A059960 -id:A059960 - cp's OEIS frontend

A075589 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 9.

89, 151, 233, 257, 263, 271, 353, 367, 373, 503, 541, 571, 587, 601, 647, 653, 727, 733, 751, 977, 991, 1013, 1181, 1291, 1321, 1433, 1453, 1621, 1753, 1861, 2281, 2371, 2377, 2671, 3061, 3079, 3203, 3323, 3793, 4051, 4073, 4283, 4357, 4519, 4591, 4639

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

			For p = 233, the next prime number is 239. The numbers between 233 and 237 and the prime divisors are respectively 234 {2, 3, 13}, 235 {5, 47}, 236 {2, 59}, 237 {3, 79 }, 238 {2, 7, 17}. The set of prime divisors is {2, 3, 5, 7, 13, 17, 47, 59, 79} and has 9 elements, so 233 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075586, A075587, A075588.

a:=[]; for p in PrimesInInterval(2,5000) do b:={}; for s in [p..NextPrime(p)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 9 then Append(~a,p); end if; end for; a; // Marius A. Burtea, Sep 26 2019

More terms from Matthew Conroy, Apr 30 2003

A060213 Lesser of twin primes whose average is 6 times a prime.

Original entry on oeis.org

11, 17, 29, 41, 101, 137, 281, 617, 641, 821, 1697, 1877, 2081, 2237, 2381, 2657, 2801, 3461, 3557, 3917, 4637, 4721, 5441, 6197, 6701, 8537, 8597, 9677, 10937, 12161, 12377, 12821, 12917, 13217, 13721, 13757, 13997, 14081, 16061, 17417

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Lowest factor-density among all positive consecutive integer triples; for p > 41, last digit of p can be only 1 or 7 (see Alexandrov link, p. 15). - Lubomir Alexandrov, Nov 25 2001

			102197 is here because 102198 = 17033*6 and 17033 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998, see p. 15.

Cf. A002822, A058383, A059960, A027856, A001454, A001359, A060212.

map(t -> 6*t-1, select(p -> isprime(p) and isprime(6*p-1) and isprime(6*p+1), [2,seq(i,i=3..10000,2)]));

Transpose[Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]] == 2 && PrimeQ[Mean[#]/6]&]][[1]] (* Harvey P. Dale, May 04 2014 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 6) && isprime((n+1)/6); \\ Michel Marcus, Dec 14 2013

a(n) = 6 * A060212(n) - 1. - Sean A. Irvine, Oct 31 2022

Offset changed to 1 by Michel Marcus, Dec 14 2013

A060210 Largest prime factor of 1+smaller term of twin primes.

Original entry on oeis.org

2, 3, 3, 3, 5, 7, 5, 3, 17, 3, 23, 5, 5, 3, 11, 19, 5, 5, 47, 13, 29, 7, 3, 11, 29, 19, 5, 103, 107, 11, 5, 137, 23, 13, 7, 17, 43, 7, 59, 13, 3, 41, 71, 43, 31, 11, 17, 11, 19, 31, 67, 5, 139, 283, 41, 149, 13, 313, 23, 13, 37, 13, 347, 29, 11, 71, 17, 373, 7, 11, 13, 397, 17

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Also: Largest prime factor of the average, or the sum, of twin prime pairs. - M. F. Hasler, Jan 03 2011

			101 is the 9th lesser twin, 102 = 2*3*17, and its max p factor is 17=a(9).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A001359, A001454, A002822, A059960, A027856.

FactorInteger[1+#][[-1,1]]&/@Select[Partition[Prime[Range[500]],2,1], #[[2]]- #[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 16 2017 *)

p=3; for(n=1,1e3, until(o+2==p,p=nextprime(2+o=p)); print1(vecmax(factor(p-1)[,1])","))  \\ M. F. Hasler, Jan 03 2011

A078883 Lesser member p of a twin prime pair such that p+1 is 3-smooth.

Original entry on oeis.org

3, 5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767

Offset: 1

Reinhard Zumkeller, Dec 11 2002

nonn

			A000040(20) = 71 and 71+1 = 72 = 2^3*3^2 = A003586(17) and 71+2 = 73 = A000040(21), therefore 71 is a term.

Ray Chandler, Table of n, a(n) for n = 1..62 (terms < 10^1000)

Cf. A000040, A001359, A014574, A003586, A027856, A078884.

Apart from initial terms, same as A059960.

seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A027856(n)-1 = A078884(n)-2.

