A081640 -id:A081640 - cp's OEIS frontend

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

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

Offset: 1

Robert G. Wilson v, Jan 31 2001

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

Cf. A005113, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430.

Cf. A082449, A129246, A081640, A129248.

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

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

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

A129248 a(n) = n-th prime of class 14- according to the Erdős-Selfridge classification.

Original entry on oeis.org

377982107, 437391349, 716174549, 742922699, 1385934359, 1603768277, 1780127639, 1790436371, 1895437139, 1968261067, 2066951933, 2109424013, 2148523669, 2150787839, 2238778847, 2299583987, 2334899909, 2368121663

Offset: 1

M. F. Hasler, Apr 05 2007, Apr 21 2007

nonn

a[1..2] calculated using A081641[1..11]; a[3] <= 716174549.

M. F. Hasler, Table of n, a(n) for n=1..339

Cf. A056637, A081640, A081641, A129249, A129250.

A129248 = nextclassminus(A081641) /* cf. A081640 */

{ a(n) } = { p = 2*m*A081641(k)+1 | k=1,2,...,oo and m=1,2,... such that p is prime and m has no factor of class > 13- }

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

Robert G. Wilson v, Mar 23 2003

nonn

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

M. F. Hasler, Table of n, a(n) for n=1..1641

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

Cf. A056637, A081640, A129248.

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 &]]

A081641 = nextclassminus(A081640) /* cf. A081640 - M. F. Hasler, Apr 05 2007 */

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

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

M. F. Hasler, Apr 21 2007

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

Index entries for sequences related to the Erdos-Selfridge classification

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

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)

{ 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- }.

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

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A129248 a(n) = n-th prime of class 14- according to the Erdős-Selfridge classification.

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

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A129250 Primes of Erdős-Selfridge class 16-.

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