A325264 -id:A325264 - cp's OEIS frontend

A325249 Sum of the omega-sequence of n.

0, 1, 1, 3, 1, 5, 1, 4, 3, 5, 1, 8, 1, 5, 5, 5, 1, 8, 1, 8, 5, 5, 1, 9, 3, 5, 4, 8, 1, 7, 1, 6, 5, 5, 5, 7, 1, 5, 5, 9, 1, 7, 1, 8, 8, 5, 1, 10, 3, 8, 5, 8, 1, 9, 5, 9, 5, 5, 1, 12, 1, 5, 8, 7, 5, 7, 1, 8, 5, 7, 1, 10, 1, 5, 8, 8, 5, 7, 1, 10, 5, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

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).

			The omega-sequence of 180 is (5,3,2,2,1) with sum 13, so a(180) = 13.

Positions of m's are A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A118914, A181819, A182850, A323023, A325274.
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).

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

a(n) = A056239(A325248(n)).

a(n!) = A325274(n).

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 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, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Feb 10 2003

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
is(n)=omega(n)>1 || isprimepower(n)>2
```

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Definition clarified by Harvey P. Dale, Jul 23 2013

A383017 Numbers k such that A382883(k) = -1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 36, 37, 41, 42, 43, 47, 53, 59, 61, 64, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 100, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179

Offset: 1

Peter Luschny, Apr 12 2025

nonn

See the comments in A382883.

Cf. A382883, A382943, A383016, A000040, A325264.

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; Select[Range[200], f[#] == -1 &] (* Amiram Eldar, Apr 29 2025 *)

def A383017List(upto): return [n for n in srange(1, upto) if A382883(n) == -1]

A382883 = this sequence union A383016 union A382943.

A000040 is a subsequence.

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

Gus Wiseman, Apr 18 2019

nonn

			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

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).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325277.
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).

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

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A325249 Sum of the omega-sequence of n.

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A080257 Numbers having at least two distinct or a total of at least three prime factors.

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A383017 Numbers k such that A382883(k) = -1.

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A325265 Numbers with sum of omega-sequence > 4.

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