A175204 Smallest index m such that omega(m) + omega(m+1) + omega(m+2) = n.

1, 2, 4, 10, 20, 68, 154, 644, 2210, 6578, 35308, 92378, 310154, 1042404, 5617820, 35515634, 184055430, 1082950218, 5386096364, 19304763268, 254772473240, 1383442606194

Offset: 2

Author

Michel Lagneau, Mar 04 2010

Keywords

• nonn
• more

Comments

The arithmetic function omega(m) + omega(m+1) + omega(m+2) = Sum_{j=0..2} A001221(m+j) starts 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 4, 4, 5, 5 (m >= 1).

The sequence is a "first-serve" inverse of this function.

a(24) <= 6127197154440. [Donovan Johnson, Oct 22 2010]

Examples

			For n=2, m=1 and omega(1) + omega(2) + omega(3) = 0 + 1 + 1 = 2.
For n=3, m=2 and omega(2) + omega(3) + omega(4) = 1 + 1 + 1 = 3.
For n=4, m=4 and omega(4) + omega(5) + omega(6) = 1 + 1 + 2 = 4.
For n=5, m=10 and omega(10) + omega(11) + omega(12) = 2 + 1 + 2 = 5.
For n=6, m=20 and omega(20) + omega(21) + omega(22) = 2 + 2 + 2 = 6.
For n=7, m=68 and omega(68) + omega(69) + omega(70) = 2 + 2 + 3 = 7.
		

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.
• Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Crossrefs

Cf. A001221, A144494.

Programs

Maple

with(numtheory): for k from 1 to 20 do :indic:=0: for n from 1 to 2000 do :
s1:= ifactors(n)[2] :u1 :=s1[i][1], i=1..nops(s1):uu1:= nops(s1): s2:= ifactors(n+1)[2] :u2 :=s2[i][1], i=1..nops(s2): uu2:= nops(s2): s3:= ifactors(n+2)[2] :u3 :=s3[i][1], i=1..nops(s3): uu3:= nops(s3): if uu1+uu2+uu3 = k and indic=0 then print(n): indic:=1:else fi:od:od:

Extensions

Added punctuation to the examples. Corrected and edited by Michel Lagneau, Apr 25 2010

Use of variables adapted to OEIS standards by R. J. Mathar, Oct 12 2010

a(16) corrected and a(19)-a(23) from Donovan Johnson, Oct 22 2010