A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

2, 3, 61, 193, 15121, 838561, 807905281, 19896463921, 3059220303001, 3931520917431241

Offset: 1

Author

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

Keywords

• hard
• nice
• nonn
• more

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 are counted with multiplicity. - Harvey P. Dale, Mar 09 2021

Examples

			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.
		

References

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

Links

• alt.math.recreational thread, Consecutive numbers with counting prime factors.
• Carlos Rivera, Puzzle 425. Consecutive numbers, increasing quantity of prime factors, The Prime Puzzles & Problems Connection.

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).

Programs

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[n-1]+n-1, k <= kmax, k++, If[AllTrue[Range[0, n-1], PrimeOmega[k+#] == #+1&], Return[k] ] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Sep 06 2017 *)

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

Cross-references and editing by Charles R Greathouse IV, Apr 20 2010