A046660 -id:A046660 - cp's OEIS frontend

A376366 The number of non-unitary prime divisors of the cubefree numbers.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, On the number of prime factors with a given multiplicity over h-free and h-full numbers, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; arXiv preprint, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.2.

Cf. A002117, A004709, A046660, A056170, A088453, A368779, A369427, A376362, A376365, A382419.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Count[e, 2], Nothing]]; Array[f, 150]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#select(x -> x == 2, e), ", ")));}

a(n) = A056170(A004709(n)).
a(n) = A369427(A004709(n)).
Sum_{A004709(k) <= x} a(k) = c * x + O(sqrt(x)/log(x)), where c = (1/zeta(3)) * Sum_{p prime} ((p-1)/(p^3-1)) = 0.24833233043359932037... (Das et al., 2025).
a(n) = log_2(A382419(n)). - Amiram Eldar, Mar 25 2025
Sum_{k=1..n} a(k) ~ c * n, where c = Sum_{p prime} ((p-1)/(p^3-1)) = 0.29850959207541746... - Vaclav Kotesovec, Mar 25 2025 (according to the above formula)
From Amiram Eldar, Apr 05 2025: (Start)
a(n) = A046660(A004709(n)).
a(n) = A368779(n) - A376365(n). (End)

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

A261256 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.

Original entry on oeis.org

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

Offset: 1

Carlos Eduardo Olivieri, Aug 12 2015

nonn

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

			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.

Charlie Neder, Table of n, a(n) for n = 1..500

Cf. A001221, A001222.
Cf. A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093.
Cf. A046660, A257851, A264959.

a261256 n = a257851 n (n - 1)  -- Reinhard Zumkeller, Nov 29 2015

OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq

a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++);); ik;} \\ Michel Marcus, Oct 06 2015

a(n+1) > 2*a(n).
a(n) >= 2^prime(n) for n < 5.
a(n) = A257851(n,n-1). - Reinhard Zumkeller, Nov 29 2015
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

a(17)-a(21) from Jon E. Schoenfield, Sep 12 2015

More terms from Charlie Neder, Jan 31 2019

A264959 a(n) = A257851(n,n).

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Nov 29 2015

nonn

Cf. A046660, A257851, A261256.

Haskell
```
a264959 n = a257851 n n
```

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;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 03 2021 *)

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

a(21)-a(25) from Michel Marcus, Feb 05 2022

More terms from Jinyuan Wang, Feb 18 2022

A325200 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0, 0, 3, 2, 0, 2, 0, 1, 0, 6, 2, 0, 2, 0, 0, 4, 3, 4, 2, 0, 2, 0, 0, 6, 2, 6, 4, 2, 0, 2, 0, 0, 4, 9, 5, 4, 4, 2, 0, 2, 0, 1, 0, 15, 6, 8, 4, 4, 2, 0, 2, 0, 0, 5, 12, 12, 9, 6, 4, 4, 2, 0, 2, 0, 0, 10, 6, 21, 10, 12, 6, 4, 4, 2, 0, 2, 0

Offset: 0

Gus Wiseman, Apr 11 2019

			Triangle begins:
  1
  1  0
  0  2  0
  1  0  2  0
  0  3  0  2  0
  0  3  2  0  2  0
  1  0  6  2  0  2  0
  0  4  3  4  2  0  2  0
  0  6  2  6  4  2  0  2  0
  0  4  9  5  4  4  2  0  2  0
  1  0 15  6  8  4  4  2  0  2  0
  0  5 12 12  9  6  4  4  2  0  2  0
  0 10  6 21 10 12  6  4  4  2  0  2  0
  0 10 12 20 18 13 10  6  4  4  2  0  2  0
  0  5 27 20 23 16 16 10  6  4  4  2  0  2  0
  1  0 38 22 32 22 19 14 10  6  4  4  2  0  2  0
  0  6 34 38 34 35 20 22 14 10  6  4  4  2  0  2  0
  0 15 22 57 44 40 34 23 20 14 10  6  4  4  2  0  2  0
  0 20 20 71 55 54 45 32 26 20 14 10  6  4  4  2  0  2  0
  0 15 43 70 71 66 60 44 35 24 20 14 10  6  4  4  2  0  2  0
  0  6 74 64 99 83 70 65 42 38 24 20 14 10  6  4  4  2  0  2  0
Row n = 9 counts the following partitions (empty columns not shown):
  (432)   (333)    (54)      (63)      (72)       (81)        (9)
  (3321)  (441)    (621)     (6111)    (711)      (21111111)  (111111111)
  (4221)  (522)    (22221)   (222111)  (2211111)
  (4311)  (531)    (51111)   (411111)  (3111111)
          (3222)   (321111)
          (5211)
          (32211)
          (33111)
          (42111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Row sums are A000041. Column k = 1 is A325191. Column k = 2 is A325199.
T(n,k) = A325189(n,k) - A325188(n,k).
Cf. A046660, A065770, A071724, A243055, A325169, A325178, A325195, A325196, A325197, A366157, A368986.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]-otb[#]==k&]],{n,0,20},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(b=#p,c=0); for(i=1, #p, my(x=#p-i+p[i]); b=min(b,x); c=max(c,x)); r[c-b+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A366157(n) - A368986(n). - Andrew Howroyd, Jan 13 2024

