A006862 -id:A006862 - cp's OEIS frontend

A297891 Numbers that divide exactly two Euclid numbers.

277, 1051, 1381, 1657, 1867, 3001, 3373, 3499, 4637, 4877, 5147, 6673, 7547, 10859, 10987, 14797, 17291, 18749, 19531, 25939, 27337, 27953, 31013, 32203, 32983, 33547, 34123, 34591, 35747, 38047, 38197, 38711, 44293, 44357, 47059, 47569, 48809, 51151, 51437

Offset: 1

Jon E. Schoenfield, Jan 07 2018

nonn

The k-th Euclid number, A006862(k), is 1 plus the product of the first k primes, i.e., 1 + A002110(k). A113165 lists the numbers (> 1) that divide at least one Euclid number; a(1) = 277 = A113165(19); a(2) = 1051 = A113165(41); a(53) = 92143 = A113165(995).
Up to N = 10^5, roughly 5% of the terms in A113165 are also in this sequence. Does that ratio continue to hold as N increases?
It appears that the vast majority of terms in A113165 are prime, but that sequence contains a number of composite numbers as well, beginning with A113165(59) = 1843 = 19*97, A113165(125) = 5263 = 19*277, A113165(195) = 10147 = 73*139, and A113165(231) = 12629 = 73*173. But do any composites divide more than one Euclid number?

			a(1) = 277 because 277 is the smallest number that divides exactly two Euclid numbers: 1 + 2*3*5*7*11*13*17 = 510511 and 1 + 2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59 = 1922760350154212639071.

Giovanni Resta, Table of n, a(n) for n = 1..1000

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

A066542 Nonnegative integers all of whose anti-divisors are either 2 or odd.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

John W. Layman, Jan 07 2002

nonn

See A066272 for definition of anti-divisor.
The following conjectures have been proved by Bob Selcoe. - Michael Somos, Feb 28 2014
Additional conjectures suggested by computational experiments:
1) Numbers all of whose anti-divisors (AD's) are odd => {2^k} (A000079).
2) Numbers with AD 2, all other AD's odd => primes (A000040).
3) Numbers none of whose AD's are multiples of 3 => 3*2^k (A007283).
4) Numbers all of whose AD's are even => 3*A002822 = A040040 (except for a(0)=1), both related to twin prime pairs.
Calculations suggest the following conjecture. This sequence consists of all odd primes and nonnegative powers of 2 and no other terms. This has been verified for to n=100000. Robert G. Wilson v extended the conjecture out to 2^20.
From Bob Selcoe, Feb 24 2014: (Start)
The sequence consists of all odd primes and powers of two (>=2^2) and no other terms.
Proof: Denote the even anti-divisors of n as ADe(n). ADe(n) is defined as the set of numbers x satisfying the equation n(mod x)=x/2. Substitute x = 2n/y, since it can be shown that ADe(n) => 2n divided by the odd divisors of n when n>1 (This is because 2j anti-divides only numbers of the form 3j+2j*k; j>=1, k>=0.  For example: j=7; 14 anti-divides only 21,35,49,63.... So in other words, even numbers anti-divide only odd multiples (>=3) of themselves, divided by 2). Therefore, ADe(n) is n(mod [2n/y])=n/y, and y must be an odd divisor of n and 2n, y>1.  Since y is the only odd divisor of n when y>1 iff n is prime, then ADe(n) => 2 when n is prime.  Since 2n has no odd divisors when n=2^k, then ADe(n) is null when n=2^k.  Therefore, the only numbers whose anti-divisors are either 2 or odd must be primes and powers of 2.
Similarly, for odd anti-divisors (ADo(n)): Given 2j+1 (odd numbers) anti-divide only numbers of the forms [(3j+1)+(2j+1)*k] and [(3j+2)+(2j+1)*k]; j>=1, k>=0. (For example: j=6; 13 anti-divides only 19,20, 32,33, 45,46...). Since odd n divided by its odd divisors ARE its odd divisors, then ADo(n) => the divisors of 2n-1 and 2n+1 (except 1, 2n-1 and 2n+1).
By extension:
1) Numbers all of whose anti-divisors (AD's) are odd => {2^k} (A000079).
2) Numbers with ADe(n)=2, all other AD's odd => primes (A000040).
3) Numbers none of whose AD's are multiples of j => j*2^k.
4) When 2n-1 and 2n+1 are twin primes, (A040040, except for a(0)=1) then n has only even AD's.
(End)
If 1 and 2 are included, this sequence contains all positive integers not contained in A111774. - Bob Selcoe, Sep 09 2014 [corrected by Wolfdieter Lang, Nov 06 2020]

