A162510 -id:A162510 - cp's OEIS frontend

A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 32, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 32, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 64, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 32, 16, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4

Offset: 1

Henry Bottomley, May 29 2001

The inverse Möbius transform of A162510. - R. J. Mathar, Feb 09 2011

			a(100)=16 since 100=2*2*5*5 and so a(100)=2*2*2*2.

R. Zumkeller, Table of n, a(n) for n = 1..10005
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011-2012. See eq. (2.12).
Index entries for sequences computed from exponents in factorization of n

Cf. A000079, A001222, A001316, A034444, A069205 (partial sums), A123667, A124508, A156552.

with(numtheory): seq(2^bigomega(n),n=1..95);

Table[2^PrimeOmega[n], {n, 1, 95}] (* Jean-François Alcover, Jun 08 2013 *)

a(n)=direuler(p=1,n,1/(1-2*X))[n] /* Ralf Stephan, Mar 28 2015 */

a(n) = 2^bigomega(n); \\ Michel Marcus, Aug 08 2017

a(n) = Sum_{d divides n} 2^(bigomega(d)-omega(d)) = Sum_{d divides n} 2^(A001222(d) - A001221(d)). - Benoit Cloitre, Apr 30 2002
a(n) = A000079(A001222(n)), i.e., a(n)=2^bigomega(n). - Emeric Deutsch, Feb 13 2005
Totally multiplicative with a(p) = 2. - Franklin T. Adams-Watters, Oct 04 2006
Dirichlet g.f.: Product_{p prime} 1/(1-2*p^(-s)). - Ralf Stephan, Mar 28 2015
a(n) = A001316(A156552(n)). - Antti Karttunen, May 29 2017
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} 1/(1 - 1/(p^s - 1)^2). - Vaclav Kotesovec, Mar 14 2023

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

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

A245195 a(n) = 2^A014081(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 4, 4, 4, 8, 8, 8, 16, 32, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16, 2, 2, 2, 4, 2

Offset: 0

N. J. A. Sloane, Jul 24 2014

This sequence provides a bridge between A245180 (and, presumably, A160239) and A014081.
See A245196 for more about this class of sequences.
Run length transform of A011782: 1,1,2,4,8,16,32,64,... - Chai Wah Wu, Oct 19 2016

Chai Wah Wu and Robert Israel, Table of n, a(n) for n = 0..10000
Robert Israel, Proof that A277560 is the same as A245195 [This will be modified to reflect the fact that the two sequences have now been merged]
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A014081, A245180, A160239, A048896, A245196, A005725, A005940, A011782, A106737, A162510.

# This Maple program applies more generally to a sequence where the recurrence across a block is as follows. The parameters to be set are the sequence G(0), G(1), G(2), ... (the final terms in the blocks), and the multiplier m.
# For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
# (if j=0) a(2^k-2^r) = G(k-r-1),
# (if j>0) a(2^k-2^r+j) = m*G(k-r-1)*a(j).
# Since Maple gives its lists an offset of 1, it is necessary to add 1 to the arguments of G.
# For the present sequence, G(n)=2^n and m=1.
G:=[seq(2^n,n=0..30)];
m:=1;
f:=proc(n) option remember; global m,G; local k,r,j,np;
if n <= 2 then G[0+1] elif n=3 then G[1+1]
elif n=4 then G[0+1] elif n=5 then m*G[0+1] elif n=6 then G[1+1] elif n=7 then G[2+1]
else
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r-1+1]; else m*G[k-r-1+1]*f(j); fi;
fi;
end;
[seq(f(n),n=1..520)]:
# Setting G(n) = A083424(n) and m = 8 gives A245180. Setting G(n) = 2^n and m = 2 gives A048896.
A245195:=n->add(binomial(n,2*k)*binomial(n,k) mod 2, k=0..floor(n/2)): seq(A245195(n), n=0..200); # Wesley Ivan Hurt, Nov 01 2016