A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 32, 49, 64, 81, 121, 128, 169, 256, 289, 361, 512, 529, 625, 841, 961, 1024, 1331, 1369, 1444, 1681, 1849, 2048, 2116, 2197, 2209, 2809, 3481, 3721, 3844, 4232, 4489, 4913, 5041, 5324, 5329, 5476, 6241, 6859, 6889, 7396, 7569, 7688, 7921

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

			The initial terms of A280292 are (0,0,0,2,0,0,0,4,3,0,0,2,0,0,0,6,0,3,0,2,0,0,0,4,5,0,6,2,...), wherein a value appears for the first time at positions 1, 4, 8, 9, 16, 25, ...

Michel Marcus, Table of n, a(n) for n = 1..391

For length instead of sum we have A151821.
The unsorted version is A280286, firsts of A280292.
For indices instead of factors we have A380957 (unsorted A380956), firsts of A380955.
A multiplicative version is A380988 (unsorted A380987), firsts of A290106.
For prime multiplicities instead of factors see A380989, firsts of A380958.
For product instead of sum we have A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A364916 counts partitions by (sum minus sum of distinct parts).
Cf. A000720, A001414, A008472, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

f(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2] - f[j, 1]); \\ A280292
lista(nn) = my(v=Set(vector(nn, i, f(i))), list=List()); for (i=1, #v, my(k=1); while(f(k) != v[i], k++); listput(list, k)); vecsort(Vec(list)); \\ Michel Marcus, Apr 15 2025

Sorted positions of first appearances in A001414 - A008472.

A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1

Offset: 1

Gerard P. Michon, Jul 05 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Gérard P. Michon, Multiplicative functions.

Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.

A162511 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*(-1)^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after Reinhard Zumkeller *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ Charles R Greathouse IV, Mar 09 2015

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a,b:~(a^b), factorint(n).values(),0)&1 else 1 # Chai Wah Wu, Jan 01 2023

Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
From Reinhard Zumkeller, Jul 08 2009 (Start)
a(n) = (-1)^(A001222(n)-A001221(n)).
a(A162644(n)) = +1; a(A162645(n)) = -1. (End)
a(n) = A076479(n) * A008836(n). - R. J. Mathar, Mar 30 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023

A195069 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.

Original entry on oeis.org

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

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0003698..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 25 2024

			14336 = 2^11 * 7^1, so it has 12 prime factors (counted with multiplicity) and 2 distinct prime factors, and 12-2 = 10.

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

Cf. A025487, A060687, A195087, A195088, A195089, A195090, A195091, A195092, A195093.

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

a195069 n = a195069_list !! (n-1)
a195069_list = filter ((== 10) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

op(select(n->bigomega(n)-nops(factorset(n))=10, [$1..104448])); # Paolo P. Lava, Jul 03 2018

Select[Range[200000], PrimeOmega[#] - PrimeNu[#] == 10&]

isok(n) = bigomega(n) - omega(n) == 10; \\ Michel Marcus, Jul 03 2018

A046660(a(n)) = 10. - Reinhard Zumkeller, Nov 29 2015

cp's OEIS Frontend

A376366 The number of non-unitary prime divisors of the cubefree numbers.

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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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A261256 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.

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A264959 a(n) = A257851(n,n).

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A325200 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.

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