A006862 -id:A006862 - cp's OEIS frontend

A279097 Numbers k such that prime(k) divides primorial(j) + 1 for some j.

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 59, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 177, 179, 180, 186, 205, 207, 208, 210, 211, 218, 221, 233, 243, 249, 256, 260, 261, 262

Offset: 1

Jon E. Schoenfield, Mar 24 2017

nonn

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).

Primorial(j) + 1 is the j-th Euclid number, A006862(j).

			1 is in the sequence because primorial(0) + 1 = 1 + 1 = 2 is divisible by prime(1) = 2.
4 is in the sequence because primorial(2) + 1 = 2*3 + 1 = 7 is divisible by prime(4) = 7.
8 is in the sequence because primorial(7) + 1 = 2*3*5*7*11*13*17 + 1 = 510511 is divisible by prime(8) = 19.
59 is in the sequence because primorial(7) + 1 = 510511 is divisible by prime(59) = 277 (and primorial(17) + 1 = 1922760350154212639071 is divisible by prime(59) as well).
5 is not in the sequence because there is no number j such that primorial(j) + 1 is divisible by prime(5) = 11:
    primorial(1) + 1 = 2       + 1 =   3 == 3 (mod 11)
    primorial(2) + 1 = 2*3     + 1 =   7 == 7 (mod 11)
    primorial(3) + 1 = 2*3*5   + 1 =  31 == 9 (mod 11)
    primorial(4) + 1 = 2*3*5*7 + 1 = 211 == 2 (mod 11)
and primorial(j) + 1 = 2*...*11*... + 1  == 1 (mod 11) for all j >= 5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A002110, A006862, A113165, A279098, A279099, A283928.

np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] > 0 &] (* Giovanni Resta, Mar 29 2017 *)

A279098 Numbers k such that prime(k) divides primorial(j) + 1 for exactly one integer j.

Original entry on oeis.org

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 179, 180, 186, 205, 207, 208, 210, 211, 218, 233, 243, 249, 256, 261, 262, 297, 308, 316, 318

Offset: 1

Jon E. Schoenfield, Mar 24 2017

nonn

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).

Primorial(j) + 1 is the j-th Euclid number, A006862(j).

			59 is not in this sequence because both primorial(7) + 1 = 510511 and primorial(17) + 1 = 1922760350154212639071 are divisible by prime(59) = 277.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Subsequence of A279097 (which also includes numbers k such that prime(k) divides primorial(j) + 1 for more than one integer j).

Cf. A000040, A002110, A006862, A113165, A279099, A283928.

np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] == 1 &] (* Giovanni Resta, Mar 29 2017 *)

A279099 Numbers k such that prime(k) divides primorial(j) + 1 for exactly two integers j.

Original entry on oeis.org

59, 177, 221, 260, 285, 431, 476, 489, 625, 653, 686, 860, 957, 1320, 1334, 1734, 1987, 2140, 2215, 2854, 2991, 3051, 3341, 3455, 3535, 3591, 3645, 3695, 3798, 4020, 4032, 4079, 4612, 4614, 4856, 4904, 5019, 5234, 5263, 5842, 6178, 6184, 6491, 6639, 6745, 7151

Offset: 1

Jon E. Schoenfield, Mar 24 2017

nonn

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).

Primorial(j) + 1 is the j-th Euclid number, A006862(j).

			59 is in this sequence because prime(59) = 277 divides primorial(j) + 1 for exactly two integers j: 7 and 17.
436 is not in this sequence because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j: 206, 263, and 409.

Giovanni Resta, Table of n, a(n) for n = 1..5000

Subsequence of A279097 (which includes all numbers k such that prime(k) divides primorial(j) + 1 for one or more integers j); cf. A279098 (exactly one integer j).

Cf. A000040, A002110, A006862, A113165, A279098, A283928.

np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[1000], np[#] == 2 &] (* Giovanni Resta, Mar 29 2017 *)

A283928 Numbers k such that prime(k) divides primorial(j) + 1 for exactly three integers j.

