A299151 -id:A299151 - cp's OEIS frontend

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

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

A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 5, 27, 5, 11, 9, 13, 7, 15, 35, 17, 27, 19, 15, 21, 11, 23, 15, 75, 13, 135, 21, 29, 15, 31, 63, 33, 17, 35, 81, 37, 19, 39, 25, 41, 21, 43, 33, 135, 23, 47, 105, 147, 75, 51, 39, 53, 135, 55, 35, 57, 29, 59, 45, 61, 31, 189, 231, 65, 33

Offset: 1

Gus Wiseman, Feb 03 2018

Dirichlet convolution of a(n)/A046644(n) with itself yields A000265. - Antti Karttunen, Aug 30 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 Andrew Howroyd)
Vaclav Kotesovec, Graph - the asymptotic ratio (65537 terms)
Wikipedia, Dirichlet convolution.

Cf. A000010, A000265, A003958, A007431, A018804, A023900, A029935, A046643, A046644, A165825, A257098, A298971, A299119, A299150 (denominators), A299151, A317848, A318319, A318321, A318649.

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
odd[n_] := n/2^IntegerExponent[n, 2]; f[p_, e_] := odd[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]); numerator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018

\\ 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 09 2018

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

a(n) = numerator(n*A317848(n)/A165825(n)) = A000265(n*A317848(n)). - Andrew Howroyd, Aug 09 2018

Sum_{k=1..n} A299149(k)/A299150(k) ~ n^2 / (2*sqrt(Pi*log(n))) * (1 + (1-gamma) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

Keyword:mult added by Andrew Howroyd, Aug 09 2018

A317831 Numerators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

Original entry on oeis.org

1, 3, 2, 19, 3, 3, 4, 63, 9, 9, 6, 19, 7, 6, 6, 867, 9, 27, 10, 57, 8, 9, 12, 63, 11, 21, 11, 19, 15, 9, 16, 3069, 12, 27, 12, 171, 19, 15, 14, 189, 21, 12, 22, 57, 27, 18, 24, 867, 41, 33, 18, 133, 27, 33, 18, 63, 20, 45, 30, 57, 31, 24, 18, 22199, 21, 18, 34, 171, 24, 18, 36, 567, 37, 57, 22, 95, 24, 21, 40, 2601, 227, 63, 42, 19, 27, 33

Offset: 1

Antti Karttunen, Aug 10 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences related to sigma(n)

Cf. A317832 (gives the denominators).

Cf. also A000203, A299151.

A317831perA317832(n) = if(1==n,n,(sigma(n)-sumdiv(n,d,if((d>1)&&(dA317831perA317832(d)*A317831perA317832(n/d),0)))/2);
A317831(n) = numerator(A317831perA317832(n));

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/((1-p*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) * (A000203(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317831(k) / A317832(k) ~ n^2 * sqrt(Pi/(24*log(n))) * (1 - (gamma - 1 + 6*zeta'(2)/Pi^2) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

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]]

A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.

Original entry on oeis.org

1, 3, 2, 19, 4, 2, 11, 63, 6, 3, 19, 13, 23, 17, 5, 867, 16, 4, 35, 5, 17, 25, 21, 11, 31, 29, 13, 113, 27, 13, 57, 3069, 13, 9, 23, 25, 71, 41, 14, 69, 79, 33, 41, 169, 9, 25, 89, 615, 259, 53, 17, 197, 51, 25, 29, 389, 20, 31, 113, 59, 117, 67, 10, 22199, 18, 14, 131, 31, 51, 71, 69, 11, 143, 77, 22, 281, 91, 35, 153, 489, 71, 85, 81, 151, 19

Offset: 1

Antti Karttunen, Aug 11 2018

The first negative term is a(330) = -21.

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

Cf. A005187, A317928 (denominators).

Cf. also A297111, A300244, A299151, A317931.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317927perA317928(n) = if(1==n,n,(A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2);
A317927(n) = numerator(A317927perA317928(n));

\\ Memoized implementation:
memo = Map();
A317927perA317928(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2); mapput(memo,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A005187(n) - Sum_{d|n, d>1, d 1.

A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 3, 5, 3, 3, 5, 3, 5, 3, 1, 35, 5, 3, 7, 9, 5, 5, 7, 5, 19, 5, 5, 9, 7, 1, 5, 63, 1, 5, 9, 9, 11, 7, 5, 15, 11, 5, 13, 15, 13, 7, 9, 35, 27, 19, 7, 15, 13, 5, 7, 15, 3, 7, 11, 3, 9, 5, -7, 231, -1, 1, 11, 15, 7, 9, 13, 15, 15, 11, 47, 21, 19, 5, 13, 105, 27, 11, 19, 15, 27, 13, 11, 25, 17, 13, 23, 21, 11, 9, 1, 63

Offset: 1

Antti Karttunen, Aug 11 2018

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

Cf. A002487, A317932 (denominators, conjectured).

Cf. also A299151, A317927, A317839, A317843.

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);
A317931(n) = numerator(A317931perA317932(n));

\\ Memoized implementation:
memo = Map();
A317931perA317932(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2); mapput(memo,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A002487(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

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

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A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

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A317831 Numerators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

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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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A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.

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A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

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