Table[Sum[Mod[Binomial[n, 2 k] Binomial[n, k], 2], {k, 0, n}], {n, 0, 85}] (* Michael De Vlieger, Oct 21 2016 *)

a(n) = 2^hammingweight(bitand(n, n>>1)) \\ Charles R Greathouse IV, Jul 16 2016

a(n) = sum(k=0, n, binomial(n, 2*k)*binomial(n,k) % 2); \\ Michel Marcus, Oct 21 2016

from _future_ import division
def A277560(n):
    return sum(int(not (~n & 2*k) | (~n & k)) for k in range(n//2+1))

def A245195(n): return 1<<(n&(n>>1)).bit_count() # Chai Wah Wu, Feb 11 2023

The entries may be arranged into blocks of sizes 1,2,4,8,...:
B_0: 1,
B_1: 1, 2,
B_2: 1, 1, 2, 4,
B_3: 1, 1, 1, 2, 2, 2, 4, 8,
B_4: 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16,
B_5: 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 4, 4, 4, 8, 8, 8, 16, 32,
...
Consider the block B_{k-1} containing terms a(2^(k-1)), a(2^(k-1)+1), ..., a(2^k-1). It is convenient to index the terms working backwards from the next, 2^k-th, term. For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
(if j=0) a(2^k-2^r) = 2^(k-r-1),
(if j>0) a(2^k-2^r+j) = 2^(k-r-1)*a(j).
a(n) = A162510(A005940(1+n)). - Antti Karttunen, Oct 29 2016
From Robert Israel, Nov 02 2016: (Start)
a(2*k)   = a(k).
a(4*k+1) = a(k).
a(4*k+3) = 2*a(2*k+1).
G.f. g(x) satisfies g(x) = x + (2*x+1)*g(x^2) - x*g(x^4). (End)
Also, a(n) = Sum_{k=0..floor(n/2)} ((binomial(n,2k)*binomial(n,k)) mod 2). - Chai Wah Wu, Oct 19 2016 and Robert Israel, Nov 04 2016. For proof, see the article by Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166, or the Robert Israel link.

Changed offset to 0, merged former entry A277560 from Chai Wah Wu (Oct 19 2016) with this sequence. - N. J. A. Sloane, Nov 05 2016

A162512 Dirichlet inverse of A162511.

Original entry on oeis.org

1, -1, -1, 2, -1, 1, -1, -4, 2, 1, -1, -2, -1, 1, 1, 8, -1, -2, -1, -2, 1, 1, -1, 4, 2, 1, -4, -2, -1, -1, -1, -16, 1, 1, 1, 4, -1, 1, 1, 4, -1, -1, -1, -2, -2, 1, -1, -8, 2, -2, 1, -2, -1, 4, 1, 4, 1, 1, -1, 2, -1, 1, -2, 32, 1, -1, -1, -2, 1, -1, -1, -8, -1, 1, -2, -2, 1, -1, -1, -8, 8, 1

Offset: 1

Gerard P. Michon, Jul 05 2009, Jul 06 2009

The absolute value of this sequence is A162510.

The Moebius function (A008683) can be defined in terms of this sequence: A008683(n) is equal to a(n) if a(n) is odd and zero otherwise.

Antti Karttunen, Table of n, a(n) for n = 1..10000
G. P. Michon, Multiplicative functions.

Cf. A005117, A008683, A067029, A076479, A162510, A162511.

A162512 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := -a*(-2)^(op(2,f)-1) ;
    end do:
    return a;
end proc:
seq(A162512(n),n=1..100) ; # R. J. Mathar, May 20 2017

b[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]);
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

a(n) = my(f=factor(n)); for(i=1, #f~, f[i,1]=-(-2)^(f[i,2]-1); f[i,2]=1); factorback(f); \\ Michel Marcus, May 20 2017

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [-(-2)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

(define (A162512 n) (if (= 1 n) n (* (- (expt -2 (- (A067029 n) 1))) (A162512 (A028234 n))))) ;; Antti Karttunen, May 20 2017, after the given multiplicative formula.

