A046644 -id:A046644 - cp's OEIS frontend

A318306 Additive with a(p^e) = A002487(e).

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 3, 3

Offset: 1

Antti Karttunen, Aug 29 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A002487, A077761, A318307.

Cf. also A046644.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318306(n) = vecsum(apply(e -> A002487(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318306(n): return sum(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(e)[-1:2:-1],(1,0))) for e in factorint(n).values()) # Chai Wah Wu, May 18 2023

a(n) = A007814(A318307(n)).

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) = 0.15790080909728804399..., where f(x) = -x + x * (1-x) * Product{k>=0} (1 + x^(2^k) + x^(2^(k + 1))). - Amiram Eldar, Feb 11 2024

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

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

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
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.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318499(n).

A318657 Numerators 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, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -5, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, -1

Offset: 1

Antti Karttunen, Aug 31 2018

Because the corresponding denominator sequence A318658 is equal to A046644 on all odd n, and this sequence as well as A087003 is zero on all even n, it means that also the Dirichlet convolution of a(n)/A046644(n) with itself will yield A087003. Because both A046644 and A087003 are multiplicative, this sequence is also. - Antti Karttunen, Sep 01 2018

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

Cf. A046644 or A318658 (denominators).

Cf. also A087003, A257098, 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]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d 1.

a(2n) = 0, a(2n-1) = A257098(2n-1), thus multiplicative with a(2^e) = 0, a(p^e) = A257098(p^e) for odd primes p. - Antti Karttunen, Sep 01 2018

A340141 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A160595(x) = phi(x)/gcd(phi(x), x-1).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 25, 11, 7, 1, 19, 1, 11, 7, 363, 1, 31, 1, 43, 5, 19, 1, 63, 19, 23, 61, 3, 1, 15, 1, 1335, 9, 31, 23, 189, 1, 35, 23, 139, 1, 29, 1, 115, 23, 43, 1, 867, 27, 127, 31, 11, 1, 163, 39, 279, 17, 55, 1, 73, 1, 59, 123, 9923, 5, -15, 1, 187, 21, -9, 1, 615, 1, 71, 127, 19, 29, 47, 1, 1875, 1363, 79, 1, 203, 31

Offset: 1

Antti Karttunen, Dec 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A046644 (denominators).
Cf. A160595.
Cf. also A340142, A340143.

up_to = 65537;
A160595(n) = { my(x=eulerphi(n)); x/gcd(x,n-1); };
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&&dA160595(n)));
A340141(n) = numerator(v340141rat[n]);

A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 1, 7, 1, 5, 1, 9, 7, 5, 1, 15, 1, 5, 1, 43, 1, 15, 1, 7, 3, 3, 1, 5, 3, 5, 9, 15, 1, 9, 1, 87, 3, 5, 1, 1, 1, 5, 3, 13, 1, 11, 1, 11, 15, 3, 1, 187, 7, 19, 1, 15, 1, 5, 3, 21, 3, 3, 1, -1, 1, 3, 11, 387, 1, 9, 1, 7, 1, 13, 1, 119, 1, 7, 19, 23, 3, 19, 1, 139, -21, 7, 1, 21, 1, 5, 1, 39, 1, 67, 3, 3, 5, 3, 5, 451, 1, 15, 19, 69, 1, 13, 1, -27, 7

Offset: 1

Antti Karttunen, Aug 14 2018

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

Cf. A000010, A303757, A046644 (denominators).

Cf. also A317830, A317833, A317834, A317937.

A303757[n_] := If[n == 2, 1, Count[EulerPhi[Range[n]] - EulerPhi[n], 0]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A303757[n] -
     Sum[If[1Jean-François Alcover, Dec 20 2021 *)

up_to = 65537;
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; };
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.
v303757 = ordinal_transform(vector(up_to,n,if(1==n,0,eulerphi(n))));
v317847 = DirSqrt(vector(up_to, n, v303757[n]));
A317847(n) = numerator(v317847[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A303757(n) - Sum_{d|n, d>1, d 1.

A317848 Multiplicative with a(p^e) = binomial(2*e, e).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 20, 6, 4, 2, 12, 2, 4, 4, 70, 2, 12, 2, 12, 4, 4, 2, 40, 6, 4, 20, 12, 2, 8, 2, 252, 4, 4, 4, 36, 2, 4, 4, 40, 2, 8, 2, 12, 12, 4, 2, 140, 6, 12, 4, 12, 2, 40, 4, 40, 4, 4, 2, 24, 2, 4, 12, 924, 4, 8, 2, 12, 4, 8, 2, 120, 2, 4, 12, 12, 4, 8, 2, 140

Offset: 1

Andrew Howroyd, Aug 08 2018

The Dirichlet convolution square of this sequence is A165825.

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

Cf. A000265, A000984, A006519, A037445, A046643, A046644, A165825, A299149, A299150.

f[p_, e_] := Binomial[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n)={my(v=factor(n)[,2]); prod(i=1, #v, binomial(2*v[i], v[i]))}

\\ 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&&d

A317848(n) = factorback(apply(e -> binomial(e+e,e),factor(n)[,2])); \\ Antti Karttunen, Sep 17 2018

A037445(n) = A006519(a(n)).
A046643(n) = numerator(a(n)/A165825(n)) = A000265(a(n)).
A046644(n) = denominator(a(n)/A165825(n)) = A165825(n)/A037445(n).
A299149(n) = numerator(n*a(n)/A165825(n)) = A000265(n*a(n)).
A299150(n) = denominator(n*a(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)).

