1,1

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.

John W. Layman (layman(AT)math.vt.edu) 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.

2*a(15)-1 = 47738922361 < a(16) <= 429650301257 = 9*2*a(15)-1 - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 02 2007

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 (Maximilian.Hasler(AT)gmail.com), Apr 09 2007

a(16) <= 288310406533, with equality if 144155203267 is the 12th 15+ prime; 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 (Maximilian.Hasler(AT)gmail.com), Apr 09 2007

a(16) calculated using A129475(n) up to n=19. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 16 2007

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).

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 (Maximilian.Hasler(AT)gmail.com), Apr 02 2007

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.

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]; if( n[ #n] <= 3, return(1)); if( #p <= 1 | n[ #n] < p[ #p], return(2)); n[1]=p[ #p]; p=vecextract(p, "^-1"); forstep( i=#n, 2, -1, if( n[i] < n[1], break); if( checkclass(n[i], p) > #p, return(2+#p))); 0 }; A005113(n, p, a=[])={ while( #a<n, until( checkclass(p, a) > #a, p=nextprime(p+1)); a=concat(a, p); p=a[ #a]*2-2); a }; A005113(11) /* < 10 sec @ 2 GHz */ - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 02 2007

(PARI) class( n, s=+1 /* +1 for n+ class, -1 for n- class */ ) = { if( isprime(n), if(( n=factor(n+s)[, 1] ) & n[ #n]>3, vecsort(vector(#n, i, class(n[i], s)))[ #n]+1, 1), 0) }; 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 (Maximilian.Hasler(AT)gmail.com), Apr 09 2007

Cf. A056637, A005105, A005106, A005107, A005108, A019268.

Cf. A081633 - A081639, A084071, A090468, A129474 - A129476, A129469.

Sequence in context: A063092 A034011 A085497 this_sequence A072857 A119535 A011919

Adjacent sequences: A005110 A005111 A005112 this_sequence A005114 A005115 A005116

more,nonn

N. J. A. Sloane (njas(AT)research.att.com).

Extended through a(12) by Robert G. Wilson v (rgwv(AT)rgwv.com).

a(13) from John W. Layman (layman(AT)math.vt.edu).

a(14) from Don Reble, Apr 11, 2003. 4294967296 < a(15) <= 23869461181.

a(15) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006

a(7) corrected by Tomas Oliveira e Silva, Oct 27 2006

a(16) from M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 16 2007