A123066 -id:A123066 - cp's OEIS frontend

A321962 Where the zeros in A123066 occur.

1, 51, 53, 7955, 7959, 7961, 7985, 7987, 8245, 8805, 8807, 8809, 8813, 8815, 8817, 8819, 8821, 8825, 8829, 8847, 8851, 8853, 8855, 8857, 8859, 8873, 8877, 8879, 8969, 8973, 8975, 9063, 9079, 9081, 9083, 9089, 9091, 9093, 9095, 9097, 9163, 9165, 9215, 9219

Offset: 1

Peter Luschny, Dec 21 2018

nonn

Let pp(n) be the prime parity of n, defined as 1 if the number of distinct primes dividing n is odd and -1 if it is even; by convention pp(1) = 0. The cumulative sum of pp is A123066. We call the initial segment of the integers [1..n] balanced with respect to prime parity if the cumulative sum of pp(n) is 0. [1..a(n)] give the balanced segments.

Amiram Eldar, Table of n, a(n) for n = 1..8908 (terms below 10^10)
Harald Helfgott and Adrián Ubis, Primos, paridad y análisis, Publicaciones Matemáticas del Uruguay, Vol. 18 (2023), pp. 205-283; arXiv preprint, arXiv:1812.08707 [math.NT], 2018-2019.

Cf. A001222, A123066, A030230, A030231.

a_list := proc(len) local omega, c, L, j; c := 0; L := 1;
omega := n -> nops(numtheory[factorset](n));
for j from 2 to len do
   c := c + (-1)^omega(j);
   if c = 0 then L := L,j fi
od; L end: a_list(10000);

A123066[n_] := Join[{0}, Accumulate[Table[-(-1)^PrimeNu[j], {j,2,n}]]];
A321962List[n_] := Position[A123066[n], 0] // Flatten;
A321962List[10000]

A030231 Numbers with an even number of distinct prime factors.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119

Offset: 1

David W. Wilson

Gcd(A008472(a(n)), A007947(a(n)))=1; see A014963. - Labos Elemer, Mar 26 2003
Superset of A007774. - R. J. Mathar, Oct 23 2008
A076479(a(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Union of the rows of A125666 with even indices. - R. J. Mathar, Jul 19 2023

T. D. Noe, Table of n, a(n) for n = 1..1000
H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

Cf. A028260, A030230, A123066.
Cf. A008472, A007947, A014963.
Cf. A007774, A076479.
Cf. A008683, A000005, A001221, A023900.

a030231 n = a030231_list !! (n-1)
a030231_list = filter (even . a001221) [1..]
-- Reinhard Zumkeller, Mar 26 2013

Select[Range[200],EvenQ[PrimeNu[#]]&] (* Harvey P. Dale, Jun 22 2011 *)

j=[]; for(n=1,200,x=omega(n); if(Mod(x,2)==0,j=concat(j,n))); j

is(n)=omega(n)%2==0 \\ Charles R Greathouse IV, Sep 14 2015

From Benoit Cloitre, Dec 08 2002: (Start)
k such that Sum_{d|k} mu(d)*A000005(d) = (-1)^omega(k) = +1 where mu(d)=A008683(d), and omega(d)=A001221(d).
k such that A023900(k) > 0. (End)
Union of A007774, A033993, A074969,... - R. J. Mathar, Jul 22 2025

Corrected by Dan Pritikin (pritikd(AT)muohio.edu), May 29 2002

A030230 Numbers that have an odd number of distinct prime divisors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 42, 43, 47, 49, 53, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 89, 90, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 120, 121, 125, 126, 127, 128, 130, 131, 132, 137, 138, 139, 140, 149

Offset: 1

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 1..1000
Mats Granvik, Mathematica program to compute the relation to the Dirichlet inverse of the Euler totient function
H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

Cf. A030231, A123066.
Cf. A008472, A007947, A014963.
Cf. A076479.
Cf. A008683, A000005, A001221, A023900.

a030230 n = a030230_list !! (n-1)
a030230_list = filter (odd . a001221) [1..]
-- Reinhard Zumkeller, Aug 14 2011

q:= n-> is(nops(ifactors(n)[2])::odd):
select(q, [$1..150])[];  # Alois P. Heinz, Feb 12 2021