A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

1, 3, 5, 23, 9, 15, 13, 91, 59, 27, 21, 115, 25, 39, 45, 1451, 33, 177, 37, 207, 65, 63, 45, 455, 179, 75, 353, 299, 57, 135, 61, 5797, 105, 99, 117, 1357, 73, 111, 125, 819, 81, 195, 85, 483, 531, 135, 93, 7255, 363, 537, 165, 575, 105, 1059, 189, 1183, 185, 171, 117, 1035, 121, 183, 767, 46355, 225, 315, 133, 759, 225, 351, 141, 5369

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Because A018804 gets only odd values on primes, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A018804, A046644 (denominators).

Cf. also A318444.

a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Numerator;
Array[a, 72] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
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.
v318443aux = DirSqrt(vector(up_to, n, A018804(n)));
A318443(n) = numerator(v318443aux[n]);

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

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A018804(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318443(k) / A046644(k) ~ sqrt(3/2)*n^2/Pi. - Vaclav Kotesovec, May 10 2025

A318656 The 2-adic valuation of ratio A318649(n)/A318512(n); a(n) = 2*A007814(n) - A046645(n).

Original entry on oeis.org

0, 1, -1, 1, -1, 0, -1, 2, -3, 0, -1, 0, -1, 0, -2, 1, -1, -2, -1, 0, -2, 0, -1, 1, -3, 0, -4, 0, -1, -1, -1, 2, -2, 0, -2, -2, -1, 0, -2, 1, -1, -1, -1, 0, -4, 0, -1, 0, -3, -2, -2, 0, -1, -3, -2, 1, -2, 0, -1, -1, -1, 0, -4, 2, -2, -1, -1, 0, -2, -1, -1, -1, -1, 0, -4, 0, -2, -1, -1, 0, -7, 0, -1, -1, -2, 0, -2, 1, -1, -3, -2, 0

Offset: 1

Antti Karttunen, Sep 02 2018

sign

Also the 2-adic valuation of ratio A318681(n)/A299150(n) [which is equal to A318649(n)/A318512(n), but not represented in lowest terms], as well as the 2-adic valuation of A318680(n)/A299150(n) = A318511(n)/A318512(n).

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

Cf. A007814, A046645, A299150, A318440, A318511, A318512, A318513, A318649, A318655, A318680, A318681.

Cf. A318654 (positions of positive terms).

A007814(n) = valuation(n,2);
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
A318656(n) = ((2*A007814(n))-A007814(A046644(n)));

a(n) = A318655(n) - A318513(n).
a(n) = A007814(n) - A318440(n).
a(n) = 2*A007814(n) - A046645(n) = A007814(n^2) - A046645(n).

A340144 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 25, 11, 7, 1, 19, 1, 11, 3, 363, 1, 31, 1, 43, 5, 19, 1, 63, 19, 23, 61, 3, 1, 19, 1, 1335, 9, 31, 11, 189, 1, 35, 11, 139, 1, 29, 1, 115, 7, 43, 1, 867, 27, 127, 15, 11, 1, 163, 19, 279, 17, 55, 1, 93, 1, 59, 51, 9923, 5, -15, 1, 187, 21, 3, 1, 615, 1, 71, 55, 19, 29, 59, 1, 1875, 1363, 79, 1, 203

Offset: 1

Antti Karttunen, Dec 29 2020

			For n = 561 = 3*11*17, its divisors d are: 1, 3, 11, 17, 33, 51, 187, 561.
For this sequence, the corresponding terms a(d) are: 1, 1, 1, 1, 9, 15, 79, -99.
For A046644, the corresponding terms are:            1, 2, 2, 2, 4,  4,  4,   8.
Convolving these ratios as Sum_{d|561} r(d)*r(n/d) = 2*((1/1)*(-99/8) + (1/2)*(79/4) + (1/2)*(15/4) + (1/2)*(9/4)) yields 1 as expected, because 561 is Carmichael number (A002997) and A247074 obtains value 1 on all of them.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A046644 (denominators).
Cf. A247074.
Cf. also A340141, A340145, A340146.

up_to = 65537;
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
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&&dA247074(n)));
A340144(n) = numerator(v340144rat[n]);

A383657 Numerator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

Original entry on oeis.org

1, 3, 3, 15, 3, 9, 3, 35, 15, 9, 3, 45, 3, 9, 9, 315, 3, 45, 3, 45, 9, 9, 3, 105, 15, 9, 35, 45, 3, 27, 3, 693, 9, 9, 9, 225, 3, 9, 9, 105, 3, 27, 3, 45, 45, 9, 3, 945, 15, 45, 9, 45, 3, 105, 9, 105, 9, 9, 3, 135, 3, 9, 45, 3003, 9, 27, 3, 45, 9, 27, 3, 525, 3

Offset: 1

Vaclav Kotesovec, May 04 2025

In general, for m > 0, if Dirichlet g.f. is zeta(s)^m, then Sum_{j=1..n} a(j) ~ n*log(n)^(m-1)/Gamma(m) * (1 + (m-1)*(m*gamma - 1)/log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A383658.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

coeff=CoefficientList[Series[1/(1-x)^(3/2),{x,0,20}]//Normal,x];dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Numerator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,73] (* Shenghui Yang, May 04 2025 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(3/2))[n]), ", "))

Sum_{k=1..n} A383657(k)/A383658(k) ~ 2*n*sqrt(log(n)/Pi) * (1 - (1 - 3*gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

A318306 Additive with a(p^e) = A002487(e).

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A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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A318657 Numerators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

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A340141 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A160595(x) = phi(x)/gcd(phi(x), x-1).

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A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).

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A317848 Multiplicative with a(p^e) = binomial(2*e, e).

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A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

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A318656 The 2-adic valuation of ratio A318649(n)/A318512(n); a(n) = 2*A007814(n) - A046645(n).

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