A052297 -id:A052297 - cp's OEIS frontend

A061214 Product of composite numbers between the n-th and (n+1)st primes.

1, 4, 6, 720, 12, 3360, 18, 9240, 11793600, 30, 45239040, 59280, 42, 91080, 311875200, 549853920, 60, 1072431360, 328440, 72, 2533330800, 531360, 4701090240, 60072730099200, 970200, 102, 1157520, 108, 1367520, 1063186156509747740870400000, 2146560, 43191973440

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

			a(4) = 8 * 9 * 10 = 720. 7 is the fourth prime and 11 is the fifth prime. a(5) = 12 as 11 and 13 both are primes.

T. D. Noe, Table of n, a(n) for n = 1..2000

Cf. A006530, A052248, A052297, A072472, A361760, A361761.

Cf. A046933 and A054265 (number and sum of these composites).

a061214 n = a061214_list !! (n-1)
a061214_list = f a000040_list where
   f (p:ps'@(p':ps)) = (product [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011

A061214 := proc(n)
    local k ;
    product(k,k=ithprime(n)+1..ithprime(n+1)-1) ;
end proc: # R. J. Mathar, Apr 23 2013

Table[Times@@Range[Prime[n]+1,Prime[n+1]-1],{n,30}] (* Harvey P. Dale, Jun 14 2011 *)
Times@@Range[#[[1]]+1,#[[2]]-1]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Apr 23 2022 *)

{ n=0; q=2; forprime (p=3, prime(2001), a=1; for (i=q + 1, p - 1, a*=i); q=p; write("b061214.txt", n++, " ", a) ) } \\ Harry J. Smith, Jul 19 2009

v=primes(100);for(i=1,#v-1,v[i]=prod(j=v[i]+1,v[i+1]-1,j));vecextract(v,"1..-2") \\ Charles R Greathouse IV, Feb 27 2012

from math import prod
from sympy import prime
def A061214(n): return prod(i for i in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 10 2022

A006530(a(n)) = A052248(n) for n > 1. - Reinhard Zumkeller, Jun 22 2011

More terms from James Sellers, Apr 24 2001

Better definition from T. D. Noe, Jan 21 2008

A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

Original entry on oeis.org

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

Labos Elemer, Mar 02 2001

nonn

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

Primes p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is 2. - Amarnath Murthy, Sep 26 2002

			a(11)+1 = 2*2*2*3*3*3*3*3*3*3*3*3*3 = 472392.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000, first 49 terms from T. D. Noe)

Cf. A014574, A002822, A033845, A058383, A027856.
Cf. A052297, A075581, A075580, A075583, A075584, A075585, A075586, A075587, A075588, A075589.
Apart from initial terms, same as A078883.

nn=10^15; Sort[Reap[Do[n=2^i 3^j; If[n<=nn && PrimeQ[n-1] && PrimeQ[n+1], Sow[n-1]], {i, Log[2, nn]}, {j, Log[3, nn]}]][[2, 1]]]
Select[Select[Partition[Prime[Range[38*10^5]],2,1],#[[2]]-#[[1]]==2&][[All,1]],FactorInteger[#+1][[All,1]]=={2,3}&] (* The program generates the first 15 terms of the sequence. *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 1, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

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

A075580 Smallest prime p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 53, 73, 89, 359, 139, 401, 181, 113, 211, 293, 661, 863, 773, 523, 1933, 1831, 1069, 1381, 887, 1637, 1129, 1669, 1951, 4027, 3469, 4177, 6397, 2477, 2971, 5531, 1327, 4297, 4831, 5351, 5591, 9973, 11743, 13187, 8467, 27851, 18803

Offset: 0

Amarnath Murthy, Sep 26 2002

nonn

a(10) > a(11).

Donovan Johnson, Table of n, a(n) for n = 0..257 (first 200 terms from T. D. Noe)
XIAO Gang, Factoris, factors huge integers.

Cf. A052297, A075581, A059960, A075583, A075584, A075585, A075586, A075587, A075588, A075589.

Table[i=1; While[Length[Union[Flatten[Table[First/@FactorInteger[j],{j,(x=Prime[i])+1,NextPrime[x]-1}]]]]!=n,i++]; x,{n,0,47}] (* Jayanta Basu, May 25 2013 *)

Corrected and extended by Sam Alexander, Oct 20 2003

A075583 Primes p such that the composite numbers between p and the next prime together contain exactly three distinct prime factors.

Original entry on oeis.org

7, 29, 41, 59, 101, 137, 149, 179, 197, 227, 239, 269, 281, 311, 347, 521, 599, 617, 641, 809, 821, 827, 881, 1031, 1061, 1277, 1451, 1487, 1607, 1619, 1667, 1697, 1787, 1871, 1877, 1997, 2027, 2081, 2087, 2111, 2237, 2267, 2381, 2657, 2687, 2711, 2801, 2999

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

p+2 is prime for all members p except 7. - David Wasserman, Jan 20 2005

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

Cf. A052297, A075580, A059960, A075584, A075585, A075586, A075587, A075588, A075589.

