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 1-4 of 4 results.

A056637 a(n) is the least prime of class n-, according to the Erdős-Selfridge classification of primes.

Original entry on oeis.org

2, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107, 1432349099, 22111003847, 110874748763
Offset: 1

Views

Author

Robert G. Wilson v, Jan 31 2001

Keywords

Comments

A prime p is in class 1- if p-1 has no prime factor larger than 3. If p-1 has other prime factors, p is in class (c+1)-, where c- is the largest class of its prime factors. See also A005109.
1432349099 < a(16) <= 25782283783.
a(18) <= 619108107719, a(19) <= 19811459447009, a(20) <= 152772264735359. These upper limits can be found by generating class (n+1)- primes from a list of n- class primes; if the latter is sufficiently complete, one can deduce that there is no smaller (n+1)- prime. - M. F. Hasler, Apr 05 2007

Crossrefs

Cf. A005113, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.
Cf. A082449, A129246, A081640, A129248.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; a = Table[0, {15}]; a[[1]] = 2; k = 5; Do[c = ClassMinusNbr[ k]; If[ a[[c]] == 0, a[[c]] = k]; k = NextPrime[k], {n, 3, 7223000}]; a

Formula

a(n+1) >= 2*a(n)+1, since a(n+1)-1 is even and must have a factor of class n- which is odd (n>1) and >= a(n). a(n+1) <= min { p = 2*k*a(n)+1 | k=1,2,3... such that p is prime }, since a(n) is a prime of class n-. - M. F. Hasler, Apr 05 2007

Extensions

Extended by Robert G. Wilson v, Mar 20 2003
More terms from Don Reble, Apr 11 2003
a(16) and a(17) from M. F. Hasler, Apr 21 2007

A081641 a(n) = n-th prime of class 13- according to the Erdős-Selfridge classification.

Original entry on oeis.org

36449279, 53065907, 59681213, 69096887, 132756479, 135388367, 164255999, 179043637, 188991053, 207290663, 241560239, 279709259, 309550999, 364492781, 372993983, 377982103, 398007431, 406165099, 425633717, 445901987, 447609067, 516737983
Offset: 1

Views

Author

Robert G. Wilson v, Mar 23 2003

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.

Links

Crossrefs

Cf. A005113, A005105, A005106, A005107, A005108, A081633, A081633, A081635, A081636, A081637, A081638.
Cf. A056637, A081640, A129248.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[23733333], ClassMinusNbr[ Prime[ # ]] == 12 &]]
  • PARI
    A081641 = nextclassminus(A081640) /* cf. A081640 - M. F. Hasler, Apr 05 2007 */

Extensions

Edited by N. J. A. Sloane, May 14 2008 at the suggestion of R. J. Mathar.

A129249 Primes of Erdos-Selfridge class 15-.

Original entry on oeis.org

1432349099, 1749565397, 2771868719, 3790874279, 5288908679, 5804138567, 6273146879, 8123301983, 8594094589, 11055501923, 11809566403, 11914176299, 12647401799, 13432673083, 13925010299, 14208729979
Offset: 1

Views

Author

M. F. Hasler, Apr 16 2007, Apr 21 2007

Keywords

Comments

The fastest way to calculate the complete list of class N primes up to P is to use the given formula (or PARI function nextclass()) with the list of class (N-1) primes up to P/2.

Links

Programs

  • PARI
    class(n, s=1) = { if(!isprime(n),0, if(!(n=factor(n+s)[,1]) || n[ #n]<=3,1, for(i=2,#n,n[1]=max(class(n[i],s)+1,n[1]));n[1]))}
nextclass(a,s=-1,p,n=[])={if(!p,p=nextprime(a[ #a]+1)); print("producing primes of class ",1+class(a[1],s),["+","-"][1+(s<0)]," up to 2*",p); for(i=1,#a,for(k=1,p/a[i],if(isprime(2*k*a[i]-s),n=concat(n,2*k*a[i]-s))));vecsort(n)}
A129249=nextclass(A129248)

Formula

{ a(n) } = { p = 2*m*A129248(k)+1 | k=1,2,3... and m=1,2,3... such that p is prime and m has no factor of class > 14- }

A129250 Primes of Erdős-Selfridge class 16-.

Original entry on oeis.org

22111003847, 25782283783, 34824831403, 42970472971, 44905511759, 45490491349, 52486961911, 54560052479, 55437374381, 65803884467, 66333011539
Offset: 1

Views

Author

M. F. Hasler, Apr 21 2007

Keywords

Comments

Knowledge of a(k), k=1..9 allows us to establish A056637(17) = 1 + 2*a(9) = 110874748763.

Links

Crossrefs

Cf. A056637, A081640, A081641, A129248, A129249.

Programs

  • PARI
    nextclass( a, s=-1, p, n=[] )={ if( !p, p=nextprime(a[ #a]+1)); print("Computing all primes of next class up to ",2*p-s ); for( i=1,#a, for( k=1,p/a[i], if( is/*pseudo*/prime(2*k*a[i]-s), n=concat(n,2*k*a[i]-s); ) ) ); vecsort(n) }; A129250=nextclass(A129249)

Formula

{ a(n) } = { p=1+2*k*A129249(n); n=1,2,3..., k=1,2,3... such that p is prime and k has no factor of class > 15- }.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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