(* Prior to version 7.0 *) littleOmega[n_] := Length[FactorInteger[n]]; Select[ Range[2, 149], (-1)^littleOmega[#] == -1 &] (* Jean-François Alcover, Nov 30 2011, after Benoit Cloitre *)
(* Version 7.0+ *) Select[Range[2, 149], (-1)^PrimeNu[#] == -1 &]
Select[Range[1000],OddQ[PrimeNu[#]]&] (* Harvey P. Dale, Nov 27 2012 *)

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

From Benoit Cloitre, Dec 08 2002: (Start)
k such that Sum_{d|k} mu(d)*tau(d) = (-1)^omega(k) = -1 where mu(d) = A008683(d), tau(d) = A000005(d) and omega(d) = A001221(d).
k such that A023900(k) < 0. (End)
gcd(A008472(a(n)), A007947(a(n))) > 1; see A014963. - Labos Elemer, Mar 26 2003
A076479(a(n)) = -1. - Reinhard Zumkeller, Jun 01 2013

A346617 Irregular triangle T(n,m) read by rows (n >= 1, 1 <= m <= Max(A001221([1..n]))): T(n,m) = number of integers in [1,n] with m distinct prime factors.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 1, 5, 1, 6, 1, 7, 1, 7, 2, 8, 2, 8, 3, 9, 3, 9, 4, 9, 5, 10, 5, 11, 5, 11, 6, 12, 6, 12, 7, 12, 8, 12, 9, 13, 9, 13, 10, 14, 10, 14, 11, 15, 11, 15, 12, 16, 12, 16, 12, 1, 17, 12, 1, 18, 12, 1, 18, 13, 1, 18, 14, 1, 18, 15, 1, 18, 16, 1, 19, 16, 1, 19, 17, 1

Offset: 1

N. J. A. Sloane, Aug 19 2021

Column k >= 1 of the triangle gives the number of numbers i in the range 1 <= i <= n with omega(i) = A001221(i) = k.

A285577 is a similar triangle which has an extra column on the left for k = 0.

			Rows 1 through 12 are:
1 [0]
2 [1]
3 [2]
4 [3]
5 [4]
6 [4, 1]
7 [5, 1]
8 [6, 1]
9 [7, 1]
10 [7, 2]
11 [8, 2]
12 [8, 3]
13 [9, 3]

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 52-56.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, Vol. 1, p. 211, Eq. (5).

Alois P. Heinz, Rows n = 1..10000, flattened
N. J. A. Sloane, The first 100 rows.

Cf. A001221, A013939, A069811, A123066, A174863, A285577.

Row lengths give A111972 (for n>1).

omega := proc(n) nops(numtheory[factorset](n)) end proc: # # A001221
A:=Array(1..20,0);
ans:=[[0]];
mx:=0;
for n from 2 to 100 do
k:=omega(n);
if k>mx then mx:=k; fi;
A[k]:=A[k]+1;
ans:=[op(ans),[seq(A[i],i=1..mx)]];
od:
ans;
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 0,
      b(n-1)+x^nops(ifactors(n)[2]))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..max(1, degree(p))))(b(n)):
seq(T(n), n=1..40);  # Alois P. Heinz, Aug 19 2021

T[n_] := If[n == 1, {0},
     Range[n] // PrimeNu // Tally // Rest // #[[All, 2]]&];
Array[T, 40] // Flatten (* Jean-François Alcover, Mar 08 2022 *)

For fixed k, T(n,k) ~ (1/(k-1)!) * n * (log log n)^(k-1) / log n [Landau].
From Alois P. Heinz, Aug 19 2021: (Start)
Sum_{k>=1} k * T(n,k) = A013939(n).
Sum_{k>=1} k^2 * T(n,k) = A069811(n).
Sum_{k>=1} (-1)^(k-1) * T(n,k) = A123066(n).
Sum_{k>=1} (-1)^k * T(n,k) = -1 + A174863(n).
Sum_{k>=1} T(n,k) = n - 1. (End)

cp's OEIS Frontend

A321962 Where the zeros in A123066 occur.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A030231 Numbers with an even number of distinct prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A030230 Numbers that have an odd number of distinct prime divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A346617 Irregular triangle T(n,m) read by rows (n >= 1, 1 <= m <= Max(A001221([1..n]))): T(n,m) = number of integers in [1,n] with m distinct prime factors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula