A046644 -id:A046644 - cp's OEIS frontend

A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

0, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 7, 1, 4, 1, 4, 2, 2, 1, 5, 3, 2, 4, 4, 1, 3, 1, 8, 2, 2, 2, 6, 1, 2, 2, 5, 1, 3, 1, 4, 4, 2, 1, 8, 3, 4, 2, 4, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 4, 10, 2, 3, 1, 4, 2, 3, 1, 7, 1, 2, 4, 4, 2, 3, 1, 8, 7, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 4, 2

Offset: 1

N. J. A. Sloane

A268375 gives numbers n for which a(n) = A289617(n) = A005187(A001222(n)). - Antti Karttunen, Jul 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..65536
Index entries for sequences computed from exponents in factorization of n.

See A046643, A046644 for more details.

Cf. A001222, A005187, A007814, A077761, A268375, A289617, A289618, A293442, A293447.

f[p_, e_] := 2*e - DigitCount[2*e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

A007814(n) = (valuation(n,2));
A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); } \\ After the Maple-program given in A046643.
A046645(n) = A007814(denominator(A046643perA046644(n))); \\ Antti Karttunen, Jul 08 2017

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046645(n) = vecsum(apply(e -> A005187(e), factorint(n)[, 2])); \\ A faster implementation. - Antti Karttunen, Jul 08 2017

a(n) = A007814(A046644(n)). - Michel Marcus, Apr 16 2015
Additive with a(p^n) = A005187(n). - Antti Karttunen, Jul 08 2017
a(n) = A293447(A293442(n)). - Antti Karttunen, Nov 10 2017
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 1.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 29 2023

A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

Original entry on oeis.org

1, 2, 1, 8, 2, 2, 2, 16, 2, 4, 2, 8, 2, 4, 2, 128, 2, 4, 2, 16, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 2, 256, 2, 4, 4, 16, 2, 4, 2, 32, 2, 4, 2, 16, 4, 4, 2, 128, 8, 16, 2, 16, 2, 4, 4, 32, 2, 4, 2, 16, 2, 4, 4, 1024, 4, 4, 2, 16, 2, 8, 2, 32, 2, 4, 8, 16, 4, 4, 2, 256, 8, 4, 2, 16, 4, 4, 2, 32, 2, 8, 4, 16, 2, 4, 4, 256, 2, 16, 4, 64, 2, 4, 2, 32, 4

Offset: 1

Antti Karttunen, Aug 11 2018

These are denominators for rational valued sequences that are obtained as "Dirichlet Square Roots" of sequences b that satisfy the condition b(3) = 2, and b(p) = odd number for any other primes p. For example, A064989, A065769 and A234840. - Antti Karttunen, Aug 31 2018

The original definition was: Denominators of the rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence. However, this definition depends on the conjecture given in A261179.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A317930, A318319, A318669 (some of the numerator sequences), A317931 (conjectured, for A002487).
Cf. A305439 (the 2-adic valuation), A318666.
Cf. also A046644, A261179, A299150, A237649, A299150, A317928, A317839, A317843, A317934, A318319.

\\ Original program, based on conjectural formula:
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317931perA317932(n) = if(1==n,n,(A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2);
A317932(n) = denominator(A317931perA317932(n));

\\ New fast program implementing the new definition:
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
A317932(n) = (A046644(n)/2^valuation(n,3)); \\ Antti Karttunen, Aug 31 2018

A011371(n) = (A005187(n)-n);
A317932(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i,1], m *= 2^(A011371(f[i,2])), m *= 2^A005187(f[i,2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

a(n) = A046644(n)/A318666(n) = 2^A305439(n).
a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d 1, where b can be A064989, A065769 or A234840 for example, conjecturally also A002487.
Multiplicative with a(3^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for any other primes. - Antti Karttunen, Sep 03 2018

Definition changed, the original (now conjectured alternative definition) moved to the comments section by Antti Karttunen, Aug 31 2018

Keyword:mult added by Antti Karttunen, Sep 03 2018

A317940 Numerators of sequence whose Dirichlet convolution with itself yields A046644.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 9, 7, 1, 1, 7, 1, 1, 1, 427, 1, 7, 1, 7, 1, 1, 1, 9, 7, 1, 9, 7, 1, 1, 1, 471, 1, 1, 1, 49, 1, 1, 1, 9, 1, 1, 1, 7, 7, 1, 1, 427, 7, 7, 1, 7, 1, 9, 1, 9, 1, 1, 1, 7, 1, 1, 7, 4099, 1, 1, 1, 7, 1, 1, 1, 63, 1, 1, 7, 7, 1, 1, 1, 427, 427, 1, 1, 7, 1, 1, 1, 9, 1, 7, 1, 7, 1, 1, 1, 471, 1, 7, 7, 49, 1, 1, 1, 9, 1

Offset: 1

Antti Karttunen, Aug 14 2018

Multiplicative because A046644 is.

