A195069 -id:A195069 - cp's OEIS frontend

A316499 Intersection of A001694 and A195069.

2048, 9216, 13824, 20736, 25600, 31104, 46656, 50176, 64000, 69984, 104976, 115200, 123904, 157464, 160000, 172800, 173056, 175616, 177147, 225792, 236196, 259200, 288000, 338688, 388800, 400000, 432000, 508032, 557568, 583200, 614656, 627200, 648000, 681472, 720000, 762048, 778752, 790272

Offset: 1

Robert Israel, Jul 04 2018

nonn

Powerful(1) numbers k such that A046660(k) = 10.

These are the "primitive" members of A195069, in the sense that A195059 is the set of numbers k*m where k is in this sequence and m is squarefree and coprime to k.

			a(3)=13824 is a member because 13824= 2^9*3^3, 9 and 3 are both greater than 1 and (9-1)+(3-1)=10.

Robert Israel, Table of n, a(n) for n = 1..2528

Cf. A001694, A046660, A195069.

N:= 10^6: # to get all terms <= N
p:= 1:
for i from 1 to 10 do F[i]:= {} od:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  for i from min(10,floor(log[p](N))-1) to 2 by -1 do F[i]:= F[i] union
    select(`<=`,`union`({p^(i+1)},seq(map(t -> p^(i+1-j)*t, F[j]),j=1..i-1)),N)
  od;
  F[1]:= F[1] union {p^2};
od:
sort(convert(F[10],list));

A046660 Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0

Offset: 1

N. J. A. Sloane

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3, 1).
a(n) = 0 for squarefree n.
A162511(n) = (-1)^a(n). - Reinhard Zumkeller, Jul 08 2009
a(n) = the number of divisors of n that are each a composite power of a prime. - Leroy Quet, Dec 02 2009
a(A005117(n)) = 0; a(A060687(n)) = 1; a(A195086(n)) = 2; a(A195087(n)) = 3; a(A195088(n)) = 4; a(A195089(n)) = 5; a(A195090(n)) = 6; a(A195091(n)) = 7; a(A195092(n)) = 8; a(A195093(n)) = 9; a(A195069(n)) = 10. - Reinhard Zumkeller, Nov 29 2015

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 51-52.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.

Not the same as A066301.

Cf. A001222, A001221, A003557, A056170, A136141, A257851, A261256, A264959, A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069, A275699, A275812.

import Math.NumberTheory.Primes.Factorisation (factorise)
a046660 n = sum es - length es where es = snd $ unzip $ factorise n
-- Reinhard Zumkeller, Nov 28 2015, Jan 09 2013

with(numtheory); A046660 := n -> bigomega(n)-nops(factorset(n)):
seq(A046660(k), k=1..100); # Wesley Ivan Hurt, Oct 27 2013
# Or:
with(NumberTheory): A046660 := n -> NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct'):  # Peter Luschny, Jul 14 2023

Table[PrimeOmega[n] - PrimeNu[n], {n, 50}] (* or *) muf[n_] := Module[{fi = FactorInteger[n]}, Total[Transpose[fi][[2]]] - Length[fi]]; Array[muf, 50] (* Harvey P. Dale, Sep 07 2011. The second program is several times faster than the first program for generating large numbers of terms. *)

a(n)=bigomega(n)-omega(n) \\ Charles R Greathouse IV, Nov 14 2012

a(n)=my(f=factor(n)[,2]); vecsum(f)-#f \\ Charles R Greathouse IV, Aug 01 2016

from sympy import factorint
def A046660(n): return sum(e-1 for e in factorint(n).values()) # Chai Wah Wu, Jul 18 2023

a(n) = Omega(n) - omega(n) = A001222(n) - A001221(n).
Additive with a(p^e) = e - 1.
a(n) = Sum_{k = 1..A001221(n)} (A124010(n,k) - 1). - Reinhard Zumkeller, Jan 09 2013
G.f.: Sum_{p prime, k>=2} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Jan 06 2017
Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141). - Amiram Eldar, Jul 28 2020
a(n) = Sum_{p|n} A286563(n/p,p), where p is prime. - Ridouane Oudra, Sep 13 2023
a(n) = A275812(n) - A056170(n). - Amiram Eldar, Jan 09 2024
a(n) = A001222(A003557(n)). - Peter Munn, Feb 06 2024

More terms from David W. Wilson

A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

Original entry on oeis.org

8, 24, 27, 36, 40, 54, 56, 88, 100, 104, 120, 125, 135, 136, 152, 168, 180, 184, 189, 196, 225, 232, 248, 250, 252, 264, 270, 280, 296, 297, 300, 312, 328, 343, 344, 351, 375, 376, 378, 396, 408, 424, 440, 441, 450, 456, 459, 468, 472, 484, 488

