A325251 - OEIS

%I #8 Aug 22 2019 09:53:36

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,28,29,

%T 31,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,60,

%U 61,62,63,65,67,68,69,71,73,74,75,76,77,79,82,83,84,85,86

%N Numbers whose omega-sequence covers an initial interval of positive integers.

%C We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

%C The enumeration of these partitions by sum is given by A325260.

%e The sequence of terms together with their omega sequences begins:

%e    1:              31: 1             63: 3 2 2 1

%e    2: 1            33: 2 2 1         65: 2 2 1

%e    3: 1            34: 2 2 1         67: 1

%e    4: 2 1          35: 2 2 1         68: 3 2 2 1

%e    5: 1            37: 1             69: 2 2 1

%e    6: 2 2 1        38: 2 2 1         71: 1

%e    7: 1            39: 2 2 1         73: 1

%e    9: 2 1          41: 1             74: 2 2 1

%e   10: 2 2 1        43: 1             75: 3 2 2 1

%e   11: 1            44: 3 2 2 1       76: 3 2 2 1

%e   12: 3 2 2 1      45: 3 2 2 1       77: 2 2 1

%e   13: 1            46: 2 2 1         79: 1

%e   14: 2 2 1        47: 1             82: 2 2 1

%e   15: 2 2 1        49: 2 1           83: 1

%e   17: 1            50: 3 2 2 1       84: 4 3 2 2 1

%e   18: 3 2 2 1      51: 2 2 1         85: 2 2 1

%e   19: 1            52: 3 2 2 1       86: 2 2 1

%e   20: 3 2 2 1      53: 1             87: 2 2 1

%e   21: 2 2 1        55: 2 2 1         89: 1

%e   22: 2 2 1        57: 2 2 1         90: 4 3 2 2 1

%e   23: 1            58: 2 2 1         91: 2 2 1

%e   25: 2 1          59: 1             92: 3 2 2 1

%e   26: 2 2 1        60: 4 3 2 2 1     93: 2 2 1

%e   28: 3 2 2 1      61: 1             94: 2 2 1

%e   29: 1            62: 2 2 1         95: 2 2 1

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];

%t Select[Range[100],normQ[omseq[#]]&]

%Y Positions of normal numbers (A055932) in A325248.

%Y Cf. A000430, A055932, A056239, A112798, A181819, A323023, A325247, A325260, A325261, A325277.

%Y Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 16 2019