371610] (* Wesley Ivan Hurt, Oct 02 2021 *)

divisors(69371610) \\ Charles R Greathouse IV, Jun 21 2017

A177709 Sums of 4 distinct primorials.

Original entry on oeis.org

39, 219, 243, 247, 248, 2319, 2343, 2347, 2348, 2523, 2527, 2528, 2551, 2552, 2556, 30039, 30063, 30067, 30068, 30243, 30247, 30248, 30271, 30272, 30276, 32343, 32347, 32348, 32371, 32372, 32376, 32551, 32552, 32556, 32580, 510519, 510543

Offset: 1

Jonathan Vos Post, May 11 2010

This is to numbers that are the sum of 4 different primes (A177708) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 4 distinct primorials begins: 2347, 2551, 30271, 32371, 510751. The subsequence of nontrivial powers a^b with b>1 begin: a(3) = 243, a(24) = 30276 = 30030+210+30+6 = 2^2 x 3^2 x 29^2.

			a(1) = 39 = 30+6+2+1
a(2) = 219 = 210+6+2+1
a(3) = 243 = 210+30+2+1 = 3^5
a(4) = 247 = 210+30+6+1
a(5) = 248 = 210+30+6+2.

Cf. A000040, A002110, A006862, A038609, A124867, A177689, A177697, A177708.

{a(n)} = {A002110(i) + A002110(j) + A002110(k)+ A002110(L) for distinct i, j, k, L}.

Corrected (2348 inserted) by R. J. Mathar, May 15 2010

A191623 Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).

Original entry on oeis.org

3, 19, 11251, 2980024297506569894680811251

Offset: 1

Jonathan Vos Post, Jun 09 2011

nonn

Primes of the form 1 + A191554(k), associated with positions k = 1, 2, 3, and 5 there. The next one (if it exists) occurs at k >= 15 and has > 53500 digits. [Edited by R. J. Mathar, Jun 17 2011 and Joerg Arndt, Jun 21 2011]

This connects A191554 and A191555, which are deeply about primes and monic polynomial irreducible by Eisenstein's Criterion, to primes by another way, connecting additive and multiplicative number theory analogously to the relationship in Primorial primes: A014545, n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime.

			a(1) = 1 + 2^1 = 1 + 2 = 3 is prime.
a(2) = 1 + (2^1 * 3^2) = 1 + 18 = 19 is prime.
a(3) = 1 + (2^1 * 3^2 * 5^4) = 1 + 11250 = 11251 is prime.

Cf. A000040, A191554.

Select[Array[1 + Product[Prime[k]^(2^(k - 1)), {k, #}] &, 5], PrimeQ] (* Michael De Vlieger, Feb 15 2020 *)

A225236 Primes a(n) = 1 + p_1 + p_1p_2 + p_1p_2p_3 + ... + p_1p_2...p_n where p_1 < p_2 < ... < p_n are the smallest possible primes, with p_1 = 2.

Original entry on oeis.org

3, 13, 83, 853, 32423, 3599833, 535143923, 89303006953, 15446143311143, 3962154101487973, 1112987090349177203, 352673891880866663113, 151875423856593903090323, 74549545656530694788850433, 62494194870870101337941582723, 53556130238578900142519833155253

Offset: 1

Michel Lagneau, May 03 2013

nonn

a(n) = 1 + sum of product of primes p_i from i = 1 to n such that p_i < p_(i+1) and a(n) is prime.
The corresponding primes p_i are in A227613.
a(n) == 3 (mod 10) because p_1*p_2*...*p_n == 0 (mod 10).
Property of this sequence:
b(n) = (a(n) - 1)/2 = 6, 41, 426, 16211, 1799916,... is alternatively even == 6 mod 10 and odd == 1 mod 10.
An easy way to write and compute these numbers is the nested product 1 + p_1*(1 + p_2*(1 + p_3*(...))). - T. D. Noe, May 03 2013

Cf. A006862, A143293, A227613.

with(numtheory) : a1:=3:p0:=3:p1:=2:k0:=2:for n from 1 to 50 do:ii:=0:for k from k0 to 1000 while(ii=0) do:p:=ithprime(k):pp:=p1*p: ppp:=p0+pp:if type(ppp,prime)=true then p0:=ppp:p1:=pp: k0:=k+1:ii:=1:printf(`%d, `,ppp):else fi:od:od:

A264665 Integers n such that A002110(n) + 2 is the sum of 2 nonzero squares.

Original entry on oeis.org

2, 3, 4, 5, 6, 11, 15, 33, 49, 50, 52, 53, 54, 57, 60, 61, 64, 68

Offset: 1

Altug Alkan, Nov 20 2015

Integers n such that A006862(n) + 1 is the sum of 2 nonzero squares.

			a(1) = 2 because 2*3 + 2 = 2^2 + 2^2.
a(2) = 3 because 2*3*5 + 2 = 4^2 + 4^2.
a(3) = 4 because 2*3*5*7 + 2 = 14^2 + 4^2.

Cf. A000404, A002110, A006862.

Rest@ Select[Range@ 36, SquaresR[2, Product[Prime@ k, {k, #}] + 2] > 0 &] (* Michael De Vlieger, Nov 23 2015 *)

a(n) = prod(k=1, n, prime(k)) + 2;
is(n) = { for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)) }
for(n=1, 1e5, if(is(a(n)), print1(n, ", ")))

A297893 Numbers that divide exactly three Euclid numbers.