Original entry on oeis.org

436, 2753, 13396, 19960, 24293, 26157, 58492, 58723, 61935, 121992, 136592, 145803, 149027, 159752, 179811, 180776, 184575, 194499, 262321, 268645, 280911, 315198, 327876, 339951, 364307, 390394, 413010, 433626, 444744, 492661, 510412, 518156, 541925, 542177

Offset: 1

Jon E. Schoenfield, Mar 24 2017

nonn

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).

Primorial(j) + 1 is the j-th Euclid number, A006862(j).

			436 is in this sequence because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j: 206, 263, and 409.
180707 is not in this sequence because prime(180707) = 2464853 divides primorial(j) + 1 for exactly five integers j: 75366, 79914, 139731, 139990, and 175013. - _Jon E. Schoenfield_, Mar 30 2017

Giovanni Resta, Table of n, a(n) for n = 1..150

Subsequence of A279097 (which includes all numbers k such that prime(k) divides primorial(j) + 1 for one or more integers j); cf. A279098 (exactly one integer j), A279099 (exactly two).

Cf. A000040, A002110, A006862, A113165.

countReqd:=3; kMaxTest:=20000; P:=PrimesInInterval(2,NthPrime(kMaxTest)); itos:=IntegerToString; a:=[]; for k in [1..kMaxTest] do p:=P[k]; pMinus1:=p-1; primorialModp:=1; jSuccess:=[]; if primorialModp eq pMinus1 then jSuccess:=[1]; end if; for j in [1..k-1] do primorialModp:=(primorialModp*P[j]) mod p; if primorialModp eq pMinus1 then jSuccess[#jSuccess+1]:=j; end if; end for; if #jSuccess eq countReqd then a[#a+1]:=k; "a("*itos(#a)*") = " * itos(k) * "; successes at j =", jSuccess; end if; end for; a; // Jon E. Schoenfield, Mar 25 2017

a(10)-a(34) from Jon E. Schoenfield, Apr 02 2017

A049650 Compositorial numbers (A036691) + 1.

Original entry on oeis.org

2, 5, 25, 193, 1729, 17281, 207361, 2903041, 43545601, 696729601, 12541132801, 250822656001, 5267275776001, 115880067072001, 2781121609728001, 69528040243200001, 1807729046323200001, 48808684250726400001, 1366643159020339200001

Offset: 0

N. J. A. Sloane, May 05 2001

This is to Euclid numbers (A006862): 1 + product of first n consecutive primes, as nonprimes (A018252) are to primes (A000040). These numbers - 1, times the appropriate Euclid numbers - 1, are the factorials. Primes in this sequence include a(1) = 2, a(2) = 5, a(4) = 193, a(8) = 2903041, a(12) = 250822656001, a(17) = 1807729046323200001. - Jonathan Vos Post, Jun 07 2008

G. C. Greubel, Table of n, a(n) for n = 0..429
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences

Cf. A002808, A036691, A060880, A006862.

Composite[n_] := FixedPoint[n + PrimePi[#] + 1 &, n + PrimePi[n] + 1]; Table[Product[Composite[i], {i, 1, n}] + 1, {n, 0, 30}] (* G. C. Greubel, Dec 05 2017 *)

a(n) = 1 + Product_{i=1..n} A002808(i). - Jonathan Vos Post, Jun 07 2008

A060881 n-th primorial (A002110) + prime(n + 1).

Original entry on oeis.org

3, 5, 11, 37, 221, 2323, 30047, 510529, 9699713, 223092899, 6469693261, 200560490167, 7420738134851, 304250263527253, 13082761331670077, 614889782588491463, 32589158477190044789, 1922760350154212639131

Offset: 0

N. J. A. Sloane, May 05 2001

nonn

Terms are pairwise coprime with very high probability. I didn't find terms which are pairwise noncoprime, although it may be a case of the "strong law of small numbers." - Daniel Forgues, Apr 23 2012

All numbers in the range [primorial(n)+2, a(n)-1] are guaranteed to be a multiple of a prime p whose index is <= n. There are prime(n+1)-2 = A040976(n+1) such numbers. - Jamie Morken and Michel Marcus, Feb 01 2018

			a(2) = 2*3 + 5 = 11.

