A299149 -id:A299149 - cp's OEIS frontend

A318681 a(n) = n * A299149(n).

1, 2, 9, 12, 25, 18, 49, 40, 243, 50, 121, 108, 169, 98, 225, 560, 289, 486, 361, 300, 441, 242, 529, 360, 1875, 338, 3645, 588, 841, 450, 961, 2016, 1089, 578, 1225, 2916, 1369, 722, 1521, 1000, 1681, 882, 1849, 1452, 6075, 1058, 2209, 5040, 7203, 3750, 2601, 2028, 2809, 7290, 3025, 1960, 3249, 1682, 3481, 2700

Offset: 1

Antti Karttunen, Sep 02 2018

Dirichlet convolution of a(n)/A299150(n) with itself gives A000290, the squares, like gives also the self-convolution of A318649(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318649, this sequence is multiplicative as both A000027 and A299149 are multiplicative sequences (also, because A000290 and A299150 are both multiplicative).

A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A299149.

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

Cf. A000290, A299149, A299150, A318512, A318649, A318680.

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&&dA299149(n) = numerator(v299149_50[n]);
A318681(n) = (n*A299149(n));

a(n) = n * A299149(n).

a(n)/A299150(n) = A318649(n)/A318512(n).

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

A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
From Antti Karttunen, Sep 02 2018: (Start)
Proof for the above claim:
This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).
  By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:
    A318649(n)     A318681(n)     n*A299149(n)
    ----------  =  ----------  =  ------------
    A318512(n)     A299150(n)      A299150(n)
  and
    A318511(n)     A318680(n)     n*A318653(n)
    ----------  =  ----------  =  ------------
       Ay(n)       A299150(n)      A299150(n)
  where the leftmost ratios are reduced into their lowest terms.
  Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).
(End)

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

Cf. A000290, A064549, A046644, A299150, A318513, A318651, A318653, A318654, A318655, A318656, A318658, A318680, A318681.

Cf. A318649, A318511 (numerators).

f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Denominator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 65537;
A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
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&&dA064549(n)));
A318512(n) = denominator(v318511_12[n]);

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

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. [Equally, one could use A064549 in place of A000290.]
a(n) = 2^A318513(n).
a(n) = A046644(n)/A318651(n).
a(2n-1) = A046644(2n-1) = A318658(2n-1), for all n >= 1.

The main definition changed, more formulas added by Antti Karttunen, Aug 31 2018

A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

Original entry on oeis.org

1, 2, 9, 6, 25, 9, 49, 20, 243, 25, 121, 27, 169, 49, 225, 70, 289, 243, 361, 75, 441, 121, 529, 90, 1875, 169, 3645, 147, 841, 225, 961, 252, 1089, 289, 1225, 729, 1369, 361, 1521, 250, 1681, 441, 1849, 363, 6075, 529, 2209, 315, 7203, 1875, 2601, 507, 2809, 3645, 3025, 490, 3249, 841, 3481, 675, 3721, 961, 11907, 924, 4225, 1089

Offset: 1

Antti Karttunen, Aug 31 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000290, A318512 (denominators).

Cf. also A046643, A299149, A318511, A318651, A318654 (gives the positions of even terms), A318655 (the 2-adic valuation).

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&&dA318649(n) = numerator(v318649_aux[n]);

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

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * ((n^2) - Sum_{d|n, d>1, d 1.
a(n) = n*A318512(n)*A299149(n)/A299150(n).
Sum_{k=1..n} A318649(k) / A318512(k) ~ n^3/(3*sqrt(Pi*log(n))) * (1 + (1 - 3*gamma/2) / (6*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 7, 8, 14, 32, 121, 126, 248, 512, 1003, 2048, 4064, 8176, 130539, 32768, 65382, 131072, 261868, 524224, 1048064, 2097152, 4193131, 8388576, 16775168, 33554180, 67104688, 134217728, 268426672, 536870912, 8589802359, 2147482624, 4294934528, 8589934336, 17179801257, 34359738368, 68719345664, 137438949376, 274877643724, 549755813888

Offset: 1

Gus Wiseman, Feb 03 2018

Numerators of rational valued sequence f whose Dirichlet convolution with itself yields function g(n) = A000079(n-1) = 2^(n-1). - Antti Karttunen, Aug 10 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299152.

Cf. also A317831.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299151(n) = numerator(A299151perA299152(n));

More terms from Antti Karttunen, Jul 29 2018

A299152 Denominators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 03 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

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

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299151.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

up_to = 65537;
prepareA299151perA299152(up_to) = { my(vmemo = vector(up_to)); for(n=1,up_to, vmemo[n] = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299152 = prepareA299151perA299152(up_to);
A299151perA299152(n) = v299151perA299152[n];
\\ Or without memoization as:
A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299152(n) = denominator(A299151perA299152(n)); \\ Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 2, 6, 14, 40, 84, 224, 484, 1134, 2480, 5632, 12036, 26624, 56896, 122640, 261078, 557056, 1176876, 2490368, 5237360, 11008704, 23057408, 48234496, 100635144, 209714400, 436154368, 905962860, 1878931264, 3892314112, 8052800160, 16642998272, 34359209436

Offset: 1

Gus Wiseman, Feb 03 2018

nonn

For prime p, a(p) = 2^(p-2)*p. - Jon E. Schoenfield, Feb 03 2018

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A001787, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299149, A299151.

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1, n*2^(n-2)-
       add(a(d)*a(n/d), d=divisors(n) minus {1, n})/2)
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 07 2018

nn=50;
sys=Table[2^(n-1)*n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]

A383768 Numerators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

Original entry on oeis.org

1, 4, 27, 24, 125, 54, 343, 160, 2187, 250, 1331, 324, 2197, 686, 3375, 1120, 4913, 2187, 6859, 1500, 9261, 2662, 12167, 2160, 46875, 4394, 98415, 4116, 24389, 3375, 29791, 8064, 35937, 9826, 42875, 6561, 50653, 13718, 59319, 10000, 68921, 9261, 79507, 15972, 273375

Offset: 1

Vaclav Kotesovec, May 09 2025

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

Cf. A000578, A299149, A299150, A318649, A318512, A383769 (denominators).

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

Sum_{k=1..n} A383768(k) / A383769(k) ~ n^4/(4*sqrt(Pi*log(n))) * (1 + (1-2*gamma)/(8*log(n))), where gamma is the Euler-Mascheroni constant A001620.

A383769 Denominators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 09 2025

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

Cf. A000578, A299149, A299150, A318649, A318512, A383768 (numerators).

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

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)).

cp's OEIS Frontend

A318681 a(n) = n * A299149(n).

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

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A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

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A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

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A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

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A299152 Denominators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

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A299119 Positive solution to 2^(n-1) = (1/n) * Sum_{d|n} a(d) * a(n/d).

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