

A072875


Smallest start for a run of n consecutive numbers of which the ith has exactly i prime factors.


13




OFFSET

1,1


COMMENTS

By definition, each term of this sequence is prime.
a(11) <= 1452591346605212407096281241 (Frederick Schneider), see primepuzzles link.  sent by amd64(AT)vipmail.hu, Dec 21 2007
Prime factors counted with multiplicity.  Harvey P. Dale, Mar 09 2021


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 61, p. 22, Ellipses, Paris 2008.


LINKS

Table of n, a(n) for n=1..10.
alt.math.recreational thread, Consecutive numbers with counting prime factors
J. M. Bergot, Puzzle 425. Consecutive numbers, increasing quantity of prime factors


EXAMPLE

a(3)=61 because 61(prime), 62(=2*31), 63(=3*3*7) have exactly 1, 2, 3 prime factors respectively, and this is the smallest solution;
a(6)=807905281: 807905281 is prime; 807905281+1=2*403952641;
807905281+2=3*15733*17117; 807905281+3=2*2*1871*107951;
807905281+4=5*11*43*211*1619; 807905281+5=2*3*3*3*37*404357;
807905281+6=7*7*7*7*29*41*283; 807905281 is the smallest number m such that m+k is product of k+1 primes for k=0,1,2,3,4,5,6.


MATHEMATICA

(* This program is not suitable to compute a large number of terms. *) nmax = 6; kmax = 10^6; a[1] = 2; a[n_] := a[n] = For[k = a[n1]+n1, k <= kmax, k++, If[AllTrue[Range[0, n1], PrimeOmega[k+#] == #+1&], Return[k] ] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, nmax}] (* JeanFrançois Alcover, Sep 06 2017 *)


CROSSREFS

Cf. A001222, A093552, A093550, A086560, A124592.
a(1) = A000040(1), a(2) = A005383(1), a(3) = A112998(1), a(4) = A113000(1), a(5) = A113008(1), a(6) = A113150(1).
Sequence in context: A144545 A085326 A062308 * A093551 A173915 A293041
Adjacent sequences: A072872 A072873 A072874 * A072876 A072877 A072878


KEYWORD

hard,nice,nonn,more


AUTHOR

Rick L. Shepherd, Jun 30 2002 and Jens Kruse Andersen, Jul 28 2002


EXTENSIONS

a(7) found by Mark W. Lewis
a(8) and a(9) found by Jens Kruse Andersen
a(10) found by Jens Kruse Andersen; probably a(11)>10^20.  Aug 24 2002
Entry revised by N. J. A. Sloane, Jan 26 2007
Crossreferences and editing by Charles R Greathouse IV, Apr 20 2010


STATUS

approved