Harry J. Smith, Table of n, a(n) for n = 0..100
Hisanori Mishima, WIFC (World Integer Factorization Center): PI Pn + NextPrime (n = 1 to 100), PI Pn - NextPrime (n = 1 to 100), PI Pn + 1 (n = 1 to 100), PI Pn - 1 (n = 1 to 100).
Eric Weisstein's World of Mathematics, Prime Gaps

Cf. A000040, A002110, A060882, A006862, A057588.

a:= n-> mul(ithprime(k), k=1..n)+ithprime(n+1): seq(a(n), n=0..20);  # Muniru A Asiru, Feb 01 2018

Module[{nn=20,pr},pr=Prime[Range[nn+1]];Join[{3},FoldList[ Times,Most[ pr]] + Rest[pr]]] (* Harvey P. Dale, Feb 19 2016 *)
Total /@ Fold[Append[#1, {Prime[#2] #1[[-1, 1]], Prime[#2 + 1]}] &, {{1, 2}}, Range@ 17] (* Michael De Vlieger, Feb 21 2018 *)

{ n=-1; m=1; forprime (p=2, prime(101), write("b060881.txt", n++, " ", m + p); m*=p; ) } \\ Harry J. Smith, Jul 19 2009

a(n) = prod(i=1, n, prime(i)) + prime(n+1); \\ Michel Marcus, Feb 01 2018

a(n) = A002110(n) + A000040(n+1). - Michel Marcus, Feb 01 2018

Name changed by David A. Corneth, Mar 25 2018

A177689 Sums of 2 distinct primorials.

Original entry on oeis.org

3, 7, 8, 31, 32, 36, 211, 212, 216, 240, 2311, 2312, 2316, 2340, 30031, 30032, 30036, 30060, 30240, 32340, 510511, 510512, 510516, 510540, 510720, 512820, 540540, 9699691, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200

Offset: 1

Jonathan Vos Post, May 11 2010

This is to numbers that are the sum of 2 different primes (A038609) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 2 distinct primorials is the sequence of primorial primes (A018239) which is the same as the subsequence of primes among the Euclid numbers (A006862).

Cf. A000040, A002110, A006862, A038609.

{a(n)} = {A002110(i) + A002110(j) for i =/= j}.

A053341 Euclid-Pocklington primes: primes of the form Product_{i=1..k} prime(i) * prime(k+1)^m + 1 where prime(r) is the r-th prime and Product_{i=1..k} prime(i) < prime(k+1)^m.

Original entry on oeis.org

3, 5, 7, 17, 19, 151, 163, 257, 487, 751, 1459, 1471, 39367, 65537, 72031, 279511, 33820711, 86093443, 258280327, 372027811, 4092305911, 11149928791, 42638305711, 209481995953231, 411782264189299, 3561193931204911

Offset: 1

N. J. A. Sloane, Jan 05 2000

Named after the Greek mathematician Euclid (flourished c. 300 B.C.) and the English physicist and mathematician Henry Cabourn Pocklington (1870-1952). - Amiram Eldar, Jun 24 2021

			5 = 2^2+1 is of this form (with k=0).

Donovan Johnson, Table of n, a(n) for n = 1..454

Cf. A006862.

eucpock[max_] := Module[{p = 1, prod = 1, m, q, r, s = {}}, While[prod < max, prod *= p; q = NextPrime[p]; m = Max[1, Ceiling @ Log[q, prod]]; r = prod * q^m; While[r + 1 <= max, If[PrimeQ[r + 1], AppendTo[s, r + 1]]; r *= q]; p = NextPrime[p]]; Union[s]]; eucpock[10^16] (* Amiram Eldar, Jun 24 2021 *)

More terms from Jud McCranie, Jul 11 2000

A066576 Composite Euclid numbers: numbers of the form p# + 1, where p# denotes the primorial of the prime p.