No negative terms among the first 2^20 terms. Is the sequence nonnegative?

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005187, A046644, A317934 (denominators), A317941.

up_to = 65537;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
v317940aux = DirSqrt(vector(up_to, n, A046644(n)));
A317940(n) = numerator(v317940aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A046644(n) - Sum_{d|n, d>1, d 1.

A318651 a(n) = A046644(n)/A318512(n).

Original entry on oeis.org

1, 2, 1, 8, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 128, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 256, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 256, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 1024, 1, 4, 1, 16, 1, 4, 1, 64, 1, 4, 1, 16, 1, 4, 1, 256, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 512, 1, 4, 1, 16, 1, 4, 1, 32, 1

Offset: 1

Antti Karttunen, Aug 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A046644, A318512, A318652.

A318651(n) = A046644(n)/A318512(n); \\ Needs also code from those two respective entries.

a(n) = A046644(n)/A318512(n).

a(n) = 2^A318652(n).

A318652 The 2-adic valuation of A046644(n)/A318512(n) (A318651).

Original entry on oeis.org

0, 1, 0, 3, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 7, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 10, 0, 2, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 9, 0, 2, 0, 4, 0, 2, 0, 5, 0

Offset: 1

Antti Karttunen, Aug 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046644, A318512, A318651.

A318652(n) = valuation(A046644(n)/A318512(n),2); \\ Needs also code from those two respective entries.

a(n) = A007814(A318651(n)).

a(n) = A046645(n) - A318513(n).

A181117 Triangle T(n,k) read by rows. T(n,k) = A046644(A126988).

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 8, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 16, 8, 0, 2, 0, 0, 0, 1, 8, 0, 2, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 16, 4, 8, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Oct 04 2010

Denominator in fraction A181116/A181117.

			Triangle starts:
1,
2,1,
2,0,1,
8,2,0,1,
2,0,0,0,1,
4,2,2,0,0,1,
2,0,0,0,0,0,1,
16,8,0,2,0,0,0,1,
8,0,2,0,0,0,0,0,1,
4,2,0,0,2,0,0,0,0,1,
2,0,0,0,0,0,0,0,0,0,1,
16,4,8,2,0,2,0,0,0,0,0,1,
2,0,0,0,0,0,0,0,0,0,0,0,1,
4,2,0,0,0,0,2,0,0,0,0,0,0,1,
4,0,2,0,2,0,0,0,0,0,0,0,0,0,1,

Cf. A181116, A046644, A051731.

A051953 Cototient(n) := n - phi(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, 1, 38, 35, 40, 17, 54, 1, 48, 27

Offset: 1

Labos Elemer, Dec 21 1999

Unlike totients, cototient(n+1) = cototient(n) never holds -- except 2-phi(2) = 3 - phi(3) = 1 -- because cototient(n) is congruent to n modulo 2. - Labos Elemer, Aug 08 2001
Theorem (L. Redei): b^a(n) == b^n (mod n) for every integer b. - Thomas Ordowski and Robert Israel, Mar 11 2016
Let S be the sum of the cototients of the divisors of n (A001065). S < n iff n is deficient, S = n iff n is perfect, and S > n iff n is abundant. - Ivan N. Ianakiev, Oct 06 2023

			n = 12, phi(12) = 4 = |{1, 5, 7, 11}|, a(12) = 12 - phi(12) = 8, numbers not exceeding 12 and not coprime to 12: {2, 3, 4, 6, 8, 9, 10, 12}.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.
R. E. Jamison, The Helly bound for singular sums, Discrete Math., 249 (2002), 117-133.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26. For Richard Guy on his 99th birthday. May his sequence be unbounded.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Cototient

