A046644 -id:A046644 - cp's OEIS frontend

A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

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

Offset: 1

Antti Karttunen, Aug 12 2018

a(n) is the denominator of certain rational valued sequences f(n), that have been defined as f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, dA034444 and A037445.
Many of the same observations as given in A046644 apply also here. Note that A011371 shares with A005187 the property that A011371(x+y) <= A011371(x) + A011371(y), with equivalence attained only when A004198(x,y) = 0, and also the property that A011371(2^(k+1)) = 1 + 2*A011371(2^k).
The following list gives such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n).
  Numerators   Dirichlet convolution of numerator(n)/a(n) yields
  -------      -----------
  A317933      A034444
  A317941      A037445
  A317940      A046644
Expansion of Dirichlet g.f. Product_{prime} 1/(1 - 2/p^s)^(1/2) is A046643/A317934. - Vaclav Kotesovec, May 08 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)
Wikipedia, Dirichlet convolution

Cf. A011371, A034444, A317940, A317941, A317946.
Cf. A317933, A317940, A317941 (numerator-sequences).
Cf. also A046644, A299150, A299152, A317832, A317932, A317926 (for denominator sequences of other similar constructions).
Cf. A046643, A167864, A347195.

A011371(n) = (n - hammingweight(n));
A317934(n) = factorback(apply(e -> 2^A011371(e),factor(n)[,2]));

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 07 2025

for(n=1, 100, print1(denominator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = 2^A317946(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 is A034444, A037445 or A046644 for example.
Sum_{k=1..n} A046643(k)/a(k) ~ n * sqrt(A167864*log(n)/(Pi*log(2))) * (1 + (4*(gamma - 1) + 5*log(2) - 4*A347195)/(8*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 08 2025

A257099 From third root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -5, -1, 1, -1, 1, -1, 1, 1, -10, -1, 1, -1, 1, 1, 1, -1, 5, -1, 1, -5, 1, -1, -1, -1, -22, 1, 1, 1, 1, -1, 1, 1, 5, -1, -1, -1, 1, 1, 1, -1, 10, -1, 1, 1, 1, -1, 5, 1, 5, 1, 1, -1, -1, -1, 1, 1, -154, 1, -1, -1, 1, 1, -1, -1, 5, -1, 1, 1, 1, 1, -1, -1, 10, -10, 1, -1, -1, 1, 1, 1, 5, -1, -1, 1, 1, 1, 1, 1, 22, -1, 1, 1, 1

Offset: 1

Wolfgang Hintze, Apr 16 2015

Dirichlet g.f. of b(n) = a(n)/A256689(n) is (zeta(x))^(-1/3).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/3).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..500 from Wolfgang Hintze)
Vaclav Kotesovec, Graph - the asymptotic ratio (10^7 terms)

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).

Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

k = 3;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257099 *)
den = Denominator[t] (* A256689 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/3))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

with k = 3;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257099(j)/A256689(j) ~ n / (Gamma(-1/3) * log(n)^(4/3)) * (1 + 4*(gamma/3 + 1)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 9, 11, 7, 1, 3, 1, 3, 5, 171, 1, -1, 1, -5, 5, 7, 1, -1, 11, 7, 29, 35, 1, -7, 1, -41, 5, 7, 9, 93, 1, 3, 5, 11, 1, -3, 1, -5, 3, 7, 1, -61, 11, 7, 9, 27, 1, -29, 5, -1, 9, 11, 1, -29, 1, 3, 3, 771, 9, 9, 1, -5, 5, -3, 1, -73, 1, 3, 3, 19, 9, 9, 1, -141, -45, 7, 1, -53, 5, 7, 9, 43, 1, -63, 5, 11, 9, 11, 13, 1597, 1

Offset: 1

Antti Karttunen, Aug 12 2018

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

Cf. A151800, A175851, A046644 (denominators).

Cf. also A305791, A305805, A305806, A317833, A317834, A317847.

A175851[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A175851[n] - Sum[If[1 < d < n, f[d]* f[n/d], 0], {d, Divisors[n]}])];
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

A175851(n) = if(1==n,n,1 + n - precprime(n));
A317830aux(n) = if(1==n,n,(A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2);
A317830(n) = numerator(A317830aux(n));

\\ Memoized implementation:
memo317830 = Map();
A317830aux(n) = if(1==n,n,if(mapisdefined(memo317830,n),mapget(memo317830,n),my(v = (A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2); mapput(memo317830,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A175851(n) - Sum_{d|n, d>1, d 1.

