A317831 -id:A317831 - cp's OEIS frontend

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

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.

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

A317925 Numerators of rational valued sequence whose Dirichlet convolution with itself yields Euler's phi (A000010).

Original entry on oeis.org

1, 1, 1, 7, 2, 1, 3, 25, 5, 1, 5, 7, 6, 3, 2, 363, 8, 5, 9, 7, 3, 5, 11, 25, 8, 3, 13, 21, 14, 1, 15, 1335, 5, 4, 6, 35, 18, 9, 6, 25, 20, 3, 21, 35, 5, 11, 23, 363, 33, 4, 8, 21, 26, 13, 10, 75, 9, 7, 29, 7, 30, 15, 15, 9923, 12, 5, 33, 7, 11, 3, 35, 125, 36, 9, 8, 63, 15, 3, 39, 363, 139, 10, 41, 21, 16, 21, 14, 125, 44, 5, 18, 77, 15, 23

Offset: 1

Antti Karttunen, Aug 11 2018

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

Cf. A000010, A317926 (denominators).

Cf. also A046643, A317831.

f[1] = 1; f[n_] := f[n] = (EulerPhi[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; Numerator @ Array[f, 100] (* Amiram Eldar, Dec 12 2022 *)

A317925perA317926(n) = if(1==n,n,(eulerphi(n)-sumdiv(n,d,if((d>1)&&(dA317925perA317926(d)*A317925perA317926(n/d),0)))/2);
A317925(n) = numerator(A317925perA317926(n));

\\ Memoized implementation:
memo = Map();
A317925perA317926(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (eulerphi(n)-sumdiv(n,d,if((d>1)&&(dA317925perA317926(d)*A317925perA317926(n/d),0)))/2); mapput(memo,n,v); (v)));

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

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A000010(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317925(k) / A317926(k) ~ Pi^(-3/2) * n^2 * sqrt(3/(2*log(n))) * (1 + (1/2 - gamma/2 + 3*zeta'(2)/Pi^2) / (2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 10 2025

A317832 Denominators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 10 2018

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

Cf. A317831 (gives the numerators).

Cf. also A000203, A299152.

A317831perA317832(n) = if(1==n,n,(sigma(n)-sumdiv(n,d,if((d>1)&&(dA317831perA317832(d)*A317831perA317832(n/d),0)))/2);
A317832(n) = denominator(A317831perA317832(n));

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

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000203(n) - Sum_{d|n, d>1, d 1.

A317845 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001065 (sum of proper divisors) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 11, 1, 11, 1, 45, 15, 15, 1, 95, 1, 19, 17, 659, 1, 131, 1, 135, 21, 27, 1, 315, 23, 31, 89, 175, 1, 125, 1, 2319, 29, 39, 25, 901, 1, 43, 33, 455, 1, 165, 1, 255, 215, 51, 1, 3739, 31, 291, 41, 295, 1, 671, 33, 595, 45, 63, 1, 731, 1, 67, 271, 16319, 37, 245, 1, 375, 53, 237, 1, 2135, 1, 79, 335, 415, 37, 285, 1, 5419, 1979, 87, 1

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(360) = -12947.

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

Cf. A001065, A063524, A046644 (denominators).

Cf. also A317831, A317846.

A317845aux(n) = if(1==n,n,((sigma(n)-n)-sumdiv(n,d,if((d>1)&&(dA317845aux(d)*A317845aux(n/d),0)))/2);
A317845(n) = numerator(A317845aux(n));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001065(n) - Sum_{d|n, d>1, d 1.

A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A034444 is.
The first 2^20 terms are positive. Is the sequence nonnegative?
Records seem to be A001790, occurring at A000302 (apart from 4).

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

Cf. A001790, A034444, A317934 (denominators).

Cf. also A046643, A317831, A317925, A317937.

A034444(n) = (2^omega(n));
A317933perA317934(n) = if(1==n,n,(A034444(n)-sumdiv(n,d,if((d>1)&&(dA317933perA317934(d)*A317933perA317934(n/d),0)))/2);
A317933(n) = numerator(A317933perA317934(n));

up_to = 65537;
\\ Faster:
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.
v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
A317933(n) = numerator(v317933aux[n]);

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

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A034444(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317933(k) / A317934(k) ~ sqrt(6)*n/Pi. - Vaclav Kotesovec, May 10 2025

A317938 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001222 (bigomega n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 17, 7, 3, 1, 11, 1, 3, 3, 139, 1, 11, 1, 11, 3, 3, 1, 15, 7, 3, 17, 11, 1, 3, 1, 263, 3, 3, 3, 17, 1, 3, 3, 15, 1, 3, 1, 11, 11, 3, 1, 83, 7, 11, 3, 11, 1, 15, 3, 15, 3, 3, 1, -3, 1, 3, 11, 995, 3, 3, 1, 11, 3, 3, 1, 11, 1, 3, 11, 11, 3, 3, 1, 83, 139, 3, 1, -3, 3, 3, 3, 15, 1, -3, 3, 11, 3, 3, 3, 189, 1, 11, 11, 17, 1, 3, 1, 15, 3

Offset: 1

Antti Karttunen, Aug 12 2018

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

Cf. A001222, A063524, A046644 (denominators).

Cf. also A317831, A317925, A317933, A317845, A317846, A317937.

A317938aux(n) = if(1==n,n,(bigomega(n)-sumdiv(n,d,if((d>1)&&(dA317938aux(d)*A317938aux(n/d),0)))/2);
A317938(n) = numerator(A317938aux(n));

\\ Memoized implementation:
memo317938 = Map();
A317938aux(n) = if(1==n,n,if(mapisdefined(memo317938,n),mapget(memo317938,n),my(v = (bigomega(n)-sumdiv(n,d,if((d>1)&&(dA317938aux(d)*A317938aux(n/d),0)))/2); mapput(memo317938,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001222(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

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

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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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A317925 Numerators of rational valued sequence whose Dirichlet convolution with itself yields Euler's phi (A000010).

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A317832 Denominators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

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A317845 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001065 (sum of proper divisors) + A063524 (1, 0, 0, 0, ...).

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A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

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A317938 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A001222 (bigomega n) + A063524 (1, 0, 0, 0, ...).

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