			From _Bob Selcoe_, Feb 24 2014: (Start)
ADe(420): Odd divisors of 420 are: 3,5,7,15,21,35, 105. ADe(420) => 840/{3,5,7,15,21,35,105} = 8,24,40,56,120,168 and 280.
ADo(420) => the divisors of 839 and 841, which are (a) for 839: null (839 is prime); and (b) for 841: 29 (841 is 29^2).
All AD's (AD(420)) => 8,24,29,40,56,120,168 and 280 (End)

Cf. A000040, A000079, A002822, A007283, A040040.

Cf. A014545, A006862, A111774.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 &], 2n / Select[ Divisors[2*n], OddQ[ # ] && # != 1 &]]], # < n & ]; f[n_] := Select[ antid[n], EvenQ[ # ] && # > 2 & ]; Select[ Range[3, 300], f[ # ] == {} & ]

A085725 Numbers k such that 1 + (product of first k primes) is a semiprime.

Original entry on oeis.org

6, 8, 9, 14, 18, 19, 21, 23, 26, 34, 36, 40, 48, 54, 76, 88, 105, 122, 135

Offset: 1

Jason Earls, Jul 20 2003

See A250293 for more information.

Cf. A001358 (semiprimes), A002110, A006862 (Euclid numbers), A250293.

isok(k) = bigomega(1 + prod(j=1, k, prime(j))) == 2; \\ Michel Marcus, Mar 30 2019

a(n) = primepi(A250293(n)). - Hugo Pfoertner, Feb 05 2021

More terms from David Wasserman, Feb 09 2005
a(14)=54 from Donovan Johnson, Feb 21 2008
Offset changed to 1 by Alois P. Heinz, Jul 02 2015
a(15) added by Daniel Suteu, Mar 30 2019
a(16)-a(18) using terms of A250293 from Hugo Pfoertner, Feb 05 2021
Missing 105 inserted by Sean A. Irvine, Mar 03 2023

A125191 Primes of the form k# + (k+1)# +- 1, where k# = A002110(k) = primorial(k).

Original entry on oeis.org

2, 7, 37, 239, 241, 2521, 32341, 540539, 540541, 232792559, 232792561, 207030183359, 311671001662019, 41287621429375723111588738861, 5801527386969669153864265802424086050777441586253956297278498679

Offset: 1

Tomas Xordan, Jan 12 2007

nonn

Prime numbers of the form (prime(k+1) + 1)*k# +- 1.

			Let k = 1; then 1#+2# = 2+6 = 8, 8-1 = 7 is prime (hence a term of the sequence) but 8+1 = 9 is nonprime.
Let k = 3; then 3#+4# = 30+210 = 240, 240-1 = 239 is prime and 240+1 = 241 is also prime, so both are terms.

Cf. A002110 (primorial numbers), A006862 (Euclid numbers), A057588 (Kummer numbers).

A002110 := 1 : A000040 := 2 : for n from 1 to 38 do if isprime(A002110*(1+A000040)-1) then printf("%d,",A002110*(1+A000040)-1) ; fi ; if isprime(A002110*(1+A000040)+1) then printf("%d,",A002110*(1+A000040)+1) ; fi ; A002110 := A002110*A000040 : A000040 := nextprime(A000040) : od : # R. J. Mathar, Jan 26 2007

plim=45;k= FoldList[Times, 1, Prime[Range[plim]]];m=Table[k[[l]]+k[[l+1]],{l,plim}];Sort[Select[Join[m+1,m-1],PrimeQ]] (* James C. McMahon, Dec 15 2024 *)
Join[{2},Select[Sort[Flatten[#+{1,-1}&/@(Total/@Partition[FoldList[Times,Prime[Range[40]]],2,1])]],PrimeQ]] (* Harvey P. Dale, Jul 15 2025 *)

