cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 17 results. Next

A075582 Duplicate of A059960.

Original entry on oeis.org

5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311
Offset: 1

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Author

Keywords

A052248 Greatest prime divisor of all composite numbers between p and next prime.

Original entry on oeis.org

2, 3, 5, 3, 7, 3, 11, 13, 5, 17, 19, 7, 23, 17, 29, 5, 31, 23, 3, 37, 41, 43, 47, 11, 17, 53, 3, 37, 61, 43, 67, 23, 73, 5, 31, 79, 83, 43, 89, 5, 61, 3, 97, 11, 103, 109, 113, 19, 29, 79, 5, 83, 127, 131, 89, 5, 137, 139, 47, 97, 151, 103, 13, 157, 163, 167, 173, 29, 13
Offset: 2

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Author

N. J. A. Sloane

Keywords

Comments

Or, largest of all prime factors of the numbers between prime(n) and prime(n+1).
a(n) = 3, 5, 7, 11, 13 iff prime(n) is in A059960, A080185, A080186, A080187, A080188 respectively. This sequence defines a mapping f of primes > 2 to primes (cf. A080189) and f(p) < p holds for all p > 2. - Klaus Brockhaus, Feb 10 2003
a(n) = A006530(A061214(n)). - Reinhard Zumkeller, Jun 22 2011

Examples

			a(8) = 11 since 20 = 2*2*5, 21 = 3*7, 22 = 2*11 are the numbers between prime(8) = 19 and prime(9) = 23.
For n=9, n-th prime is 23, composites between 23 and next prime are 24 25 26 27 29 of which largest prime divisor is 13, so a(9)=13.

Links

Crossrefs

Cf. A006530, A059960, A080185, A080186, A080187, A080188, A080189, A052180, A065091.

Programs

  • Haskell
    a052248 n = a052248_list !! (n-2)
a052248_list = f a065091_list where
   f (p:ps'@(p':ps)) = (maximum $ map a006530 [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011
  • Mathematica
    g[n_] := Block[{t = Range[Prime[n] + 1, Prime[n + 1] - 1]}, Max[First /@ Flatten[ FactorInteger@t, 1]]]; Table[ g[n], {n, 2, 72}] (* Robert G. Wilson v, Feb 08 2006 *)
cmp[{a_,b_}]:=Max[Flatten[FactorInteger/@Range[a+1,b-1],1][[All,1]]]; cmp/@ Partition[ Prime[Range[2,80]],2,1] (* Harvey P. Dale, May 16 2020 *)
  • PARI
    forprime(p=3,360,q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

Formula

a(n) = max(prime(n) < k < prime(n+1), A006530(k)).

A060212 Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.

Original entry on oeis.org

2, 3, 5, 7, 17, 23, 47, 103, 107, 137, 283, 313, 347, 373, 397, 443, 467, 577, 593, 653, 773, 787, 907, 1033, 1117, 1423, 1433, 1613, 1823, 2027, 2063, 2137, 2153, 2203, 2287, 2293, 2333, 2347, 2677, 2903, 3257, 3307, 3407, 3413, 3593, 3623, 3673, 3923
Offset: 1

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Author

Labos Elemer, Mar 20 2001

Keywords

Comments

Primes in A182521. Also all primes p for which A182481(p)=1. - Vladimir Shevelev, May 03 2012
Conjecture: a(n) ~ n*log(n)*log(n*log(n))*log(log(n)). - Carl R. White, Nov 16 2023

Links

Crossrefs

Cf. A001359, A002822, A014574, A027856, A058383, A059960, A182481, A182521, A294731.

Programs

  • Mathematica
    lst={}; Do[p=Prime[n]; If[PrimeQ[6*p-1] && PrimeQ[6*p+1], AppendTo[lst,p]], {n,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)
  • PARI
    forprime(p=2, 9999, if(isprime(6*p+1) & isprime(6*p-1), print(p))) \\ David Radcliffe, Apr 02 2016
  • Python
    from sympy import *; print([p for p in primerange(2,9999) if isprime(6*p-1) and isprime(6*p+1)]) # David Radcliffe, Apr 02 2016

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

Views

Author

Amarnath Murthy, Sep 26 2002

Keywords

Comments

a(10) > a(11).

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Sep 26 2002

Keywords

Comments

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

Links

Crossrefs

Cf. A052297, A075580, A059960, A075584, A075585, A075586, A075587, A075588, A075589.
See A080899 for another version.

Programs

  • Magma
    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

Extensions

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

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Author

Amarnath Murthy, Sep 26 2002

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    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

Extensions

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

Views

Author

Amarnath Murthy, Sep 26 2002

Keywords

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Sep 26 2002

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Sep 26 2002

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Sep 26 2002

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Magma
    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
  • Mathematica
    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 *)

Extensions

More terms from Matthew Conroy, Apr 30 2003
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