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A005113 Smallest prime in class n (sometimes written n+) according to the Erdős-Selfridge classification of primes.
(Formerly M2057)
41
2, 13, 37, 73, 1021, 2917, 15013, 49681, 532801, 1065601, 8524807, 68198461, 545587687, 1704961513, 23869461181, 288310406533 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=1..16.

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

(Contribution from M. F. Hasler, Apr 02 2007) (Start)

(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. \\ (End)

(Contribution from M. F. Hasler, Apr 09 2007) (Start)

(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])) \\ (End)

CROSSREFS

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

Cf. A081633 - A081639, A084071, A090468, A129474, A129475, A129469.

Sequence in context: A034011 A085497 A265775 * A239196 A072857 A119535

Adjacent sequences:  A005110 A005111 A005112 * A005114 A005115 A005116

KEYWORD

more,nonn

AUTHOR

N. J. A. Sloane.

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

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Last modified August 19 07:12 EDT 2017. Contains 290794 sequences.