A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 7, 1, 7, 1, 25, 11, 11, 1, 43, 1, 15, 13, 363, 1, 71, 1, 67, 17, 23, 1, 139, 19, 27, 61, 91, 1, 57, 1, 1335, 25, 35, 21, 365, 1, 39, 29, 215, 1, 81, 1, 139, 131, 47, 1, 1875, 27, 199, 37, 163, 1, 367, 29, 291, 41, 59, 1, 235, 1, 63, 171, 9923, 33, 129, 1, 211, 49, 137, 1, 1055, 1, 75, 235, 235, 33, 153, 1, 2883, 1363, 83, 1, 335, 41

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(420) = -1269.

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

Cf. A051953, A063524, A046644 (denominators).

Cf. also A317845, A317925.

A317846aux(n) = if(1==n,n,((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(dA317846aux(d)*A317846aux(n/d),0)))/2);
A317846(n) = numerator(A317846aux(n));

\\ Memoized implementation:
memo317846 = Map();
A317846aux(n) = if(1==n,n,if(mapisdefined(memo317846,n),mapget(memo317846,n),my(v = ((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(dA317846aux(d)*A317846aux(n/d),0)))/2); mapput(memo317846,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A051953(n) - Sum_{d|n, d>1, d 1.

A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 25, 11, 1, 1, 7, 1, 1, 1, 363, 1, 11, 1, 7, 1, 1, 1, 25, 19, 1, 61, 7, 1, 1, 1, 1335, 1, 1, 1, 77, 1, 1, 1, 25, 1, 1, 1, 7, 11, 1, 1, 363, 27, 19, 1, 7, 1, 61, 1, 25, 1, 1, 1, 7, 1, 1, 11, 9923, 1, 1, 1, 7, 1, 1, 1, 275, 1, 1, 19, 7, 1, 1, 1, 363, 1363, 1, 1, 7, 1, 1, 1, 25, 1, 11, 1, 7, 1, 1, 1, 1335, 1, 27, 11, 133, 1, 1, 1, 25, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A003557 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
Wikipedia, Dirichlet convolution

Cf. A003557, A046644 (denominators).

Cf. also A300717, A300719, A318317.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]--); factorback(f); }; \\ From A003557
A317935aux(n) = if(1==n,n,(A003557(n)-sumdiv(n,d,if((d>1)&&(dA317935aux(d)*A317935aux(n/d),0)))/2);
A317935(n) = numerator(A317935aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003557(n) - Sum_{d|n, d>1, d 1.

A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Note that A318314 differs from A318454 at exactly those n where A001227 differs from A068068, the numbers in A038838. - Antti Karttunen, Sep 07 2018

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

Cf. A005187, A011371, A068068, A318313 (numerators), A318315.

Cf. also A006519, A046644, A299150, A317932, A317934, A317940, A318454, A318662.

a35[n_] := (1 - (-1)^n)/2;
a120[n_] := DigitCount[n, 2, 1];
a[n_] := Product[{p, e} = pe; 2^(((2 - a35[p])*e) - a120[e]), {pe, FactorInteger[n]}];
a /@ Range[100] (* Jean-François Alcover, Sep 19 2019 *)

up_to = 16384;
A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
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.
v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
A318313(n) = numerator(v318313_15[n]);
A318314(n) = denominator(v318313_15[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318315(n).
From Antti Karttunen, Sep 03-07 2018: (Start, conjectured formulas)
a(n) = A006519(n) * A317934(n), thus multiplicative with a(2^e) = 2^A005187(e), a(p^e) = 2^A011371(e) for odd primes p.
Equally, multiplicative with a(p^e) = 2^(((2-A000035(p))*e)-A000120(e)) for all primes p.
(End)

A318658 Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 31 2018

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

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

Cf. A005187, A087003, A318657 (numerators), A318659.

up_to = 65537;
A087003(n) = ((n%2)*moebius(n)); \\ I.e. a(n) = A000035(n)*A008683(n).
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&&dA087003(n)));
A318657(n) = numerator(v318657_18[n]);
A318658(n) = denominator(v318657_18[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318659(n).
a(2n) = 1, a(2n-1) = A046644(2n-1) = A318512(2n-1), for all n >= 1.