Offset: 1

Harvey P. Dale, Sep 08 2011

nonn

From Amiram Eldar, Nov 07 2020: (Start)
Numbers whose powerful part (A057521) is either a cube of a prime (A030078) or a square of a squarefree semiprime (A085986).
The asymptotic density of this sequence is (6/Pi^2) * (Sum_{p prime} 1/(p^2*(p+1)) + Sum_{p=4} (-1)^(k+1)*(k-1)*P(k) + (Sum_{k>=2} (-1)^k*P(k))^2)/2 = 0.0963023158..., where P is the prime zeta function. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
Index entries for sequences computed from exponents in factorization of n

Subsequence of A048108.
Subsequences: A030078 and A085986.
Cf. A001221, A001222, A025487, A057521, A060687, A195069, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A046660, A257851, A261256, A264959.

a195086 n = a195086_list !! (n-1)
a195086_list = filter ((== 2) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[500],PrimeOmega[#]-PrimeNu[#]==2&]

is(n)=bigomega(n)-omega(n)==2 \\ Charles R Greathouse IV, Sep 14 2015

is(n)=my(f=factor(n)[,2]); vecsum(f)==#f+2 \\ Charles R Greathouse IV, Aug 01 2016

A001222(a(n)) - A001221(a(n)) = 2.

A046660(a(n)) = 2. - Reinhard Zumkeller, Nov 29 2015

A195087 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 3.

Original entry on oeis.org

16, 48, 72, 80, 81, 108, 112, 162, 176, 200, 208, 240, 272, 304, 336, 360, 368, 392, 405, 464, 496, 500, 504, 528, 540, 560, 567, 592, 600, 624, 625, 656, 675, 688, 752, 756, 792, 810, 816, 848, 880, 891, 900, 912, 936, 944, 968, 976

Offset: 1

Harvey P. Dale, Sep 08 2011

nonn

The asymptotic density of this sequence is (Sum_{p prime} 1/(p^3*(p+1)) + Sum_{p != q primes} 1/(p^2*(p+1)*q*(q+1)) + Sum_{p < q < r primes} 1/(p*(p+1)*q*(q+1)*r*(r+1)))/zeta(2) = 0.04761... . - Amiram Eldar, Sep 03 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001222, A060687, A195069, A195086, A195088, A195089, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A257851, A261256, A264959.

a195087 n = a195087_list !! (n-1)
a195087_list = filter ((== 3) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[1000],PrimeOmega[#]-PrimeNu[#]==3&]

is(n)=bigomega(n)-omega(n)==3 \\ Charles R Greathouse IV, Sep 14 2015

A001222(a(n)) - A001221(a(n)) = 3.

A046660(a(n)) = 3. - Reinhard Zumkeller, Nov 29 2015

A195089 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5.

Original entry on oeis.org

64, 192, 288, 320, 432, 448, 648, 704, 729, 800, 832, 960, 972, 1088, 1216, 1344, 1440, 1458, 1472, 1568, 1856, 1984, 2000, 2016, 2112, 2160, 2240, 2368, 2400, 2496, 2624, 2752, 3008, 3024, 3168, 3240, 3264, 3392, 3520, 3600, 3645, 3648, 3744, 3776, 3872, 3904

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0118439..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195089 n = a195089_list !! (n-1)
a195089_list = filter ((== 5) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[4000],PrimeOmega[#]-PrimeNu[#]==5&]

is(n)=bigomega(n)-omega(n)==5 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 5. - Reinhard Zumkeller, Nov 29 2015

A195091 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7.

Original entry on oeis.org

256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 3200, 3328, 3840, 3888, 4352, 4864, 5376, 5760, 5832, 5888, 6272, 6561, 7424, 7936, 8000, 8064, 8448, 8640, 8748, 8960, 9472, 9600, 9984, 10496, 11008, 12032, 12096, 12672, 12960, 13056, 13122, 13568

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0029589..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195091 n = a195091_list !! (n-1)
a195091_list = filter ((== 7) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[14000],PrimeOmega[#]-PrimeNu[#]==7&]

is(n)=bigomega(n)-omega(n)==7 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 7. - Reinhard Zumkeller, Nov 29 2015

A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

Original entry on oeis.org

1, 4, 9, 8, 24, 27, 16, 48, 72, 80, 32, 96, 144, 160, 216, 64, 192, 288, 320, 432, 448, 128, 384, 576, 640, 864, 896, 1296, 256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 1024, 3072, 4608, 5120, 6912, 7168, 10368, 11264, 12800, 13312

Offset: 0

Reinhard Zumkeller, Nov 29 2015

At the suggestion of Michel Marcus's remark in Carlos Eduardo Olivieri's A261256.

