A303555 -id:A303555 - cp's OEIS frontend

A325272 Adjusted frequency depth of n!.

0, 1, 3, 4, 5, 4, 6, 6, 6, 4, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 8, 7, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

			Recursively applying A181819 starting with 120 gives 120 -> 20 -> 6 -> 4 -> 3, so a(5) = 5.

a(n) = A001222(A325275(n)).
Cf. A000142, A006939, A303555, A323023, A325238, A325273, A325274, A325276, 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).

fd[n_]:=Switch[n,1,0,?PrimeQ,1,,1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
Table[fd[n!],{n,30}]

a(n) = A323014(n!).

A325238 First positive integer with each omega-sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 96, 120, 128, 192, 210, 216, 240, 256, 360, 384, 420, 480, 512, 720, 768, 840, 900, 960, 1024, 1260, 1296, 1440, 1536, 1680, 1920, 2048, 2310, 2520, 2880, 3072, 3360, 3840, 4096, 4620, 5040, 5760, 6144, 6720

Offset: 1

Gus Wiseman, Apr 14 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = frequency depth of n, and the k-th part is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, given by red(n = p^i*...*q^j) = prime(i)*...*prime(j), i.e., the product of primes indexed by the prime exponents of n.

			The sequence of terms together with their omega-sequences begins:
    1:
    2: 1
    4: 2 1
    6: 2 2 1
    8: 3 1
   12: 3 2 2 1
   16: 4 1
   24: 4 2 2 1
   30: 3 3 1
   32: 5 1
   36: 4 2 1
   48: 5 2 2 1
   60: 4 3 2 2 1
   64: 6 1
   96: 6 2 2 1
  120: 5 3 2 2 1
  128: 7 1
  192: 7 2 2 1
  210: 4 4 1
  216: 6 2 1
  240: 6 3 2 2 1
  256: 8 1
  360: 6 3 3 1
  384: 8 2 2 1
  420: 5 4 2 2 1

Cf. A001221, A001222, A007916, A011784, A070175, A071625, A118914, A181819, A181821, A303555, A304465, A323014, A323023, A325238, A325239.

tomseq[n_]:=If[n<=1,{},Most[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]];
omseqs=Table[Total/@tomseq[n],{n,1000}];
Sort[Table[Position[omseqs,x][[1,1]],{x,Union[omseqs]}]]

A325273 Prime omicron of n!.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

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 prime omicron of n (A304465) is 0 if n is 1, 1 if n is prime, and otherwise the second-to-last part of the omega-sequence of n. For example, the prime omicron of 180 is 2.
Conjecture: all terms after a(10) = 4 are less than 4.
From James Rayman, Apr 17 2021: (Start)
The conjecture is false. a(3804) = 4. In fact, there are 91 values of n < 10000 such that a(n) = 4.
The first value of n such that a(n) = 5 is 37934. For any other n < 5*10^5, a(n) < 5. (End)

James Rayman, Table of n, a(n) for n = 0..10000

a(n) = A055396(A325275(n)/2).
Cf. A000142, A006939, A303555, A323023, A325238, A325272, A325274, A325275, A325276, 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&]];
omicron[n_]:=Switch[n,1,0,?PrimeQ,1,,omseq[n][[-2]]];
Table[omicron[n!],{n,0,100}]

from sympy.ntheory import *
def red(v):
    r = {}
    for i in v: r[i] = r.get(i, 0) + 1
    return r
def omicron(v):
    if len(v) == 0: return 0
    if len(v) == 1: return v[0]
    else: return omicron(list(red(v).values()))
f, a_list = {}, []
for i in range(101):
    a_list.append(omicron(list(f.values())))
    g = factorint(i+1)
    for k in g: f[k] = f.get(k, 0) + g[k]
print(a_list) # James Rayman, Apr 17 2021

More terms from James Rayman, Apr 17 2021

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

A307409 a(n) = (A001222(n) - 1)*A001221(n).

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Apr 07 2019

nonn

a(n) + 2 appears to differ from A000005 at n=1 and when n is a term of A320632. Verified up to n=3000.
If A320632 contains the numbers such that A001222(n) - A051903(n) > 1, then this sequence contains precisely the numbers p^k and p^k*q for distinct primes p and q. The comment follows, since d(p^k) = k+1 = (k-1)*1 + 2 and d(p^k*q) = 2k+2 = ((k+1)-1)*2 + 2. - Charlie Neder, May 14 2019
Positions of first appearances are A328965. - Gus Wiseman, Nov 05 2019
Regarding Neder's comment above, see also my comments in A322437. - Antti Karttunen, Feb 17 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Two less than A307408.
A113901(n) is bigomega(n) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A070175, A071625, A113901, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965, A322437.

a[n_] := (PrimeOmega[n] - 1)*PrimeNu[n];
aa = Table[a[n], {n, 1, 104}];

a(n) = (bigomega(n) - 1)*omega(n); \\ Michel Marcus, May 15 2019

a(n) = (A001222(n) - 1)*A001221(n).
a(n) = binomial(A001222(n) - 1, 1)*binomial(A001221(n), 1).
a(n) = A307408(n) - 2.

