cp's OEIS Frontend

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

%I A072875 #36 Aug 30 2023 07:29:50
%S A072875 2,3,61,193,15121,838561,807905281,19896463921,3059220303001,
%T A072875 3931520917431241
%N A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.
%C A072875 By definition, each term of this sequence is prime.
%C A072875 a(11) <= 1452591346605212407096281241 (Frederick Schneider), see primepuzzles link. - sent by amd64(AT)vipmail.hu, Dec 21 2007
%C A072875 Prime factors are counted with multiplicity. - _Harvey P. Dale_, Mar 09 2021
%D A072875 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 61, p. 22, Ellipses, Paris 2008.
%H A072875 alt.math.recreational thread, <a href="http://groups.google.com/groups?dq=&amp;hl=da&amp;lr=&amp;ie=UTF-8&amp;threadm=ahtgsf%24vvg45%241%40ID-54909.news.dfncis.de&amp;prev=/groups%3Fhl%3Dda%26lr%3D%26ie%3DUTF-8%26group%3Dalt.math.recreational">Consecutive numbers with counting prime factors</a>.
%H A072875 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_425.htm">Puzzle 425. Consecutive numbers, increasing quantity of prime factors</a>, The Prime Puzzles & Problems Connection.
%e A072875 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;
%e A072875 a(6)=807905281: 807905281 is prime; 807905281+1=2*403952641;
%e A072875 807905281+2=3*15733*17117; 807905281+3=2*2*1871*107951;
%e A072875 807905281+4=5*11*43*211*1619; 807905281+5=2*3*3*3*37*404357;
%e A072875 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.
%t A072875 (* 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 *)
%Y A072875 Cf. A001222, A093552, A093550, A086560, A124592.
%Y A072875 a(1) = A000040(1), a(2) = A005383(1), a(3) = A112998(1), a(4) = A113000(1), a(5) = A113008(1), a(6) = A113150(1).
%K A072875 hard,nice,nonn,more
%O A072875 1,1
%A A072875 _Rick L. Shepherd_, Jun 30 2002 and _Jens Kruse Andersen_, Jul 28 2002
%E A072875 a(7) found by Mark W. Lewis
%E A072875 a(8) and a(9) found by _Jens Kruse Andersen_
%E A072875 a(10) found by _Jens Kruse Andersen_; probably a(11) > 10^20. - Aug 24 2002
%E A072875 Entry revised by _N. J. A. Sloane_, Jan 26 2007
%E A072875 Cross-references and editing by _Charles R Greathouse IV_, Apr 20 2010