			0:    1
1:    4     9
2:    8    24      27
3:   16    48      72    80
4:   32    96     144   160     216
5:   64   192     288   320     432   448
6:  128   384     576   640     864   896    1296
7:  256   768    1152  1280    1728  1792    2592   2816
8:  512  1536    2304  2560    3456  3584    5184   5632    6400
--  ------------------------------------------------------------
0:  1
1:  2^2   3^2
2:  2^3 2^3*3     3^3
3:  2^4 2^4*3 2^3*3^2 2^4*5
4:  2^5 2^5*3 2^4*3^2 2^5*5 2^3*3^3
5:  2^6 2^6*3 2^5*3^2 2^6*5 2^4*3^3 2^6*7
6:  2^7 2^7*3 2^6*3^2 2^7*5 2^5*3^3 2^7*7 2^4*3^4
7:  2^8 2^8*3 2^7*3^2 2^8*5 2^6*3^3 2^8*7 2^5*3^4 2^8*11
8:  2^9 2^9*3 2^8*3^2 2^9*5 2^7*3^3 2^9*7 2^6*3^4 2^9*11 2^8*5^2

Reinhard Zumkeller, Rows n = 0..20 of triangle, flattened

Cf. A046660, A151821, A261256.

Cf. A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069.

a257851 n k = a257851_tabl !! n !! k
a257851_row n = a257851_tabl !! n
a257851_tabl = map
   (\x -> take (x + 1) $ filter ((== x) . a046660) [1..]) [0..]

T[n_] := Reap[For[m = 1; k = 1, k <= n+1, If[PrimeOmega[m] - PrimeNu[m] == n, Sow[m]; k++]; m++]][[2, 1]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Sep 17 2021 *)

T(n,0) = A151821(n+1);
T(n,n-1) = A261256(n) for n > 0;
T(n,n) = A264959(n).
T(0,0) = A005117(1);
T(1,k) = A060687(k+1), k = 0..1;
T(2,k) = A195086(k+1), k = 0..2;
T(3,k) = A195087(k+1), k = 0..3;
T(4,k) = A195088(k+1), k = 0..4;
T(5,k) = A195089(k+1), k = 0..5;
T(6,k) = A195090(k+1), k = 0..6;
T(7,k) = A195091(k+1), k = 0..7;
T(8,k) = A195092(k+1), k = 0..8;
T(9,k) = A195093(k+1), k = 0..9;
T(10,k) = A195069(k+1), k = 0..10.

A195088 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4.

Original entry on oeis.org

32, 96, 144, 160, 216, 224, 243, 324, 352, 400, 416, 480, 486, 544, 608, 672, 720, 736, 784, 928, 992, 1000, 1008, 1056, 1080, 1120, 1184, 1200, 1215, 1248, 1312, 1376, 1504, 1512, 1584, 1620, 1632, 1696, 1701, 1760, 1800, 1824, 1872, 1888, 1936, 1952

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0237194..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195089, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195088 n = a195088_list !! (n-1)
a195088_list = filter ((== 4) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[2000],PrimeOmega[#]-PrimeNu[#]==4&]

is(n)=bigomega(n)-omega(n)==4 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 4. - Reinhard Zumkeller, Nov 29 2015

A195090 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6.

Original entry on oeis.org

128, 384, 576, 640, 864, 896, 1296, 1408, 1600, 1664, 1920, 1944, 2176, 2187, 2432, 2688, 2880, 2916, 2944, 3136, 3712, 3968, 4000, 4032, 4224, 4320, 4374, 4480, 4736, 4800, 4992, 5248, 5504, 6016, 6048, 6336, 6480, 6528, 6784

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0059189..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195090 n = a195090_list !! (n-1)
a195090_list = filter ((== 6) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

op(select(n->bigomega(n)-nops(factorset(n))=6, [$1..6784])); # Paolo P. Lava, Jul 03 2018

Select[Range[7000],PrimeOmega[#]-PrimeNu[#]==6&]

is(n)=bigomega(n)-omega(n)==6 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 6. - Reinhard Zumkeller, Nov 29 2015

A195092 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 8.

Original entry on oeis.org

512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 6656, 7680, 7776, 8704, 9728, 10752, 11520, 11664, 11776, 12544, 14848, 15872, 16000, 16128, 16896, 17280, 17496, 17920, 18944, 19200, 19683, 19968, 20992, 22016, 24064, 24192

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0014793..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195091, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195092 n = a195092_list !! (n-1)
a195092_list = filter ((== 8) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[25000],PrimeOmega[#]-PrimeNu[#]==8&]

is(n)=bigomega(n)-omega(n)==8 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 8. - Reinhard Zumkeller, Nov 29 2015

cp's OEIS Frontend

A316499 Intersection of A001694 and A195069.

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A046660 Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

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A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

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A195087 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 3.

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A195089 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5.

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A195091 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7.

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A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

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