cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A325265 Numbers with sum of omega-sequence > 4.

Original entry on oeis.org

6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96
Offset: 1

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Author

Gus Wiseman, Apr 18 2019

Keywords

Comments

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).

Examples

			The sequence of terms together with their omega-sequences begins:
   6: 2 2 1       46: 2 2 1         80: 5 2 2 1       112: 5 2 2 1
  10: 2 2 1       48: 5 2 2 1       81: 4 1           114: 3 3 1
  12: 3 2 2 1     50: 3 2 2 1       82: 2 2 1         115: 2 2 1
  14: 2 2 1       51: 2 2 1         84: 4 3 2 2 1     116: 3 2 2 1
  15: 2 2 1       52: 3 2 2 1       85: 2 2 1         117: 3 2 2 1
  16: 4 1         54: 4 2 2 1       86: 2 2 1         118: 2 2 1
  18: 3 2 2 1     55: 2 2 1         87: 2 2 1         119: 2 2 1
  20: 3 2 2 1     56: 4 2 2 1       88: 4 2 2 1       120: 5 3 2 2 1
  21: 2 2 1       57: 2 2 1         90: 4 3 2 2 1     122: 2 2 1
  22: 2 2 1       58: 2 2 1         91: 2 2 1         123: 2 2 1
  24: 4 2 2 1     60: 4 3 2 2 1     92: 3 2 2 1       124: 3 2 2 1
  26: 2 2 1       62: 2 2 1         93: 2 2 1         126: 4 3 2 2 1
  28: 3 2 2 1     63: 3 2 2 1       94: 2 2 1         128: 7 1
  30: 3 3 1       64: 6 1           95: 2 2 1         129: 2 2 1
  32: 5 1         65: 2 2 1         96: 6 2 2 1       130: 3 3 1
  33: 2 2 1       66: 3 3 1         98: 3 2 2 1       132: 4 3 2 2 1
  34: 2 2 1       68: 3 2 2 1       99: 3 2 2 1       133: 2 2 1
  35: 2 2 1       69: 2 2 1        100: 4 2 1         134: 2 2 1
  36: 4 2 1       70: 3 3 1        102: 3 3 1         135: 4 2 2 1
  38: 2 2 1       72: 5 2 2 1      104: 4 2 2 1       136: 4 2 2 1
  39: 2 2 1       74: 2 2 1        105: 3 3 1         138: 3 3 1
  40: 4 2 2 1     75: 3 2 2 1      106: 2 2 1         140: 4 3 2 2 1
  42: 3 3 1       76: 3 2 2 1      108: 5 2 2 1       141: 2 2 1
  44: 3 2 2 1     77: 2 2 1        110: 3 3 1         142: 2 2 1
  45: 3 2 2 1     78: 3 3 1        111: 2 2 1         143: 2 2 1
		

Crossrefs

Positions of terms > 4 in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
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), A325249 (sum).

Programs

  • Mathematica
    omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
    Select[Range[100],Total[omseq[#]]>4&]