Cf. A000010, A001065 (inverse Möbius transform), A005278, A001274, A083254, A098006, A049586, A051612, A053579, A054525, A062790 (Möbius transform), A063985 (partial sums), A063986, A290087.
Records: A065385, A065386.
Number of zeros in the n-th row of triangle A054521. - Omar E. Pol, May 13 2016
Cf. A063740 (number of k such that cototient(k) = n). - M. F. Hasler, Jan 11 2018

a051953 n = n - a000010 n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory); A051953 := n->n-phi(n);

Table[n - EulerPhi[n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

A051953(n) = n - eulerphi(n); \\ Michael B. Porter, Jan 28 2010

from sympy.ntheory import totient
print([i - totient(i) for i in range(1, 101)]) # Indranil Ghosh, Mar 17 2017

a(n) = n - A000010(n).
Equals Mobius transform (A054525) of A001065. - Gary W. Adamson, Jul 11 2008
a(A006881(n)) = sopf(A006881(n)) - 1; a(A000040(n)) = 1. - Wesley Ivan Hurt, May 18 2013
G.f.: sum(n>=1, A000010(n)*x^(2*n)/(1-x^n) ). - Mircea Merca, Feb 23 2014
From Ilya Gutkovskiy, Apr 13 2017: (Start)
G.f.: -Sum_{k>=2} mu(k)*x^k/(1 - x^k)^2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/zeta(s)). (End)
From Antti Karttunen, Sep 05 2018 & Apr 29 2022: (Start)
Dirichlet convolution square of A317846/A046644 gives this sequence + A063524.
a(n) = A003557(n) * A318305(n).
a(n) = A000010(n) - A083254(n).
a(n) = A318325(n) - A318326(n).
a(n) = Sum_{d|n} A062790(d) = Sum_{d|n, dA007431(d)*(A000005(n/d)-1).
a(n) = A048675(A318834(n)) = A276085(A353564(n)). [These follow from the formula below]
a(n) = Sum_{d|n, dA000010(d).
a(n) = A051612(n) - A001065(n).
(End)

A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 7, 1, 3, 3, 35, 1, 7, 1, 7, 3, 3, 1, 11, 3, 3, 5, 7, 1, 3, 1, 63, 3, 3, 3, 9, 1, 3, 3, 11, 1, 3, 1, 7, 7, 3, 1, 75, 3, 7, 3, 7, 1, 11, 3, 11, 3, 3, 1, 1, 1, 3, 7, 231, 3, 3, 1, 7, 3, 3, 1, 19, 1, 3, 7, 7, 3, 3, 1, 75, 35, 3, 1, 1, 3, 3, 3, 11, 1, 1, 3, 7, 3, 3, 3, 133, 1, 7, 7, 9, 1, 3, 1, 11, 3

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(210) = -7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A063524, A046644 (denominators).

Cf. also A317831, A317925, A317933, A317845, A317846, A317936, A317938, A317939.

A317937aux(n) = if(1==n,n,(omega(n)-sumdiv(n,d,if((d>1)&&(dA317937aux(d)*A317937aux(n/d),0)))/2);
A317937(n) = numerator(A317937aux(n));

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dAndrew Howroyd, Aug 13 2018

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001221(n) - Sum_{d|n, d>1, d 1.

A046643 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives numerator of b_n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 35, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 63, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 35, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 231, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 35, 35, 1, 1, 3, 1, 1, 1, 5, 1, 3

Offset: 1

N. J. A. Sloane, Dec 11 1999

b(n) = A046643(n)/A046644(n) is multiplicative with b(p^n) = (2n-1)!!/2^n/n!. Dirichlet g.f. of A046643(n)/A046644(n) is sqrt(zeta(x)). - Christian G. Bower, May 16 2005

That is, b(p^n) = A001147(n) / (A000079(n)*A000142(n)) = A010050(n)/A000290(A000165(n)) = (2n)!/((2^n*n!)^2). - Antti Karttunen, Jul 08 2017

			b_1, b_2, ... = 1, 1/2, 1/2, 3/8, 1/2, 1/4, 1/2, 5/16, 3/8, 1/4, 1/2, 3/16, ...

Cf. A000079, A000165, A001147, A001790, A005187, A010050, A028234, A067029.