See A080899 for another version.

a:=[]; for k in PrimesInInterval(2,3000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 3 then  Append(~a,k); end if; end for; a; // Marius A. Burtea, Sep 26 2019

More terms from David Wasserman, Jan 20 2005

A075584 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 4.

Original entry on oeis.org

13, 79, 419, 461, 569, 659, 857, 1019, 1049, 1091, 1229, 1289, 1301, 1319, 1427, 1481, 1721, 1931, 1949, 2129, 2141, 2339, 2549, 2789, 2969, 3119, 3299, 3329, 3359, 3389, 3539, 3821, 3929, 4001, 4019, 4091, 4157, 4217, 4229, 4241, 4259, 4421, 4649, 4787

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

It seems that for n > 2, a(n) + 2 is prime. Any counterexample p must have p > 3^1000000 and p+4 prime, and {p+1, p+2, p+3} must contain a power of 2 or 3. (The case where p+1 and p+3 are 3-smooth case can be ruled out via Catalan's conjecture/Mihăilescu's theorem.) In particular known Mersenne factorizations rule out the Fermat case below 2^144115188075855872 - 3, GIMPS rules out the Mersenne case below 2^36046457 - 1, and the exponents in A014224 rule out the remaining case below 3^1000000 - 2. - Charles R Greathouse IV, Jun 01 2016

			For p = 79, the next prime number is 83. The numbers between 79 and 83 and the prime divisors are respectively  80 { 2, 5 }, 81 { 3 }, 82 { 2, 41 }. The set of prime divisors is { 2, 3, 5, 41 } and has 4 elements, so 79 is a term. - _Marius A. Burtea_, Sep 26 2019

Daria Micovic, Table of n, a(n) for n = 1..10000

Cf. A052297, A075581, A075580, A059960, A075583, A075585, A075586, A075587, A075588, A075589.

a:=[]; for p in PrimesInInterval(2,4800) 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 4 then Append(~a,p); end if; end for; a; // Marius A. Burtea, Sep 26 2019

Select[Prime@ Range@ 650, Length@ Union@ Flatten@ Map[First /@ FactorInteger@ # &, Select[Range[#, NextPrime@ #], CompositeQ]] == 4 &] (* Michael De Vlieger, May 27 2016 *)
Join[{13,79},Select[Prime[Range[23,650]],PrimeQ[#+2]&&PrimeNu[#+1]==4&]] (* This program assumes the correctness of the conjecture by Charles R. Greathouse, IV, in the Comments. *) (* Harvey P. Dale, Jun 07 2019 *)

lista(nn)=forprime(p=2, nn, allp = []; forcomposite (c = p+1, nextprime(p+1), allp = Set(concat(allp, (factor(c)[,1])~));); if (#allp == 4, print1(p, ", "));); \\ Michel Marcus, May 28 2016

is(n)=if(!isprime(n), return(0)); if(isprime(n+2), return(omega(n+1)==4)); if(isprime(n+4), omega(n+1)+omega(n+2)+omega(n+3)==5, 0)
list(lim)=my(v=List(),t,p); lim\=1; for(e=4,logint(lim+2,3), p=precprime(3^e); if(isprime(p+4) && is(p), listput(v,p))); for(e=4,logint(lim+3,2), p=precprime(2^e); if(isprime(p+4) && is(p), listput(v,p))); p=2; forprime(q=3,lim+2, if(q-p==2 && omega(p+1)==4, listput(v,p)); p=q); Set(v) \\ Charles R Greathouse IV, Jun 01 2016

More terms from Matthew Conroy, Apr 30 2003
Name edited by Michel Marcus, May 28 2016
Typo in name fixed by Daria Micovic, Jun 01 2016

A075585 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 5.

Original entry on oeis.org

19, 23, 37, 43, 97, 127, 223, 499, 673, 1213, 2309, 2729, 6089, 6269, 7589, 8969, 9239, 9281, 10709, 11549, 11969, 12539, 13397, 14321, 15329, 16829, 17489, 18059, 19139, 19379, 19469, 19889, 20747, 21317, 21839, 22109, 22619, 23369, 23561, 24179

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

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

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075586, A075587, A075588, A075589.

a:=[]; for k in PrimesInInterval(2,25000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 5 then  Append(~a,k); end if; end for; a; // Marius A. Burtea, Sep 26 2019

dpd5Q[p_]:=Length[Union[Flatten[FactorInteger[#][[All,1]]&/@Range[ p+1,NextPrime[ p]-1]]]]==5; Select[Prime[Range[3000]],dpd5Q] (* Harvey P. Dale, Aug 11 2021 *)

More terms from Sam Alexander, Oct 21 2003

A075586 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 6.

