A099812 -id:A099812 - cp's OEIS frontend

A092523 Number of distinct prime factors of n-th odd number.

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

Offset: 1

Reinhard Zumkeller, Apr 07 2004

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Odd Number.

Cf. A001221, A077761, A091304, A099812.

Table[PrimeNu[n],{n,1,211,2}] (* Harvey P. Dale, May 25 2015 *)

a(n) = omega(2*n-1); \\ Amiram Eldar, Sep 21 2024

a(n) = A001221(2*n-1).

Sum_{k=1..n} a(k) = n * (log(log(n)) + B - 1/2) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A129597 Central diagonal of array A129595.

Original entry on oeis.org

1, 4, 6, 16, 10, 24, 14, 64, 54, 40, 22, 96, 26, 56, 90, 256, 34, 216, 38, 160, 126, 88, 46, 384, 250, 104, 486, 224, 58, 360, 62, 1024, 198, 136, 350, 864, 74, 152, 234, 640, 82, 504, 86, 352, 810, 184, 94, 1536, 686, 1000, 306, 416, 106, 1944, 550, 896, 342

Offset: 1

Antti Karttunen, May 01 2007, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

nonn

These are the positions of first appearances of each positive integer in A346704. - Gus Wiseman, Oct 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000

a(n) = A129595(n,n).
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n) + 1 for n > 1.
The version for odd indices is A342768(n) = a(n)/2 for n > 1.
Except the first term, the sorted version is 2*A346635.
These are the positions of first appearances in A346704.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A346633 adds up the even bisection of standard compositions (odd: A209281).
A346698 adds up the even bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A037143, A329888, A344606, A345957, A346697, A346700, A346701.

Table[If[n==1,1,2*n^2/FactorInteger[n][[-1,1]]],{n,100}] (* Gus Wiseman, Aug 10 2021 *)

A129597(n) = if(1==n, n, my(f=factor(n)); (2*n*n)/f[#f~, 1]); \\ Antti Karttunen, Oct 16 2021

From Gus Wiseman, Aug 10 2021: (Start)
For n > 1, A001221(a(n)) = A099812(n).
If g = A006530(n) is the greatest prime factor of n > 1, then a(n) = 2n^2/g.
a(n) = A100484(A000720(n)) = 2n iff n is prime.
a(n > 1) = 2*A342768(n).
(End)

A100008 Number of unitary divisors of 2n.

Original entry on oeis.org

2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 8, 2, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 2, 8, 4, 8, 4, 4, 4, 8, 4, 4, 8, 4, 4, 8, 4, 4, 4, 4, 4, 8, 4, 4, 4, 8, 4, 8, 4, 4, 8, 4, 4, 8, 2, 8, 8, 4, 4, 8, 8, 4, 4, 4, 4, 8, 4, 8, 8, 4, 4, 4, 4, 4, 8, 8, 4, 8, 4, 4, 8, 8, 4, 8, 4, 8, 4, 4, 4, 8, 4, 4, 8, 4, 4, 16

Offset: 1

N. J. A. Sloane, Nov 20 2004

b(n) = a(n)/a(1) is multiplicative with b(2^e) = 1, b(p^e) = 2 otherwise. - David W. Wilson, Jun 12 2005

			a(6)=4 because among the six divisors of 12 only 1,3,4 and 12 are unitary.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Bisection of A034444, twice A068068.

Cf. A000079, A001221, A099812.

with(numtheory): for n from 1 to 120 do printf(`%d,`,2^nops(ifactors(2*n)[2])) od: # Emeric Deutsch, Dec 24 2004

a[n_] := 2^PrimeNu[2*n]; Array[a, 100] (* Amiram Eldar, Jan 28 2023 *)

A100008(n) = 2^omega(2*n); \\ Antti Karttunen, Sep 14 2017

a(n) = A000079(A099812(n)) = A000079(A001221(2n)) = 2*A068068(n). - Antti Karttunen, Sep 14 2017
Dirichlet g.f.: 2*zeta(s)^2/(zeta(2*s)*(1+1/2^s)). - Amiram Eldar, Jan 28 2023
Sum_{k=1..n} a(k) ~ 8*n*((log(n) - 1 + 2*gamma + log(2)/3)/Pi^2 - 12*zeta'(2)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 28 2023

More terms from Emeric Deutsch, Dec 24 2004

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A092523 Number of distinct prime factors of n-th odd number.

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A129597 Central diagonal of array A129595.

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A100008 Number of unitary divisors of 2n.

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