A081639 -id:A081639 - cp's OEIS frontend

A129470 Primes p such that the largest prime factor of p+1 has Erdős-Selfridge class+ < N-1 if p is of class N+.

883, 1747, 2417, 2621, 3181, 3301, 3533, 3571, 3691, 3853, 4027, 4133, 4513, 4783, 4861, 4957, 5303, 5381, 5393, 5563, 5641, 5821, 6067, 6577, 6991, 7177, 7253, 7331, 8059, 8093, 8377, 8731, 8839, 8929, 8969, 9221, 9281, 9397, 9613, 9931

Offset: 1

M. F. Hasler, Apr 16 2007

In practice the class+ of a prime p is most often given by 1 + the class of the largest prime factor of p+1; terms of this sequence are counterexamples to this "rule". Terms of this sequence are at least of class 3+, since primes of class 1+ and 2+ have all prime factors of p+1 of class 1+. Terms a(k) of this sequence are >= -1 + 2*A005113(N-1) * nextprime(A005113(N-1)), where N is the class of a(k).

			a(3) = 883 = -1 + 2*13*17 is a prime of class 3+ since 13 is of class 2+, but the largest divisor of 883+1 is 17 which is only of class 1+.

Cf. A005105, A005106, A005107, A005108, A005113, A081633, A081634, A081635, A081636, A081637, A081638, A081639, A084071, A090468, A129469, A129471, A129472, A129473, A129474, A129475, A129477, A129478.

class(n,s=1)={n=factor(n+s)[,1];if(n[ #n]<=3,1,for(i=2,#n,n[1]=max(class(n[i],s)+1,n[1]));n[1])}; A129470(n=100,p=1,a=[])={ local(f); while( #a 3, f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f,2,-1, f[i]=class( f[i] ); if( f[i] > f[ #f], a=concat(a,p); /*print(#a," ",p);*/ break))); a}

A129474 Primes of Erdos-Selfridge class 14+.

Original entry on oeis.org

1704961513, 7281416041, 7638227617, 9462536833, 11934730597, 13237911481, 13282423003, 13522629793, 13942983841, 14185279861, 16029089501, 16221987853, 17434233041, 18171787987, 19639505461, 20717555041

Offset: 1

M. F. Hasler, Apr 16 2007

nonn

Primes of class r (or r+) are by definition the primes p for which p + 1 has all factors of a lower class < r, but at least one factor of class r - 1. See A005113 for more information.

a(1..149) calculated using A090468 up to 37.5e9, which gives A129474(150) > 75e9.

			a(1) = A005113[14] = 1704961513 = -1+2*852480757, where 852480757 = A090468[2]

M. F. Hasler, Table of n, a(n) for n = 1..149

Cf. A005113, A005105-A005108, A081633-A081639, A084071, A090468, A129475.

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)};
A129474=nextclass(A090468,1)

{ a(n) } = { p = 2*m*A090468(k)-1 | k=1,2,3... and m=1,2,3... such that p is prime and m has no factor of class > 13+ }

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

Jonathan Vos Post, Dec 16 2004

nonn

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

			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.

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

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

Cf. A084071, A090468, A129474, A129475.

More terms from David Wasserman, Mar 26 2008

cp's OEIS Frontend

A129470 Primes p such that the largest prime factor of p+1 has Erdős-Selfridge class+ < N-1 if p is of class N+.

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A129474 Primes of Erdos-Selfridge class 14+.

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

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