A097977 Smallest prime p such that p+n is the product of exactly n distinct primes.

2, 13, 67, 1151, 9277, 1616609, 1874723, 111546427, 2751478721, 862410107549, 747543645019, 3080843115635273, 1006366256282297, 2679162624135569701, 166366498382137547479, 993969333554296364249, 8302567374394710807373

Offset

1,1

Comments

The sequence goes on with a(18, 19, 20, ...) = 112733682549950000276752997, 32674073817649531442865671, 376147205196163170923414109829, ... - M. F. Hasler, Jan 14 2012

The terms are of the form a(n)=A002110(n+k)/Q - n, where Q is the product of k among the first n+k-1 primes, and these k primes include all prime factors of n, cf. examples. - M. F. Hasler, Jan 14 2012

This sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Jan 16 2012

Links

Table of n, a(n) for n=1..17.

Example

a(4)=1151, for example, is the first of a sequence of primes that continues with 1361, 2141, 2411, 2801, 3251, 3881, 3923, ...with the property 1151+4=3*5*7*11, 1361+4=3*5*7*13, 2141+5=3*5*11*13, 2801+4=3*5*11*17, 3251+4=3*5*7*31, 3881+4=3*5*7*37, 3923+4=3*7*11*17, ...
The terms can be written a(n)=A002110(n+k)/Q-n (cf. comment) as follows:
a(1)=A002110(2)/2 - 1
a(2)=A002110(3)/2 - 2
a(3)=A002110(4)/3 - 3
a(4)=A002110(5)/2 - 4
a(5)=A002110(7)/(5*11) - 5
a(6)=A002110(8)/(2*3) - 6
a(7)=A002110(9)/(7*17) - 7
a(8)=A002110(9)/2 - 8
a(9)=A002110(12)/(3*29*31) - 9
a(10)=A002110(14)/(2*5*37*41) - 10
a(11)=A002110(13)/(11*37) - 11
a(12)=A002110(16)/(2*3*41*43) - 12
a(13)=A002110(14)/13 - 13
a(14)=A002110(18)/(2*7*53*59) - 14
a(15)=A002110(18)/(3*5*47) - 15
a(16)=A002110(18)/(2*59) - 16
a(17)=A002110(20)/(17*59*67) - 17
a(18)=A002110(22)/(2*3*67*71) - 18
etc.

Mathematica

Table[k := 1; While[Not[Length[FactorInteger[Prime[k] + n]] == n], k++ ]; Prime[k], {n, 1, 7}] (* Stefan Steinerberger, Apr 03 2006 *)

Prog

(PARI) A097977(n, show=0, LIM=9)={ my(P=A002110(n), M); for(i=0, LIM, i && P*=prime(n+i); forvec(v=vector(i, j, [1, n+i-1]), t=P/prod(j=1, #v, prime(n+i-v[j]))-n; M && t>=M && (v[#v]+=n+i) && next; isprime(t) || next; M=t; show && print([t, i, v]), 2)); M} \\ M. F. Hasler, Jan 14 2012
(Haskell)
a097977 n = head [p | p <- dropWhile (<= n) a000040_list,
a001221 (p + n) == n]
-- Reinhard Zumkeller, Jun 24 2015

Crossrefs

Cf. A000040, A001221, A002110, A191587.

Sequence in context: A106999 A054566 A038144 * A136780 A301944 A128743

Adjacent sequences: A097974 A097975 A097976 * A097978 A097979 A097980

Keyword

hard,nonn

Author

Lekraj Beedassy, Sep 07 2004

Extensions

2 more terms from Stefan Steinerberger, Apr 03 2006

a(8)-a(15) from Donovan Johnson, Oct 05 2010

a(16) from Donovan Johnson, Jan 14 2012

a(17)-a(20) from M. F. Hasler, Jan 14 2012

Status

approved