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

Original entry on oeis.org

7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633, 578415690713089, 1141260857376769, 57711166318706689

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Larger of twin primes p such that p-1 = (2^u)*(3^w), u,w >= 1.

			a(4) = 73, {71,73} are twin primes and (71 + 73)/2 = 72 = 2*2*2*3*3.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000)
Harsh Aggarwal, Table of n, a(n) for n = 62..91 (terms from 10^1000 to 10^20000)

Cf. A001454, A002822, A027856, A033845, A058383, A059960.

Take[Select[Sort[Flatten[Table[2^a 3^b,{a,250},{b,250}]]],AllTrue[#+{1,-1},PrimeQ]&]+1,23] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2019 *)

isok(p) = isprime(p) && isprime(p-2) && (vecmax(factor(p-1)[,1]) == 3); \\ Michel Marcus, Sep 05 2017

a(n) = A027856(n+1) + 1. - Amiram Eldar, Mar 17 2025

Name corrected by Sean A. Irvine, Oct 31 2022

A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.

Original entry on oeis.org

3, 5, 29, 419, 2309, 180179, 4084079, 106696589, 892371479, 103515091679, 4412330782859, 29682952539239, 22514519501013539, 313986271960080719, 22750921955774182169, 912496437361321252439, 26918644902158976946979, 1290172194953476680815969, 1901713815361424627522739779

Offset: 1

Labos Elemer, Mar 22 2001

nonn

a(349) has 1001 digits. - Michael S. Branicky, Apr 19 2025

			a(13) = -1 + (2*3*5*7*...*41)*k(13) = 304250263527210*74 and {22514519501013539, 22514519501013542} are the corresponding primes; k(13)=74 is the smallest suitable multiplier. Twin primes obtained from primorial numbers with k=1 multiplier seem to be much rarer (see A057706).
For j=1,2,3,4,5,6, a(j)=A001359(1), A059960(1), A060229(1), A060230(1), A060231(1), A060232(1) respectively.

Michael S. Branicky, Table of n, a(n) for n = 1..348

Cf. A001359, A002110, A006794, A014545, A057706, A059960, A060229, A060230, A060231, A060232.

a(n) = {my(q = prod(k=1, n, prime(k))); for(k=1, oo, if (isprime(q*k-1) && isprime(q*k+1), return(q*k-1)););} \\ Michel Marcus, Jul 10 2018

from itertools import count
from sympy import primorial, isprime
def a(n):
    p = primorial(n)
    return next(m-1 for m in count(p, p) if isprime(m-1) and isprime(m+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 18 2025

a(n) = p = k(n)*q(n)-1, where q(n)=A002110(n) and k(n)=A060256(n) is the smallest integer whose multiplication by the n-th primorial yields p+1.

a(2)=5 corrected by Ray Chandler, Apr 03 2009

a(18) and beyond from Michael S. Branicky, Apr 18 2025

A074167 Product of prime divisors of composite numbers between consecutive primes.

Original entry on oeis.org

1, 2, 6, 60, 6, 420, 6, 4620, 32760, 30, 471240, 14820, 42, 15180, 556920, 15273720, 30, 11171160, 164220, 6, 253333080, 2460, 587636280, 625757605200, 4620, 102, 289380, 6, 170940, 26848135265397670224000, 33540, 599888520, 138, 39560762839197600, 30

Offset: 1

Amarnath Murthy, Aug 30 2002

nonn

			a(4) = product of prime factors of composite numbers between 7 and 11 = 2 * 3 * (2 * 5) = 60.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A007947, A059960, A076978.

a:= n-> mul(mul(i[1], i=ifactors(j)[2]), j=ithprime(n)+1..ithprime(n+1)-1):
seq(a(n), n=1..40);  # Alois P. Heinz, May 29 2022

Array[Times @@ Flatten@ Map[FactorInteger[#][[All, 1]] &, Range[#1 + 1, #2 - 1]] & @@ Prime[{#, # + 1}] &, 35] (* Michael De Vlieger, May 29 2022 *)

a(n) = my(p=1); forcomposite(c=prime(n), prime(n+1), p*=factorback(factorint(c)[, 1])); p; \\ Michel Marcus, May 29 2022

a(n) = 6 <=> A000040(n) in { A059960 }. - Alois P. Heinz, May 29 2022

Corrected and extended by Joshua Zucker, May 08 2006

Offset corrected by Alois P. Heinz, May 29 2022

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A075589 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 9.

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A060213 Lesser of twin primes whose average is 6 times a prime.

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A060210 Largest prime factor of 1+smaller term of twin primes.

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A078883 Lesser member p of a twin prime pair such that p+1 is 3-smooth.

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A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

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A060255 Smaller of twin primes {p, p+2} whose average p+1 = k*q is the least multiple of the n-th primorial number q such that k*q-1 and k*q+1 are twin primes.

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A074167 Product of prime divisors of composite numbers between consecutive primes.

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A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.