A325247 Numbers whose omega-sequence is strict (no repeated parts).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset

1,2

Comments

First differs from A323306 in having 216.

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).

Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..61.

Example

The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    11: {5}
    13: {6}
    16: {1,1,1,1}
    17: {7}
    19: {8}
    23: {9}
    25: {3,3}
    27: {2,2,2}
    29: {10}
    31: {11}
    32: {1,1,1,1,1}
    36: {1,1,2,2}

Mathematica

omseq[n_Integer]:=If[n<=1, {}, Total/@NestWhileList[Sort[Length/@Split[#1]]&, Sort[Last/@FactorInteger[n]], Total[#]>1&]];
Select[Range[100], UnsameQ@@omseq[#]&]

Crossrefs

Positions of squarefree numbers in A325248.

Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.

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

Sequence in context: A137944 A359390 A323306 * A306013 A268082 A247199

Adjacent sequences: A325244 A325245 A325246 * A325248 A325249 A325250

Keyword

nonn

Author

Gus Wiseman, Apr 16 2019

Status

approved