{m=37;for(n=0,m,p=primorial(n)+primorial(n+1);if(isprime(a=p-1),print1(a,","));if(isprime(a=p+1),print1(a,",")))} \\ Klaus Brockhaus, Jan 25 2007

genit(maxx)={arr=List();for(n=0, maxx, p=factorback(primes(n))+factorback(primes(n+1));if(ispseudoprime(p-1),listput(arr,p-1));if(ispseudoprime(p+1),listput(arr,p+1)));arr} \\ Bill McEachen, Jun 21 2021 (from David A. Corneth's code at A002110)

Edited, corrected and extended by Klaus Brockhaus and R. J. Mathar, Jan 25 2007

A177697 Sums of 3 distinct primorials.

Original entry on oeis.org

9, 33, 37, 38, 213, 217, 218, 241, 242, 246, 2313, 2317, 2318, 2341, 2342, 2346, 2521, 2522, 2526, 2550, 30033, 30037, 30038, 30061, 30062, 30066, 30241, 30242, 30246, 30270, 32341, 32342, 32346, 32370, 32550, 510513, 510517, 510518, 510541, 510542

Offset: 1

Jonathan Vos Post, May 11 2010

This is to numbers that are the sum of 3 different primes (A124867) as primorials (A002110) are to primes (A000040). The subsequence of primes among these sums of 3 distinct primorials begins: 37, 241, 2341, 2521, 30241, 32341, 512821, 540541.

			9 = 6+2+1
33 = 30+2+1
37 = 30+6+1
38 = 30+6+2
213 = 210+2+1

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

Take[Total/@Subsets[Join[{1},FoldList[Times,Prime[Range[10]]]],{3}]// Union,40] (* Harvey P. Dale, Nov 07 2017 *)

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

A250293 Primes p such that p#+1 is a semiprime, where # is the primorial (A034386).

Original entry on oeis.org

13, 19, 23, 43, 61, 67, 73, 83, 101, 139, 151, 173, 223, 251, 383, 457, 571, 673, 761

Offset: 1

Eric Chen, Dec 24 2014

The next candidate after 571 is 859. 859# + 1 is a 359-digit composite with no known factors. - Hugo Pfoertner, Feb 05 2021

			a(1) = 13 so 13# + 1 = 30031 = 59 * 509 is a semiprime.

factordb.com, Status of 859#+1.
Hisanori Mishima, Factorizations of many number sequences, PI Pn + 1 (n = 1 to 110) (2012).

Cf. A005234, A006862, A034386, A078778, A085725, A250294.

Prime[#]&/@(Module[{pmrl=FoldList[Times,Prime[Range[50]]]},Position[ pmrl, ?(PrimeOmega[ #+1]==2&)]]//Flatten) (* _Harvey P. Dale, Apr 29 2017 *)

a(n) = prime(A085725(n)). - Hugo Pfoertner, Feb 05 2021

a(16)-a(18) using factordb.com from Hugo Pfoertner, Feb 05 2021

Missing 571 inserted by Sean A. Irvine, Mar 03 2023

A338543 Numbers k such that 1 + Product_{i

Original entry on oeis.org

1, 2, 8, 233, 431

Offset: 1

Robert Israel, Nov 01 2020

Numbers k such that A062347(k-1) == -1 (mod prime(k)).

			a(3)=8 is a term because 1 + 2*3*5*7*11*13*17 = 510511 is divisible by prime(8)=19.

Mathematics StackExchange, Is 510511 the smallest euclid number e_n that divides by the next prime p.

Cf. A002110, A006862, A081618, A062347, A341805.

select(t -> 1+mul(ithprime(i),i=1..t-1) mod ithprime(t)=0, [$1..1000]);

isok(n) = ((1+vecprod(primes(n-1))) % prime(n)) == 0; \\ Michel Marcus, Nov 03 2020

a(n) = A081618(n)+1 for n >= 2.

