Displaying 1-10 of 12 results found.
1, 9, 27, 80, 216, 448, 1296, 2816, 6400, 13312, 30720, 62208, 139264, 311296, 688128, 1474560, 2985984, 6029312, 12845056, 30408704, 65011712, 131072000, 264241152, 553648128, 1132462080, 2293235712, 4697620480, 9932111872, 20132659200, 41875931136, 88046829568
MATHEMATICA
a[n_] := a[n] = Reap[For[m = 1; k = 1, k <= n+1, If[PrimeOmega[m] - PrimeNu[m] == n, Sow[m]; k++]; m++]][[2, 1]] // Last;
PROG
(Haskell)
a264959 n = a257851 n n
(PARI) a(n) = my(nb=0, k=1); until (nb == n+1, my(f=factor(k)); if (bigomega(f) - omega(f) == n, nb++); k++; ); k-1; \\ Michel Marcus, Feb 05 2022
Excess of n = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).
+10
131
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
COMMENTS
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.
a(n) = the number of divisors of n that are each a composite power of a prime. - Leroy Quet, Dec 02 2009
FORMULA
Additive with a(p^e) = e - 1.
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
MAPLE
with(numtheory); A046660 := n -> bigomega(n)-nops(factorset(n)): # Or:
with(NumberTheory): A046660 := n -> NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jul 14 2023
MATHEMATICA
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. *)
PROG
(Haskell)
import Math.NumberTheory.Primes.Factorisation (factorise)
a046660 n = sum es - length es where es = snd $ unzip $ factorise n
(Python)
from sympy import factorint
CROSSREFS
Cf. A001222, A001221, A003557, A056170, A136141, A257851, A261256, A264959, A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069, A275699, A275812.
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.
+10
14
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
COMMENTS
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<q primes} 1/(p*(p+1)*q*(q+1))) = (1/zeta(2)) * (2*P(3) + Sum_{k>=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)
MATHEMATICA
Select[Range[500], PrimeOmega[#]-PrimeNu[#]==2&]
PROG
(Haskell)
a195086 n = a195086_list !! (n-1)
a195086_list = filter ((== 2) . a046660) [1..]
CROSSREFS
Cf. A001221, A001222, A025487, A057521, A060687, A195069, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A046660, A257851, A261256, A264959.
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 3.
+10
13
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
COMMENTS
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
MATHEMATICA
Select[Range[1000], PrimeOmega[#]-PrimeNu[#]==3&]
PROG
(Haskell)
a195087 n = a195087_list !! (n-1)
a195087_list = filter ((== 3) . a046660) [1..]
CROSSREFS
Cf. A001221, A001222, A060687, A195069, A195086, A195088, A195089, A195090, A195091, A195092, A195093.
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5.
+10
13
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
COMMENTS
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
MATHEMATICA
Select[Range[4000], PrimeOmega[#]-PrimeNu[#]==5&]
PROG
(Haskell)
a195089 n = a195089_list !! (n-1)
a195089_list = filter ((== 5) . a046660) [1..]
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7.
+10
13
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
COMMENTS
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
MATHEMATICA
Select[Range[14000], PrimeOmega[#]-PrimeNu[#]==7&]
PROG
(Haskell)
a195091 n = a195091_list !! (n-1)
a195091_list = filter ((== 7) . a046660) [1..]
Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.
+10
13
4, 24, 72, 160, 432, 896, 2592, 5632, 12800, 26624, 61440, 124416, 278528, 622592, 1376256, 2949120, 5971968, 12058624, 25690112, 60817408, 130023424, 262144000, 528482304, 1107296256, 2264924160, 4586471424, 9395240960, 19864223744, 40265318400, 83751862272
COMMENTS
S_0 would correspond to the squarefree numbers ( A005117), that is, numbers j such that A001222(j) = A001221(j). Note that S_0 is excluded from the scheme. - Michel Marcus, Sep 21 2015
FORMULA
a(n+1) > 2*a(n).
a(n) >= 2^prime(n) for n < 5.
a(n) = b(n)*2^(n+1), where b(n) consists of the values of k/2^excess(k) over odd k, sorted in ascending order. In particular, a(n) <= prime(n)*2^(n+1), with equality only when n = 2. - Charlie Neder, Jan 31 2019
EXAMPLE
For n = 1, S_1 = {4, 9, 12, 18, 20, 25, ...}, so a(1) = S_1(1) = 4.
For n = 2, S_2 = {8, 24, 27, 36, 40, 54, ...}, so a(2) = S_2(2) = 24.
For n = 3, S_3 = {16, 48, 72, 80, 81, 108, ...}, so a(3) = S_3(3) = 72.
For n = 4, S_4 = {32, 96, 144, 160, 216, 224, ...}, so a(4) = S_4(4) = 160.
For n = 5, S_5 = {64, 192, 288, 320, 432, 448, ...}, so a(5) = S_5(5) = 432.
MATHEMATICA
OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq
PROG
(PARI) a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++); ); ik; } \\ Michel Marcus, Oct 06 2015 (Haskell)
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.
+10
12
2048, 6144, 9216, 10240, 13824, 14336, 20736, 22528, 25600, 26624, 30720, 31104, 34816, 38912, 43008, 46080, 46656, 47104, 50176, 59392, 63488, 64000, 64512, 67584, 69120, 69984, 71680, 75776, 76800, 79872, 83968, 88064, 96256, 96768, 101376, 103680, 104448
COMMENTS
The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0003698..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 25 2024
EXAMPLE
14336 = 2^11 * 7^1, so it has 12 prime factors (counted with multiplicity) and 2 distinct prime factors, and 12-2 = 10.
MAPLE
op(select(n->bigomega(n)-nops(factorset(n))=10, [$1..104448])); # Paolo P. Lava, Jul 03 2018
MATHEMATICA
Select[Range[200000], PrimeOmega[#] - PrimeNu[#] == 10&]
PROG
(Haskell)
a195069 n = a195069_list !! (n-1)
a195069_list = filter ((== 10) . a046660) [1..]
(PARI) isok(n) = bigomega(n) - omega(n) == 10; \\ Michel Marcus, Jul 03 2018
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4.
+10
12
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
COMMENTS
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
MATHEMATICA
Select[Range[2000], PrimeOmega[#]-PrimeNu[#]==4&]
PROG
(Haskell)
a195088 n = a195088_list !! (n-1)
a195088_list = filter ((== 4) . a046660) [1..]
Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6.
+10
12
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
COMMENTS
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
MAPLE
op(select(n->bigomega(n)-nops(factorset(n))=6, [$1..6784])); # Paolo P. Lava, Jul 03 2018
MATHEMATICA
Select[Range[7000], PrimeOmega[#]-PrimeNu[#]==6&]
PROG
(Haskell)
a195090 n = a195090_list !! (n-1)
a195090_list = filter ((== 6) . a046660) [1..]
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