Cf. A046644, A046645.

b := proc(n) option remember; local c,i,t1; if n = 1 then 1 else c := 1; t1 := divisors(n);
for i from 2 to nops(t1)-1 do c := c-b(t1[ i ])*b(n/t1[ i ]); od; c/2; fi; end;

b[1] = 1; b[n_] := b[n] = (dn = Divisors[n]; c = 1;
Do[c = c - b[dn[[i]]]*b[n/dn[[i]]], {i, 2, Length[dn] - 1}]; c/2); a[n_] := Numerator[b[n]]; a /@ Range[90] (* Jean-François Alcover, Apr 04 2011, after Maple version *)

A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); }
A046643(n) = numerator(A046643perA046644(n)); \\ After Maple-program, Antti Karttunen, Jul 08 2017

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

(define (A046643 n) (if (= 1 n) n (* (A001790 (A067029 n)) (A046643 (A028234 n)))))
;; Or, after Christian G. Bower's May 16 2005 comment:
(definec (A046643perA046644 n) (if (= 1 n) n (* (/ (A010050 (A067029 n)) (A000290 (A000165 (A067029 n)))) (A046643perA046644 (A028234 n)))))
(define (A046643 n) (numerator (A046643perA046644 n)))
(define (A046644 n) (denominator (A046643perA046644 n)))
;; Antti Karttunen, Jul 08 2017

Sum_{b|d} b(d)b(n/d) = 1. Also b_{2^j} = A001790[ j ]/2^A005187[ j ].
From Antti Karttunen, Jul 08 2017: (Start)
Multiplicative with a(p^n) = A001790(n).
a(1) = 1; for n > 1, a(n) = A001790(A067029(n)) * a(A028234(n)).
(End)
Sum_{j=1..n} A046643(j)/A046644(j) ~ n / sqrt(Pi*log(n)) * (1 + (1 - gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 04 2025

A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 8, 2, 2, 4, 2, 2, 4, 8, 2, 8, 2, 4, 4, 2, 2, 4, 8, 2, 16, 4, 2, 4, 2, 8, 4, 2, 4, 16, 2, 2, 4, 4, 2, 4, 2, 4, 16, 2, 2, 16, 8, 8, 4, 4, 2, 16, 4, 4, 4, 2, 2, 8, 2, 2, 16, 16, 4, 4, 2, 4, 4, 4, 2, 16, 2, 2, 16, 4, 4, 4, 2, 16, 128, 2, 2

Offset: 1

Gus Wiseman, Feb 03 2018

			Sequence begins: 1, 1, 3/2, 3/2, 5/2, 3/2, 7/2, 5/2, 27/8, 5/2, 11/2, 9/4, 13/2, 7/2.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Andrew Howroyd)

Cf. A000010, A003958, A005187, A006519, A007431, A011371, A018804, A023900, A029935, A037445, A046643, A046644, A165825, A257098, A298971, A299119, A299149 (numerators), A299151, A317848, A317929, A317932, A318314, A318512, A318653, A318681.

Cf. A318440 (the 2-adic valuation).

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
f[p_, e_] := 2^((1 + Mod[p, 2])*e - DigitCount[e, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n)={my(v=factor(n)[,2]); denominator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018

A299150(n) = { my(f = factor(n), m=1); for(i=1, #f~, m *= 2^(((1+(f[i,1]%2))*f[i,2]) - hammingweight(f[i,2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 08 2025

a(n) = denominator(n*A317848(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)). - Andrew Howroyd, Aug 09 2018
a(n) = A046644(n)/A006519(n). - Andrew Howroyd and Antti Karttunen, Aug 30 2018
From Antti Karttunen, Sep 03 2018: (Start)
a(n) = 2^A318440(n).
Multiplicative with a(2^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for odd primes p.
Multiplicative with a(p^e) = 2^(((1+A000035(p))*e)-A000120(e)) for all primes p.
(End)

Keyword:mult added by Andrew Howroyd, Aug 09 2018

cp's OEIS Frontend

A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

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A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

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A317940 Numerators of sequence whose Dirichlet convolution with itself yields A046644.

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A318651 a(n) = A046644(n)/A318512(n).

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A318652 The 2-adic valuation of A046644(n)/A318512(n) (A318651).

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A181117 Triangle T(n,k) read by rows. T(n,k) = A046644(A126988).

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A051953 Cototient(n) := n - phi(n).

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A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

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