A261179 -id:A261179 - cp's OEIS frontend

A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

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

Offset: 1

Antti Karttunen, Aug 11 2018

These are denominators for rational valued sequences that are obtained as "Dirichlet Square Roots" of sequences b that satisfy the condition b(3) = 2, and b(p) = odd number for any other primes p. For example, A064989, A065769 and A234840. - Antti Karttunen, Aug 31 2018

The original definition was: Denominators of the rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence. However, this definition depends on the conjecture given in A261179.

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

Cf. A317930, A318319, A318669 (some of the numerator sequences), A317931 (conjectured, for A002487).
Cf. A305439 (the 2-adic valuation), A318666.
Cf. also A046644, A261179, A299150, A237649, A299150, A317928, A317839, A317843, A317934, A318319.

\\ Original program, based on conjectural formula:
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);
A317932(n) = denominator(A317931perA317932(n));

\\ New fast program implementing the new definition:
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
A317932(n) = (A046644(n)/2^valuation(n,3)); \\ Antti Karttunen, Aug 31 2018

A011371(n) = (A005187(n)-n);
A317932(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i,1], m *= 2^(A011371(f[i,2])), m *= 2^A005187(f[i,2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

a(n) = A046644(n)/A318666(n) = 2^A305439(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 can be A064989, A065769 or A234840 for example, conjecturally also A002487.
Multiplicative with a(3^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for any other primes. - Antti Karttunen, Sep 03 2018

Definition changed, the original (now conjectured alternative definition) moved to the comments section by Antti Karttunen, Aug 31 2018

Keyword:mult added by Antti Karttunen, Sep 03 2018

A317930 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A234840, which is a multiplicative permutation of natural numbers.

Original entry on oeis.org

1, 3, 1, 27, 19, 3, 61, 135, 3, 57, 11, 27, 281, 183, 19, 2835, 101, 9, 5, 513, 61, 33, 263, 135, 1083, 843, 5, 1647, 29, 57, 59, 15309, 11, 303, 1159, 81, 1811, 15, 281, 2565, 1091, 183, 157, 297, 57, 789, 409, 2835, 11163, 3249, 101, 7587, 541, 15, 209, 8235, 5, 87, 31, 513, 7, 177, 183, 168399, 5339, 33, 1013, 2727

Offset: 1

Antti Karttunen, Aug 23 2018

Multiplicative because A234840 is.

Question: Are all terms positive? No negative terms in range 1 .. 2^17. Also (checked for n <= 2^17) the denominators seem to be given by A317932.

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

Cf. A234840, A317932 (seems to give denominators, see A261179).

Cf. also A317929.

up_to = 16384;
A234840(n) = if(n<=1,n,my(f = factor(n)); for(i=1, #f~, if(2==f[i,1], f[i,1]++, if(3==f[i,1], f[i,1]--, f[i,1] = prime(-1+A234840(1+primepi(f[i,1])))))); factorback(f)); \\ Antti Karttunen, Aug 23 2018
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.
v317930aux = DirSqrt(vector(up_to, n, A234840(n)));
A317930(n) = numerator(v317930aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A234840(n) - Sum_{d|n, d>1, d 1.

A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 5, 5, 3, 3, 7, 3, 11, 5, 3, 35, 13, 3, 17, 9, 5, 7, 19, 5, 27, 11, 5, 15, 23, 3, 29, 63, 7, 13, 15, 9, 31, 17, 11, 15, 37, 5, 41, 21, 9, 19, 43, 35, 75, 27, 13, 33, 47, 5, 21, 25, 17, 23, 53, 9, 59, 29, 15, 231, 33, 7, 61, 39, 19, 15, 67, 15, 71, 31, 27, 51, 35, 11, 73, 105, 35, 37, 79, 15, 39, 41, 23, 35, 83, 9, 55, 57

Offset: 1

Antti Karttunen, Aug 24 2018

Multiplicative because A064989 is.

No negative terms among the first 2^20 terms.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A064989, A317932 (seems to give denominators, see A261179).

Cf. also A318321.

up_to = 16384;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); 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&&dA317937.
v318319aux = DirSqrt(vector(up_to, n, A064989(n)));
A318319(n) = numerator(v318319aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064989(n) - Sum_{d|n, d>1, d 1.

