A129472 -id:A129472 - 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}

A129471 Primes p of Erdos-Selfridge class 3+ with largest prime factor of p+1 not of class 2+.

Original entry on oeis.org

883, 1747, 2417, 2621, 3301, 3533, 3571, 3691, 3853, 4027, 4133, 4783, 4861, 5303, 5381, 5393, 5563, 5641, 5821, 6577, 6991, 7253, 7331, 8059, 8093, 8377, 8839, 8929, 8969, 9221, 9281, 9613, 9931, 10069, 10477, 10487, 10601, 10607, 10903, 11491

Offset: 1

M. F. Hasler, Apr 17 2007

See A129470

			a(1) = 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+.

Subsequence of A129470; see also A129472-A129473, A129477-A129478, A129469, A005113, A005105-A005107.

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])}; A129471(n=100,p=1,a=[])={ local(f); while( #a 3 & 2 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 3 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 3, a=concat(a,p); /*print(#a," ",p)*/)); a}

A129473 Primes p of Erdos-Selfridge class 5+ with largest prime factor of p+1 not of class 4+.

Original entry on oeis.org

15913, 18541, 22921, 36353, 47741, 49201, 52267, 55333, 60589, 64969, 66137, 66721, 69203, 72707, 73291, 74167, 75773, 78401, 79861, 80737, 82051, 84533, 90227, 90373, 95191, 95483, 95629, 97673, 99133, 101323, 103951, 104681, 104827

Offset: 1

M. F. Hasler, Apr 17 2007

See A129470

			a(1) = 15913 = -1+2*73*109 is a prime of class 5+ since 73 is of class 4+, but the largest divisor of 15913+1 is 109 which is only of class 2+.

Subsequence of A129470; see also A129471-A129472, A129477-A129478, A129469, A005113, A005105-A005108, A081633.

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])}; A129473(n=100,p=1,a=[])={ local(f); while( #a 3 & 4 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 5 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 5, a=concat(a,p); /*print(#a," ",p)*/)); a}

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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A129471 Primes p of Erdos-Selfridge class 3+ with largest prime factor of p+1 not of class 2+.

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PARI

A129473 Primes p of Erdos-Selfridge class 5+ with largest prime factor of p+1 not of class 4+.

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Author

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Examples

Crossrefs

Programs

PARI