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.

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A005113 Smallest prime in class n (sometimes written n+) according to the Erdős-Selfridge classification of primes.

Original entry on oeis.org

2, 13, 37, 73, 1021, 2917, 15013, 49681, 532801, 1065601, 8524807, 68198461, 545587687, 1704961513, 23869461181, 288310406533
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

A prime p is in class 1 if (p+1)'s largest prime factor is 2 or 3. If (p+1) has other prime factors, p's class is one more than the largest class of its prime factors. See also A005105.
John W. Layman observes that for n=10..13, the ratios r(n)= a(n)/a(n-1) are increasingly close to an integer, being 1.9999981, 7.99999906, 8.00000059 and 7.999999985.
Layman's observation is a consequence of a(n+1) = m*a(n)-1 for (n,m)=(1,7),(3,2),(4,14),(9,2),(10,8),(12,8),(14,14), while a(12) = 8 a(11)+5 is a coincidence which does not fit into that scheme. This relationship is not unusual since any N+ prime p is by definition such that p+1 = m*q where q is a (N-1)+ prime and m = (p+1)/q must be even since p,q are odd (except for q=2, allowing the odd m=7 for n=1 above) and the least N+ prime has good chances of having q equal to the least (N-1)+ prime. - M. F. Hasler, Apr 09 2007
a(n+1) >= 2*a(n)-1 since a(n+1)+1 = p*q with p of class n+ (thus >= a(n) and odd) and thus q >= 2 (even and positive). a(n+1) <= min { p = 2*k*a(n)-1 | k=1,2,3,... such that p is prime }. - M. F. Hasler, Apr 02 2007
a(17) <= 1833174628057, with equality if 916587314029 is the 10th 16+ prime; a(18) <= 3666349256113, with equality if a(17) = 1833174628057; a(19) <= 65994286610033, with equality if 41431295033731 is the third 18+ prime; a(20) <= 764276710625653, with equality if 382138355312827 is the third 19+ prime. - M. F. Hasler, Apr 09 2007

Examples

			1553 is in class 4 because 1553+1 = 2*3*7*37; 7 is in class 1 and 37 is in class 3. 37 is in class 3 because 37+1 = 2*19 and 19 is in class 2. 19 is in class 2 because 19+1 = 2*2*5 and 5 is in class 1. 5 is in class 1 because 5+1=2*3.

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A056637, A005105, A005106, A005107, A005108, A019268.
Cf. A081633 - A081639, A084071, A090468, A129474, A129475, A129469.

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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; a = Table[0, {15}]; a[[1]] = 2; k = 5; Do[c = ClassPlusNbr[ k]; If[ a[[c]] == 0, a[[c]] = k]; k = NextPrime[k], {n, 1, 28700000}]; a
  • PARI
    checkclass(n,p)={ n=factor(n+1)[,1]; n[#n] <= 3 && return(1); (#p <= 1 || n[#n] < p[#p]) && return(2); n[1]=p[#p]; p=vecextract(p,"^-1"); forstep( i=#n,2,-1, n[i] < n[1] && break; checkclass(n[i],p) > #p && return(2+#p)) }
A005113(n,p,a=[])={ while( #a #a, p=nextprime(p+1)); a=concat(a,p); p=a[#a]*2-2); a } \\ A005113(11) takes < 10 sec @ 2 GHz in 2007; less than 2.5 sec @ 2 GHz in 2013. \\ M. F. Hasler, Apr 02 2007
  • PARI
    class(n, s=+1 /* for n+ class; -1 for n- class */)={ isprime(n) || return; (( n=factor(n+s)[,1] ) && n[ #n]>3 ) || return(1); vecsort( vector( #n,i,class( n[i],s )))[#n]+1 }
someofnextclass( a, limit=0, s=0, b=[], p)={ if(!s,/* guess + or - */ s=( class(a[1]) && class(a[1])==class(a[2]) )*2-1 ); print("looking for primes of class ", 1+class( a[1], s), ["+","-"][1+(s<0)] ); for( i=1,#a, p=-s; until( p>=limit, until( isprime(p), p+=a[i]<<1 ); b=concat(b,p); if( !limit, limit=p)) ); vecsort(b) };
c=A090468; for(i=15,20,c=someofnextclass(c,9e12);print("least prime of class ",i,"+ is <= ",c[1])) \\  M. F. Hasler, Apr 09 2007

Extensions

Extended through a(12) by Robert G. Wilson v
a(13) from John W. Layman
a(14) from Don Reble, Apr 11 2003
a(15) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006
a(7) corrected by Tomás Oliveira e Silva, Oct 27 2006
a(16) calculated using A129475(n) up to n=19 by M. F. Hasler, Apr 16 2007
Edited by Max Alekseyev, Aug 17 2013

A019269 Let Dedekind's psi(m) = product of (p+1)p^(e-1) for primes p, where p^e is a factor of m. Iterating psi(m) eventually results in a number of form 2^a*3^b. a(n) is the number of steps to reach such a number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 0, 1, 2, 1, 1, 0, 1, 1, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 2, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 0, 4, 3, 2, 2, 1, 2, 2, 1, 0, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 0, 3, 2, 1
Offset: 1

Views

Author

Jud McCranie

Keywords

Comments

a(A019268(n)) = n and a(m) <> n for m < A019268(n). [Reinhard Zumkeller, Apr 12 2012]

References

  • Peter Giblin, "Primes and Programming - an Introduction to Number Theory with Computation", page 118.
  • R. K. Guy, "Unsolved Problems in Number Theory", section B41.

Links

Crossrefs

Cf. A001615, A065333.

Programs

  • Haskell
    a019269 n = snd $ until ((== 1) . a065333 . fst)
                        (\(x, i) -> (a001615 x, i+1)) (n, 0)
-- Reinhard Zumkeller, Apr 12 2012
  • Mathematica
    psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times @@ (pp+1)*Times @@ (pp^(ee-1))]];
a[n_] := Length[NestWhileList[psi, n, FactorInteger[#][[-1, 1]] > 3&]] - 1;
a /@ Range[99] (* Jean-François Alcover, Jan 18 2020 *)

A101253 a(n) = n-th prime of Erdős-Selfridge classification n+.

Original entry on oeis.org

2, 19, 113, 617, 1877, 8753, 52517, 255043, 1532173, 9287521, 48499459, 353653063, 2136716521, 18171787987, 111795382441
Offset: 1

Views

Author

Jonathan Vos Post, Dec 16 2004

Keywords

Comments

Diagonalization of the Erdős-Selfridge classification of primes n+. See A101231 for diagonalization of the Erdős-Selfridge classification of primes n-.

Examples

			a(1) = 2 because 2 is the first element of A005105.
a(2) = 19 because 19 is the 2nd element of A005106.
a(3) = 113 because 113 is the 3rd element of A005107.
a(4) = 617 because 617 is the 4th element of A005108.
a(5) = 1877 because 1877 is the 5th element of A081633.
a(6) = 8753 because 8753 is the 6th element of A081634.

References

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

Crossrefs

Cf. A101231, A005105, A005106, A005107, A005108, A081633, A081634, A081635, A081636, A081637, A081638, A081639, A005113, A019268.
Cf. A084071, A090468, A129474, A129475.

Extensions

More terms from David Wasserman, Mar 26 2008
Showing 1-3 of 3 results.

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