A366811 The number of divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 4, 8, 4, 4, 8, 2, 8, 8, 4, 8, 16, 16, 4, 4, 16, 4, 8, 4, 16, 8, 4, 16, 16, 8, 8, 32, 8, 64, 4, 8, 4, 32, 16, 16, 4, 64, 8, 16, 8, 32, 64, 128, 4, 64, 8, 32, 8, 16, 4, 64, 32, 16, 32, 8, 32, 32, 32, 8, 8, 32, 32, 64, 8, 16, 16, 128, 32, 8, 16

Offset: 0

Sean A. Irvine, Oct 23 2023

nonn

			a(6) = 4 because the divisors of 13#+1 = 30031 are {1, 59, 509, 30031}.

Amiram Eldar, Table of n, a(n) for n = 0..98

Cf. A006862, A000005, A054988, A366808, A366812, A064145.

seq(numtheory[tau](mul(ithprime(k), k=1..n)+1), n=0..30);

Map[DivisorSigma[0, #] &, 1 + FoldList[Times, 1, Prime@ Range@ 19] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma0(prime(n)#+1) = A000005(A006862(n)).

A366812 The sum of the divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

3, 4, 8, 32, 212, 2312, 30600, 544880, 9727992, 223796952, 6500793728, 200560490132, 7461870828048, 309238687200848, 13161101219883792, 615548170519961184, 33465582319854797280, 1930276657976815787040, 117814338226489513454272, 7858321551223903311137992

Offset: 0

Sean A. Irvine, Oct 23 2023

nonn

			a(6) = 30600 because the divisors of 13#+1 = 30031 are {1, 59, 509, 30031}.

Amiram Eldar, Table of n, a(n) for n = 0..98

Cf. A006862, A000230, A054988, A366809, A366811, A366757.

seq(numtheory[sigma](mul(ithprime(k), k=1..n)-1), n=0..30);

Map[DivisorSigma[1, #] &, 1 + FoldList[Times, 1, Prime@ Range@ 19] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma(prime(n)#+1) = A000230(A006862(n)).

A057230 Numbers k such that k = p+q = r+s with pq = rs = primorial number(A002110) (pq) < (rs).

Original entry on oeis.org

31, 107, 391, 467, 34049, 67973, 176413

Offset: 1

Naohiro Nomoto, Sep 19 2000

The corresponding pairs of primorials are (3#, 4#), (4#, 5#), (5#, 6#), (5#, 6#), (7#, 9#), (8#, 9#), (8#, 10#). No other terms found up to 23#. - Michel Marcus, Feb 21 2016

a(8) > 3203982595205562774973. - Sean A. Irvine, May 26 2022

			31 = 30+1 = 21+10, where 30=30*1 and 210=21*10 are primorial numbers.

Sean A. Irvine, Java program (github)

Cf. A002110, A006862, A182987.

isprimo(n) = {if (n==1, return (1)); if (!issquarefree(n), return(0)); f = factor(n); #f~ == primepi(vecmax(f[,1]));}
isok(n) = {c = 0; for (na=1, n\2, if (isprimo(na*(n - na)), c++); if (c == 2, return(1)););} \\ Michel Marcus, Feb 20 2016

cp's OEIS Frontend

A297891 Numbers that divide exactly two Euclid numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A066542 Nonnegative integers all of whose anti-divisors are either 2 or odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A085725 Numbers k such that 1 + (product of first k primes) is a semiprime.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A125191 Primes of the form k# + (k+1)# +- 1, where k# = A002110(k) = primorial(k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Extensions

A177697 Sums of 3 distinct primorials.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A250293 Primes p such that p#+1 is a semiprime, where # is the primorial (A034386).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A338543 Numbers k such that 1 + Product_{i

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A366811 The number of divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

Views

Author

Keywords

Examples

A057230 Numbers k such that k = p+q = r+s with pq = rs = primorial number(A002110) (pq) < (rs).