A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).

Original entry on oeis.org

1, 1, 1, 7, 1, 5, 1, 25, 7, 9, 1, 31, 1, 13, 5, 363, 1, 55, 1, 55, 7, 21, 1, 101, 7, 25, 33, 79, 1, 41, 1, 1335, 11, 33, 5, 305, 1, 37, 13, 177, 1, 59, 1, 127, 47, 45, 1, 1371, 7, 175, 17, 151, 1, 309, 7, 253, 19, 57, 1, 187, 1, 61, 67, 9923, 9, 95, 1, 199, 23, 113, 1, 927, 1, 73, 87, 223, 5, 113, 1, 2379, 715, 81, 1, 265, 11

Offset: 1

Antti Karttunen, Aug 10 2018

The first negative term is a(840) = -445.

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

Cf. A046644, A078898.

Cf. also A305798, A305803, A305804, A317830, A317834.

lpf[n_] := If[n == 1, 1, FactorInteger[n][[1, 1]]];
b[_] = 1;
A078898[n_] := A078898[n] = If[n == 0, 0, With[{t = lpf[n]}, b[t]++]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078898[n] - Sum[If[1 < d < n, f[d]*f[n/d], 0], {d, Divisors[n]}])]
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A317833aux(n) = if(1==n,n,(A078898(n)-sumdiv(n,d,if((d>1)&&(dA317833aux(d)*A317833aux(n/d),0)))/2);
A317833(n) = numerator(A317833aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078898(n) - Sum_{d|n, d>1, d 1.

A317834 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 17, 11, 3, 1, 19, 1, 3, 5, 139, 1, 23, 1, 19, 5, 3, 1, 39, 19, 3, 45, 19, 1, 13, 1, 263, 5, 3, 9, 77, 1, 3, 5, 55, 1, 13, 1, 19, 43, 3, 1, 387, 27, 47, 5, 19, 1, 59, 9, 71, 5, 3, 1, 43, 1, 3, 51, 995, 9, 13, 1, 19, 5, 25, 1, 87, 1, 3, 59, 19, 13, 13, 1, 707, 467, 3, 1, 59, 9, 3, 5, 71, 1, 53, 13, 19, 5, 3, 9, 1069, 1

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(216) = -97.

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

Cf. A078899, A046644 (denominators).

Cf. also A305799, A317833, A317830.

gpf[n_] := If[n == 1, 1, FactorInteger[n][[-1, 1]]];
b[_] = 1;
A078899[n_] := A078899[n] = With[{t = gpf[n]}, b[t]++];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078899[n] -
     Sum[If[1Jean-François Alcover, Dec 19 2021 *)

up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
v078899 = ordinal_transform(vector(up_to,n,A006530(n)));
A078899(n) = v078899[n];
A317834aux(n) = if(1==n,n,(A078899(n)-sumdiv(n,d,if((d>1)&&(dA317834aux(d)*A317834aux(n/d),0)))/2);
A317834(n) = numerator(A317834aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078899(n) - Sum_{d|n, d>1, d 1.

A317835 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A003415 (arithmetic derivative of n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 15, 1, 9, 1, 81, 23, 13, 1, 95, 1, 17, 15, 1499, 1, 127, 1, 151, 19, 25, 1, 393, 39, 29, 193, 207, 1, 87, 1, 6311, 27, 37, 23, 969, 1, 41, 31, 661, 1, 119, 1, 319, 259, 49, 1, 5499, 55, 295, 39, 375, 1, 769, 31, 929, 43, 61, 1, 593, 1, 65, 347, 50075, 35, 183, 1, 487, 51, 183, 1, 2751, 1, 77, 371, 543, 35, 215, 1, 9643, 5611, 85, 1

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(240) = -5067.

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

Cf. A003415, A063524, A046644 (denominators).

Cf. also A300251, A300252, A305809.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A317835aux(n) = if(1==n,n,(A003415(n)-sumdiv(n,d,if((d>1)&&(dA317835aux(d)*A317835aux(n/d),0)))/2);
A317835(n) = numerator(A317835aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003415(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

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A257099 From third root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is 1/zeta; sequence gives numerator of b(n).

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A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

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A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).

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A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

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A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

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PARI

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A318658 Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

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PARI

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A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).

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