OFFSET
1,1
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
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).
EXAMPLE
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.
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
PROG
(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<n, until( checkclass(p, 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
CROSSREFS
KEYWORD
more,nonn
AUTHOR
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
STATUS
approved