Original entry on oeis.org

30031, 510511, 9699691, 223092871, 6469693231, 7420738134811, 304250263527211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071, 117288381359406970983271

Offset: 1

Joseph L. Pe, Jan 07 2002

nonn

30031 appears in Wells's "Curious and Interesting Numbers" as the smallest composite number of the form p# + 1.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 166.

Amiram Eldar, Table of n, a(n) for n = 1..341

Cf. A014545, A006862.

[a: n in [2..19]| not IsPrime(a) where a is 1+&*[NthPrime(k):k in [1..n]]]; // Marius A. Burtea, Feb 18 2020

Do[c = Product[ Prime[i], {i, 1, n}] + 1; If[ !PrimeQ[c], Print[c]], {n, 1, 20} ]

Edited by Robert G. Wilson v, Jan 09 2002

A189409 a(n) = prime(n)#^2 + 1, where prime(n)# is the n-th primorial (A002110).

Original entry on oeis.org

2, 5, 37, 901, 44101, 5336101, 901800901, 260620460101, 94083986096101, 49770428644836901, 41856930490307832901, 40224510201185827416901, 55067354465423397733736101, 92568222856376731590410384101

Offset: 0

John M. Campbell, Apr 21 2011

A variation of Euclid numbers. It is unknown whether or not numbers in this sequence are always squarefree. It is unknown whether or not there exist infinitely many primes in this sequence. For Euclid numbers see A006862.
Comment from Abhiram R Devesh, Jan 23 2013: (Start)
(i) The last 3 digits of an entry is always either 101 or 901 (with the exception of the first 3 terms),
(ii) the thousand's place digit is an even number.
(End)

			(p_16#)^2+1 = 1062053250251407755176413469419400772901 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 0..190
E.W. Weisstein, Integer Sequence Primes
Eric W. Weisstein's World of Mathematics, Euclid's Theorem

A002110, A006862, A014545, A210482 (subsequence of primes).

Table[Product[Prime[n]^2, {n, 1, k}] + 1, {k, 0, 16}]
Join[{2},FoldList[Times,Prime[Range[20]]]^2+1] (* Harvey P. Dale, Jan 15 2019 *)

list(maxx)={n=prime(1); cnt=0;print("0  2");
while(n<=maxx,q=(prodeuler(p=1,n,p))^2+1;cnt++;
print(cnt,"  ",q); n=nextprime(n+1)); } \\ Bill McEachen, Feb 03 2014

from functools import reduce
import numpy as np
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
mul = 1
for i in range(1, 20):
    if len(factors(i))<3:
        mul *= i*i
        print(mul+1, factors(mul+1))
# Abhiram R Devesh, Jan 23 2013

a(n)=(E(n)-1)^2+1, where E(n) is the n-th Euclid number.

Typo in Mma fixed by Vincenzo Librandi, Feb 04 2014

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A279097 Numbers k such that prime(k) divides primorial(j) + 1 for some j.

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A279098 Numbers k such that prime(k) divides primorial(j) + 1 for exactly one integer j.

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A279099 Numbers k such that prime(k) divides primorial(j) + 1 for exactly two integers j.

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A283928 Numbers k such that prime(k) divides primorial(j) + 1 for exactly three integers j.

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A049650 Compositorial numbers (A036691) + 1.

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A060881 n-th primorial (A002110) + prime(n + 1).

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A177689 Sums of 2 distinct primorials.

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A053341 Euclid-Pocklington primes: primes of the form Product_{i=1..k} prime(i) * prime(k+1)^m + 1 where prime(r) is the r-th prime and Product_{i=1..k} prime(i) < prime(k+1)^m.

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