Multiplicative function with a(p^e)=-(-2)^(e-1) for any prime p and any positive exponent e.
a(n) = n/2 when n is a power of 4 (A000302).
a(n) = A008683(n) when n is a squarefree number (A005117).
Dirichlet g.f.: Product_{p prime} ((p^s + 1)/(p^s + 2)). - Amiram Eldar, Oct 26 2023

A292589 a(n) = A046523(A003557(n)) = A003557(A046523(n)); the least representative of the prime signature of {n divided by largest squarefree divisor of n}.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 4, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 1, 2, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 16, 1, 2, 2, 6, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

			For n = 18 = 2 * 3^2, A003557(18) = 3^1. The least representative of the same prime signature is 2^1, thus a(18) = 2.
For n = 27 = 3^3, A003557(27) = 9 = 3^2. The least representative of the same prime signature is 2^2, thus a(27) = 4.

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

Cf. A003557, A046523, A292582.

Differs from A162510 for the first time at n=36 where a(36) = 6, while A162510(36) = 4.

a(n) = A046523(A003557(n)) = A003557(A046523(n)).

A335073 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 18, 19, 20, 21, 29, 30, 32, 33, 35, 36, 37, 38, 42, 44, 45, 49, 51, 52, 53, 54, 70, 71, 72, 73, 77, 78, 79, 80, 84, 85, 86, 87, 89, 91, 92, 93, 101, 103, 105, 106, 108, 109, 113, 114, 118, 119, 120, 121, 123, 124, 125, 127

Offset: 1

Daniel Suteu, May 22 2020

nonn

Partial sums of A162510.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001221, A001222, A046660, A061142, A069205, A162510.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      2^(bigomega(n)-nops(factorset(n)))+a(n-1))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, May 22 2020

Accumulate[Table[2^(PrimeOmega[n]-PrimeNu[n]),{n,70}]] (* Harvey P. Dale, Aug 14 2020 *)

a(n) = sum(k=1, n, 2^(bigomega(k) - omega(k)));

a(n) = Sum_{k=1..n} A008683(k) * A069205(floor(n/k)).

a(n) = Sum_{k=1..n} A061142(k) * A002321(floor(n/k)).

A328892 If n = Product (p_j^k_j) then a(n) = Sum (2^(k_j - 1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 2, 8, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 4, 3, 1, 3, 1, 16, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 9, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 4, 1, 2, 3, 32, 2, 3, 1, 3, 2, 3, 1, 6, 1, 2, 3, 3, 2, 3, 1, 9, 8, 2, 1, 4, 2, 2, 2, 5, 1, 4

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

nonn

			a(72) = 6 because 72 = 2^3 * 3^2 and 2^(3 - 1) + 2^(2 - 1) = 6.

Index entries for sequences computed from exponents in factorization of n

Cf. A000040 (positions of 1's), A008481, A011782, A162510, A324910.

a:= n-> add(2^(i[2]-1), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 29 2019

a[1] = 0; a[n_] := Plus @@ (2^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 90}]

a(n)={vecsum([2^(k-1) | k<-factor(n)[,2]])} \\ Andrew Howroyd, Oct 29 2019

If n = Product (p_j^k_j) then a(n) = Sum ordered partition(k_j).

Additive with a(p^e) = 2^(e-1).

cp's OEIS Frontend

A061142 Replace each prime factor of n with 2: a(n) = 2^bigomega(n), where bigomega = A001222, number of prime factors counted with multiplicity.

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A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

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

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A245195 a(n) = 2^A014081(n).

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A162512 Dirichlet inverse of A162511.

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A292589 a(n) = A046523(A003557(n)) = A003557(A046523(n)); the least representative of the prime signature of {n divided by largest squarefree divisor of n}.

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A335073 a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)).

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