A305439 -id:A305439 - 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

A318440 a(n) = A046645(n) - A007814(n); the 2-adic valuation of A299150.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 02 2018

After two initial terms, all terms are positive.

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

Cf. A000120, A007814, A046645, A077761, A299150, A318656.

Cf. also A305439.

f[p_, e_] := (1 + Mod[p, 2])*e - DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

A007814(n) = valuation(n,2);
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046645(n) = vecsum(apply(e -> A005187(e),factor(n)[,2]));
A318440(n) = A046645(n) - A007814(n);

a(n) = A046645(n) - A007814(n).
a(n) = A007814(A299150(n)).
Additive with a(p^e) = (1 + (p mod 2))*e - A000120(e). - Amiram Eldar, Apr 28 2023
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -1 + Sum_{p prime} f(1/p) = 0.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 30 2023

A305438 Number of times the lexicographically least irreducible factor of (0,1)-polynomial (when factored over Q) obtained from the binary expansion of n occurs as the lexicographically least factor for numbers <= n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 4, 14, 1, 15, 1, 16, 5, 17, 2, 18, 1, 19, 6, 20, 1, 21, 1, 22, 7, 23, 1, 24, 3, 25, 8, 26, 1, 27, 1, 28, 9, 29, 1, 30, 1, 31, 10, 32, 2, 33, 1, 34, 1, 35, 1, 36, 1, 37, 11, 38, 1, 39, 1, 40, 1, 41, 1, 42, 3, 43, 1, 44, 1, 45, 1, 46, 2, 47, 4, 48, 1, 49, 12, 50, 1, 51, 1, 52, 13

Offset: 1

Antti Karttunen, Jun 09 2018

nonn

Ordinal transform of A305437.

			Binary representation of 21 is "10101", encoding (0,1)-polynomial x^4 + x^2 + 1 which factorizes over Q as (x^2 - x + 1)(x^2 + x + 1). Factor (x^2 - x + 1) is lexicographically less than factor (x^2 + x + 1) and this is also the first time factor (x^2 - x + 1) occurs as the least one, thus a(21) = 1. Note that although we have the same factor present for n=9, which encodes the polynomial x^3 + 1 = (x + 1)(x^2 - x + 1), it is not the lexicographically least factor in that case.
The next time the same factor occurs as the smallest one is for n=93, which in binary is 1011101, encoding polynomial x^6 + x^4 + x^3 + x^2 + 1 = (x^2 - x + 1)(x^4 + x^3 + x^2 + x + 1). Thus a(93) = 2.

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

Cf. A206074 (gives a subset of the positions of 1's), A305437.
Cf. A305439.
Cf. also A078898, A302788.

allocatemem(2^30);
default(parisizemax,2^31);
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; };
pollexcmp(a,b) = { my(ad = poldegree(a), bd = poldegree(b),e); if(ad != bd, return(sign(ad-bd))); for(i=0,ad,e = polcoeff(a,ad-i) - polcoeff(b,ad-i); if(0!=e, return(sign(e)))); (0); };
Aux305438(n) = if(1==n,0,my(fs = factor(Pol(binary(n)))[,1]~); vecsort(fs,pollexcmp)[1]);
v305438 = ordinal_transform(vector(up_to,n,Aux305438(n)));
A305438(n) = v305438[n];

a(2n) = n.

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

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A318440 a(n) = A046645(n) - A007814(n); the 2-adic valuation of A299150.

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A305438 Number of times the lexicographically least irreducible factor of (0,1)-polynomial (when factored over Q) obtained from the binary expansion of n occurs as the lexicographically least factor for numbers <= n; a(1) = 1.

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