A175206 Smallest number k such that omega(k)+ omega(k+1)+ omega(k+2)+ omega(k+3)= n.

1, 2, 3, 9, 12, 33, 75, 153, 492, 987, 4179, 13803, 18444, 134043, 282489, 1013724, 4289592, 12582633, 57495513, 260628717, 801621093, 3307216989, 5313193818, 62909521245, 308935340153, 611063815284

Offset: 3

Author

Michel Lagneau, Mar 04 2010

Keywords

• nonn

Comments

omega(.) = A001221(.) is the number of distinct prime divisors of the argument.

a(29) <= 3169630569180. - Donovan Johnson, Sep 17 2011

Examples

			For n=3, omega(1)+ omega(2)+ omega(3)+ omega(4)= 0 + 1 + 1 + 1 = 3.
For n=4, omega(2)+ omega(3)+ omega(4)+ omega(5)= 1 + 1 + 1 + 1 = 4.
For n=5, omega(3)+ omega(4)+ omega(5)+ omega(6)= 1 + 1 + 1 + 2 = 5.
For n=6, omega(9)+ omega(10)+ omega(11)+ omega(12)= 1 + 2 + 1 + 2 = 6.
For n=7, omega(12)+ omega(13)+ omega(14)+ omega(15)= 2 + 1 + 2 + 2 = 7.
For n=8, omega(33)+ omega(34)+ omega(35)+ omega(36)= 2 + 2 + 2 + 2 = 8.
For n=9, omega(75)+ omega(76)+ omega(77)+ omega(78)= 2 + 2 + 2 + 3 = 9.
		

References

• J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/ Cambridge University Press London 1935.

Links

• S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.

Crossrefs

Cf. A001222, A001221, A046660, A144494.

Programs

Maple

with(numtheory):for k from 1 to 20 do : indic:=0: for n from 1 to 1000000000 do :
u1 := nops(factorset(n)): u2 := nops(factorset(n+1)): u3 := nops(factorset(n+2)): u4:= nops(factorset(n+3)):
if u1+u2+u3+u4 = k and indic=0 then print(n): indic:=1: else fi: od: od:

Mathematica

n1=3; nmax=23; omega[k_] := Length[FactorInteger[k]]; Do[om[j] = omega[j], {j, 2, n1+1}]; a[n1]=1; a[n_ /; n>n1] :=  a[n] = (k = a[n-1](* assuming sequence is increasing *); While[k++; Do[om[j] = om[j+1], {j, 1, n1}]; om[n1+1] = omega[k+n1]; Sum[om[j], {j, 1, n1+1}] < n]; k); Table[Print[a[n]]; a[n], {n, n1, nmax}] (* Jean-François Alcover,  Sep 12 2011 *)

Extensions

Unspecific references removed, offset corrected, variable names standardized - R. J. Mathar, Mar 16 2010

a(24)-a(28) from Donovan Johnson, Sep 17 2011