Original entry on oeis.org

31, 47, 67, 103, 109, 163, 193, 277, 313, 349, 379, 397, 457, 463, 487, 877, 1087, 1093, 1279, 1303, 1567, 1873, 2269, 2347, 2473, 2797, 3697, 4447, 4789, 4999, 5077, 5413, 5503, 5923, 6007, 6217, 6469, 6997, 7603, 7639, 7723, 7933, 8779, 9277, 10159

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

For very large n, the probability of a(n) not being a twin prime is extremely small, unless the twin primes conjecture is false. - Sam Alexander, Oct 20 2003

			Between 31 and the next prime 37, there are 5 composite numbers whose prime divisors are respectively for 32: {2}, 33: {3,11}, 34: {2,17}, 35: {5,7} and 36: {2,3}; hence, these distinct prime divisors are {2,3,5,7,11,17}, the number of these distinct prime divisors is 6, so 31 is a term. - _Bernard Schott_, Sep 26 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
XIAO Gang, Factoris, a program that factorizes huge integers

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075587, A075588, A075589.

a:=[]; for k in PrimesInInterval(2,10000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 6 then  Append(~a,k); end if; end for; a; // Marius A. Burtea, Sep 26 2019

Select[Partition[Prime[Range[1250]],2,1],Length[Union[Flatten[ FactorInteger/@ Range[ #[[1]]+1,#[[2]]-1],1][[All,1]]]]==6&][[All,1]] (* Harvey P. Dale, May 25 2020 *)

More terms from Sam Alexander, Oct 20 2003

A075587 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 7.

Original entry on oeis.org

53, 61, 157, 229, 307, 439, 613, 757, 769, 823, 853, 859, 883, 907, 937, 967, 1009, 1297, 1423, 1429, 1447, 1483, 1489, 1549, 1597, 1663, 1693, 1993, 2083, 2137, 2203, 2239, 2389, 2437, 2659, 2689, 2707, 2749, 2833, 2857, 2953, 3019, 3037, 3163, 3187

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

Daria Micovic, Table of n, a(n) for n = 1..10000

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075586, A075588, A075589.

pd[n_]:=Transpose[FactorInteger[n]][[1]]; Transpose[Select[ Partition[ Prime[ Range[500]],2,1],Length[Union[Flatten[ pd/@Range[First[#]+1, Last[#]-1]]]] == 7&]][[1]] (* Harvey P. Dale, Jun 15 2013 *)
Select[Prime@ Range@ 500, Length@ Union@ Flatten@ Map[First /@ FactorInteger@ # &, Select[Range[#, NextPrime@ #], CompositeQ]] == 7 &] (* Michael De Vlieger, May 27 2016 *)

Corrected and extended by Matthew Conroy, Apr 30 2003

Name edited by Michel Marcus, May 28 2016

A075588 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 8.

Original entry on oeis.org

73, 83, 131, 167, 173, 251, 331, 383, 443, 563, 643, 739, 971, 1123, 1223, 1367, 1579, 1609, 1783, 1867, 1999, 2293, 2539, 2617, 2683, 3083, 3217, 3253, 3343, 3457, 3847, 4003, 4513, 4783, 4813, 4969, 5167, 5233, 5527, 5737, 5779, 5839, 5857, 6199, 6733

Offset: 1

Amarnath Murthy, Sep 26 2002

nonn

			For p = 131, the next prime number is 137. The numbers between 131 and 137 and the prime factors are respectively 132 { 2, 3, 11 }, 133 { 7, 19 }, 134 { 2, 67 }, 135 { 3, 5 }, 136 { 2, 17 }. The set of prime divisors is { 2, 3, 5, 7, 11, 17, 19, 67 } and has 8 elements, so 131 is a term. - _Marius A. Burtea_, Sep 26 2019

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

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075586, A075587, A075589.

a:=[]; for p in PrimesInInterval(2,7000) 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 8 then Append(~a,p); end if; end for; a; // Marius A. Burtea, Sep 26 2019

Select[Partition[Prime[Range[1000]],2,1],Length[Union[ Flatten[ FactorInteger[ Range[ #[[1]]+1,#[[2]]-1]],1][[All,1]]]]==8&][[All,1]] (* Harvey P. Dale, Dec 26 2019 *)

More terms from Matthew Conroy, Apr 30 2003

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

Original entry on oeis.org

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

cp's OEIS Frontend

A061214 Product of composite numbers between the n-th and (n+1)st primes.

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A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

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A075580 Smallest prime p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is n.

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A075583 Primes p such that the composite numbers between p and the next prime together contain exactly three distinct prime factors.

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Magma

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A075584 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 4.

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A075585 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 5.

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Magma

Mathematica

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A075586 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 6.

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