A325276 Irregular triangle read by rows where row n is the omega-sequence of n!.

Original entry on oeis.org

1, 2, 2, 1, 4, 2, 2, 1, 5, 3, 2, 2, 1, 7, 3, 3, 1, 8, 4, 3, 2, 2, 1, 11, 4, 3, 2, 2, 1, 13, 4, 3, 2, 2, 1, 15, 4, 4, 1, 16, 5, 4, 2, 2, 1, 19, 5, 4, 2, 2, 1, 20, 6, 4, 2, 2, 1, 22, 6, 4, 2, 1, 24, 6, 5, 2, 2, 1, 28, 6, 5, 2, 2, 1, 29, 7, 5, 2, 2, 1

Offset: 0

Gus Wiseman, Apr 18 2019

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).

			Triangle begins:
  {}
  {}
   1
   2  2  1
   4  2  2  1
   5  3  2  2  1
   7  3  3  1
   8  4  3  2  2  1
  11  4  3  2  2  1
  13  4  3  2  2  1
  15  4  4  1
  16  5  4  2  2  1
  19  5  4  2  2  1
  20  6  4  2  2  1
  22  6  4  2  1
  24  6  5  2  2  1
  28  6  5  2  2  1
  29  7  5  2  2  1
  32  7  5  2  2  1
  33  8  5  2  2  1
  36  8  5  2  2  1
  38  8  5  2  2  1
  40  8  6  2  2  1
  41  9  6  2  2  1
  45  9  6  2  2  1
  47  9  6  2  2  1
  49  9  6  3  2  2  1
  52  9  6  3  2  2  1
  55  9  6  3  2  2  1
  56 10  6  3  2  2  1
  59 10  6  3  2  2  1

Row lengths are A325272. Row sums are A325274. Row n is row A325275(n) of A112798. Second-to-last column is A325273. Column k = 1 is A022559. Column k = 2 is A000720. Column k = 3 is A071626.
Cf. A000142, A006939, A081401, A303555, A323023, A325238, A325275, 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&]];
Table[omseq[n!],{n,0,30}]

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

First differs from A084227 in having 60.

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Zeros of A328958.
The complement is A328957.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - omega(n) * Omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A000040, A005117, A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]==PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) == omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) = A001222(a(n)) * A001221(a(n)).

A325274 Sum of the omega-sequence of n!.

Original entry on oeis.org

0, 0, 1, 5, 9, 13, 14, 20, 23, 25, 24, 30, 33, 35, 35, 40, 44, 46, 49, 51, 54, 56, 59, 61, 65, 67, 72, 75, 78, 80, 83, 85, 90, 90, 95, 97, 101, 103, 105, 106, 110, 112, 115, 117, 122, 125, 127, 129, 134, 136, 139, 140, 143, 145, 149, 153, 157, 159, 160, 162

Offset: 0

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), with sum 13.

a(n) = A056239(A325275(n)).
Cf. A000142, A006939, A303555, A323023, A325238, A325272, A325273, A325276, 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&]];
Table[Total[omseq[n!]],{n,0,100}]

A325275 Heinz number of the omega-sequence of n!.

Original entry on oeis.org

1, 1, 2, 18, 126, 990, 850, 11970, 19530, 25830, 4606, 73458, 92862, 116298, 43134, 229086, 275418, 366894, 440946, 515394, 568062, 613206, 769158, 963378, 1060254, 1135602, 6108570, 6431490, 6915870, 8923590, 9398610, 10191870, 11352510, 3139866, 16458210

Offset: 0

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

A001222(a(n)) = A325272.
A055396(a(n)/2) = A325273.
A056239(a(n)) = A325274.
Row n of A325276 is row a(n) of A112798.
Cf. A000142, A006939, A056239, A112798, 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&]];
Table[Times@@Prime/@omseq[n!],{n,30}]

A328957 Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 42, 43, 47, 49, 53, 59, 61, 64, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 89, 97, 100, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 120, 121, 125, 127, 128, 130, 131, 137

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

			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}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Nonzeros of A328958.
The complement is A328956.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - Omega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]!=PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) != omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) != A001222(a(n)) * A001221(a(n)).

cp's OEIS Frontend

A325272 Adjusted frequency depth of n!.

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A325238 First positive integer with each omega-sequence.

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A325273 Prime omicron of n!.

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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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A307409 a(n) = (A001222(n) - 1)*A001221(n).

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A325276 Irregular triangle read by rows where row n is the omega-sequence of n!.

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A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

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A325274 Sum of the omega-sequence of n!.

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