A068993 -id:A068993 - 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

A113901 Product of omega(n) and bigomega(n) = A001221(n)*A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 6, 1, 4, 4, 4, 1, 6, 1, 6, 4, 4, 1, 8, 2, 4, 3, 6, 1, 9, 1, 5, 4, 4, 4, 8, 1, 4, 4, 8, 1, 9, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 8, 4, 8, 4, 4, 1, 12, 1, 4, 6, 6, 4, 9, 1, 6, 4, 9, 1, 10, 1, 4, 6, 6, 4, 9, 1, 10, 4, 4, 1, 12, 4, 4, 4, 8, 1, 12, 4, 6, 4, 4, 4, 12, 1, 6, 6, 8, 1, 9

Offset: 1

Cino Hilliard, Jan 29 2006

Positions of first appearances are A328964. - Gus Wiseman, Nov 05 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000

A307409(n) is (bigomega(n) - 1) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A066921, A066922, A068993, A070175, A071625, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965.

Table[PrimeNu[n]*PrimeOmega[n], {n,1,50}] (* G. C. Greubel, Apr 23 2017 *)

PARI
```
a(n) = omega(n)*bigomega(n);
```

a(n) = 1 iff n is prime.
A068993(a(n)) = 4. - Reinhard Zumkeller, Mar 13 2011
a(n) = A066921(n)*A066922(n). - Amiram Eldar, May 07 2025

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

A325264 Numbers whose omega-sequence sums to 7.

Original entry on oeis.org

30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, Apr 18 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 sequence of terms together with their prime indices and omega-sequences begins:
   30: {1,2,3} (3,3,1)
   36: {1,1,2,2} (4,2,1)
   42: {1,2,4} (3,3,1)
   64: {1,1,1,1,1,1} (6,1)
   66: {1,2,5} (3,3,1)
   70: {1,3,4} (3,3,1)
   78: {1,2,6} (3,3,1)
  100: {1,1,3,3} (4,2,1)
  102: {1,2,7} (3,3,1)
  105: {2,3,4} (3,3,1)
  110: {1,3,5} (3,3,1)
  114: {1,2,8} (3,3,1)
  130: {1,3,6} (3,3,1)
  138: {1,2,9} (3,3,1)
  154: {1,4,5} (3,3,1)
  165: {2,3,5} (3,3,1)
  170: {1,3,7} (3,3,1)
  174: {1,2,10} (3,3,1)
  182: {1,4,6} (3,3,1)
  186: {1,2,11} (3,3,1)
  190: {1,3,8} (3,3,1)
  195: {2,3,6} (3,3,1)
  196: {1,1,4,4} (4,2,1)

Positions of 7's 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, A325265, 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[#]]==7&]

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

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 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&]

cp's OEIS Frontend

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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A113901 Product of omega(n) and bigomega(n) = A001221(n)*A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.

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A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

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A325264 Numbers whose omega-sequence sums to 7.

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A325265 Numbers with sum of omega-sequence > 4.

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