Original entry on oeis.org

3041, 24917, 144671, 224251, 278191, 301927, 726071, 729173, 772691, 1612007, 1822021, 1954343, 2001409, 2157209, 2451919, 2465917, 2522357, 2668231, 3684011, 3779527, 3965447, 4488299, 4683271, 4869083, 5244427, 5650219, 6002519, 6324191, 6499721, 7252669

Offset: 1

Jon E. Schoenfield, Jan 07 2018

nonn

A113165 lists numbers those numbers (> 1) that divide at least one Euclid number; A297891 lists those that divide exactly two Euclid numbers.
Is this sequence infinite?
Does this sequence contain any nonprimes?
Are there any numbers > 1 that divide more than three Euclid numbers?
The first numbers that divide 4 and 5 Euclid numbers are 15415223 and 2464853, respectively. - Giovanni Resta, Jun 26 2018

			a(1) = 3041 because 3041 is the smallest number that divides exactly three Euclid numbers: 1 + A002110(206), 1 + A002110(263), and 1 + A002110(409); these numbers have 532, 712, and 1201 digits, respectively.

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002110 (primorials), A006862 (Euclid numbers), A113165 (numbers > 1 that divide Euclid numbers), A297891 (numbers > 1 that divide exactly two Euclid numbers).

a(14)-a(30) from Giovanni Resta, Jun 26 2018

A365021 a(n) is the largest prime of the form P+1 where P divides prime(n)# and p# denotes the product of all primes <= p.

Original entry on oeis.org

3, 7, 31, 211, 2311, 6007, 102103, 3233231, 17160991, 2156564411, 200560490131, 1060105447831, 27659114866111, 568815710072611, 87841397512641631, 4655594068170006391, 147904642319554818391, 6899316550553351234311, 374205788146679383613291, 24258296962030389607278931

Offset: 1

Alain Rocchelli, Aug 16 2023

nonn

Cf. A002110, A006862.

a(n) = my(P=vecprod(primes(n)), p=1); while(!ispseudoprime(floor((P/p)+1)) || gcd(P,p)<>p, p=p+2); (P/p)+1;

Conjecture: a(n) > (1/2) * prime(n-1)#.

A071554 Smallest x > 1 such that x^prime(n) == 1 mod(prime(i)) 2<=i<=n.

Original entry on oeis.org

4, 16, 106, 1156, 15016, 255256, 4849846, 111546436, 3234846616, 100280245066, 3710369067406, 152125131763606, 6541380665835016, 307444891294245706, 16294579238595022366, 961380175077106319536, 58644190679703485491636

Offset: 2

Benoit Cloitre, May 30 2002

a(n)=(1/2)*(A006862(n)+1)

A114428 Primes of the form 1 + product of the first n semiprimes.

Original entry on oeis.org

5, 2161, 30241, 453601, 4495130640001, 152834441760001, 911300420785759804800001, 19660095637340203930960075575675174251117567173124497920000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

Semiprime analog of primorial primes A005234 (primes p such that 1 + product of primes up to p is prime) as indexed by A014545 (n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime). In that sense, this sequence is indexed by (1, 4, 5, 6, 11, 12, 39, ...).

The next term has 90 digits. - Harvey P. Dale, Sep 21 2011

			a(1) = 5 = 4 + 1 = 1 + A001358(1) = 1 + A112141(1) because 4 is the first semiprime and 5 is prime.
a(2) = 2161 because 2160 + 1 = 1 + A001358(1)*A001358(2)*A001358(3)*A001358(4) = 1 + A112141(4) = 1 + (4*6*9*10) is prime.
a(3) = 1 + A112141(5).
a(4) = 1 + A112141(6).
a(5) = 1 + A112141(11).
a(6) = 1 + A112141(12).
a(7) = (4 * 6 * 9 * 10 * 14 * 15 * 21 * 22 * 25 * 26 * 33 * 34 * 35 * 38 * 39 * 46 * 49 * 51 * 55 * 57 * 58 * 62 * 65 * 69 * 74 * 77 * 82 * 85 * 86 * 87 * 91 * 93 * 94* 95 * 106 * 111 * 115 * 118 * 119) + 1 is prime.

Cf. A000040, A001358, A002110, A005234, A014545, A112141.

Select[#+1&/@FoldList[Times,1,Select[Range[200],PrimeOmega[#] == 2&]], PrimeQ] (* Harvey P. Dale, Sep 21 2011 *)

{a(n)} = {1 + A112141} INTERSECTION {A000040}.

a(7) added by Jonathan Vos Post, Dec 12 2010

cp's OEIS Frontend

A100465 Let p(1)=2, p(2)=3, p(3)=5, ... denote the primes and let E(n) = 1 + p(1) * p(2) * ... * p(n). Sequence gives primes p such that p(n+2) | E(n).

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A119988 Divisors of 69371610.

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A177709 Sums of 4 distinct primorials.

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A191623 Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).

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Mathematica

A225236 Primes a(n) = 1 + p_1 + p_1*p_2 + p_1*p_2*p_3 + ... + p_1*p_2*...*p_n where p_1 < p_2 < ... < p_n are the smallest possible primes, with p_1 = 2.

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A264665 Integers n such that A002110(n) + 2 is the sum of 2 nonzero squares.

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PARI

A297893 Numbers that divide exactly three Euclid numbers.

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A365021 a(n) is the largest prime of the form P+1 where P divides prime(n)# and p# denotes the product of all primes <= p.

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A071554 Smallest x > 1 such that x^prime(n) == 1 mod(prime(i)) 2<=i<=n.

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Formula

A114428 Primes of the form 1 + product of the first n semiprimes.

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A225236 Primes a(n) = 1 + p_1 + p_1p_2 + p_1p_2p_3 + ... + p_1p_2...p_n where p_1 < p_2 < ... < p_n are the smallest possible primes, with p_1 = 2.