A318509 Completely multiplicative with a(p) = A002487(p).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 6, 1, 5, 4, 7, 3, 6, 5, 7, 2, 9, 5, 8, 3, 7, 6, 5, 1, 10, 5, 9, 4, 11, 7, 10, 3, 11, 6, 13, 5, 12, 7, 9, 2, 9, 9, 10, 5, 13, 8, 15, 3, 14, 7, 11, 6, 9, 5, 12, 1, 15, 10, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 15, 10, 13, 3, 16, 11, 19, 6, 15, 13, 14, 5, 17, 12, 15, 7, 10, 9, 21, 2, 11, 9, 20, 9, 19, 10, 17, 5, 18

Offset: 1

Antti Karttunen, Aug 30 2018

Provided that the conjecture given in A261179 holds, then for all n >= 1, A007814(a(n)) = A007949(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A261179, A318510.

Cf. also A318307.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };

from math import prod
from functools import reduce
from sympy import factorint
def A318509(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 18 2023

A319687 a(n) = A318509(n) - A002487(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 6, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -2, -6, 0, -4, 0, 0, 0, -4, 0, -6, 0, 10, 0, 0, 0, 4, 2, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 02 2018

All terms seem to be even. See the conjecture given in A261179.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A261179, A317837, A318509, A323365.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };
A319687(n) = (A318509(n) - A002487(n));

from math import prod
from functools import reduce
from sympy import factorint
def A319687(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items())-sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) # Chai Wah Wu, May 18 2023

a(n) = A318509(n) - A002487(n).

A318510 Completely multiplicative with a(prime(k)) = A002487(prime(k+1)).

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 5, 8, 9, 6, 5, 12, 5, 10, 9, 16, 7, 18, 7, 12, 15, 10, 7, 24, 9, 10, 27, 20, 5, 18, 11, 32, 15, 14, 15, 36, 11, 14, 15, 24, 13, 30, 9, 20, 27, 14, 13, 48, 25, 18, 21, 20, 11, 54, 15, 40, 21, 10, 9, 36, 11, 22, 45, 64, 15, 30, 13, 28, 21, 30, 15, 72, 13, 22, 27, 28, 25, 30, 19, 48, 81, 26, 17, 60, 21, 18, 15, 40

Offset: 1

Antti Karttunen, Aug 30 2018

Provided that the conjecture given in A261179 holds, then for all n >= 1, A007814(a(n)) = A007814(n), i.e., then the sequence preserves the 2-adic valuation of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A003961, A318509.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318510(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(prime(1+primepi(f[i, 1])))); factorback(f); };

from math import prod
from functools import reduce
from sympy import factorint, nextprime
def A318510(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(nextprime(p))[-1:2:-1],(1,0)))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 18 2023

a(n) = A318509(A003961(n)).

A261273 Take the list of positive rationals {R(n): n>=1} in the order defined by Calkin and Wilf (Recounting the Rationals, 1999); a(n) = denominator of R(prime(n)).

Original entry on oeis.org

2, 1, 2, 1, 2, 3, 4, 3, 2, 4, 1, 7, 8, 5, 2, 8, 4, 5, 5, 4, 11, 3, 8, 12, 9, 12, 5, 8, 11, 10, 1, 6, 14, 9, 18, 7, 13, 11, 8, 18, 12, 19, 2, 11, 16, 7, 13, 3, 10, 17, 18, 4, 13, 6, 8, 6, 16, 5, 23, 22, 13, 26, 17, 10, 23, 16, 19, 29, 18, 23, 22, 12, 7, 25, 11, 2, 20, 23, 26, 29, 18, 31, 8, 27, 11, 14, 16, 27, 24, 7, 18, 4, 9, 14, 11, 6, 8, 20, 13, 21, 19, 32, 22, 30, 17, 23, 26, 40, 18, 43, 7, 41, 44, 27, 13, 20, 17, 14, 36, 30, 49, 37, 50, 34, 31, 28, 39, 12, 19, 33, 23, 16, 9, 31, 24, 15, 24, 25, 30, 50, 31, 46, 17, 22, 27, 18, 55, 50, 29, 8, 41, 36, 25, 14, 23, 10, 17, 32, 47, 40, 26, 34, 13, 22, 32, 14, 5, 27

Offset: 1

James Kirk Winkler, Aug 13 2015

nonn

The list of rationals {R(n)} is essentially given by A002487(n)/A002487(n+1).

N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

Companion sequence to A261179.

cp's OEIS Frontend

A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

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A317930 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A234840, which is a multiplicative permutation of natural numbers.

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A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.

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A318509 Completely multiplicative with a(p) = A002487(p).

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A319687 a(n) = A318509(n) - A002487(n).

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A318510 Completely multiplicative with a(prime(k)) = A002487(prime(k+1)).

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A261273 Take the list of positive rationals {R(n): n>=1} in the order defined by Calkin and Wilf (Recounting the Rationals, 1999); a(n) = denominator of R(prime(n)).

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