A365718 -id:A365718 - cp's OEIS frontend

A103391 "Even" fractal sequence for the natural numbers: Deleting every even-indexed term results in the same sequence.

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42, 22, 43, 12, 44, 23, 45, 7, 46, 24, 47, 13, 48, 25, 49, 3, 50, 26, 51, 14, 52, 27, 53, 8

Offset: 1

Eric Rowland, Mar 20 2005

A003602 is the "odd" fractal sequence for the natural numbers.

Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(A005940(i)) = A348717(A005940(j)) for all i, j >= 1. A365718 is an analogous sequence related to A356867 (Doudna variant D(3)). - Antti Karttunen, Sep 17 2023

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..10000 from Reinhard Zumkeller)

Cf. A003602, A005940, A025480, A220466, A286387, A353368 (Dirichlet inverse).
Cf. also A110962, A110963, A365718.
Differs from A331743(n-1) for the first time at n=192, where a(192) = 97, while A331743(191) = 23.
Differs from A351460.

-- import Data.List (transpose)
a103391 n = a103391_list !! (n-1)
a103391_list = 1 : ks where
   ks = concat $ transpose [[2..], ks]
-- Reinhard Zumkeller, May 23 2013

nmax := 82: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to ceil(nmax/(p+2))+1 do a((2*n-3)*2^p+1) := n od: od: a(1) := 1: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 28 2013

a[n_] := ((n-1)/2^IntegerExponent[n-1, 2] + 3)/2; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

A003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1))); \\ Antti Karttunen, Feb 05 2020

def v(n): b = bin(n); return len(b) - len(b.rstrip("0"))
def b(n): return (n//2**v(n)+1)//2
def a(n): return 1 if n == 1 else 1 + b(n-1)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, May 29 2022

def A103391(n): return (n-1>>(n-1&-n+1).bit_length())+2 if n>1 else 1 # Chai Wah Wu, Jan 04 2024

For n > 1, a(n) = A003602(n-1) + 1. - Benoit Cloitre, May 26 2007, indexing corrected by Antti Karttunen, Feb 05 2020
a((2*n-3)*2^p+1) = n, p >= 0 and n >= 2, with a(1) = 1. - Johannes W. Meijer, Jan 28 2013
Sum_{k=1..n} a(k) ~ n^2/6. - Amiram Eldar, Sep 24 2023

Data section extended up to a(105) (to better differentiate from several nearby sequences) by Antti Karttunen, Feb 05 2020

A365715 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365465(i) = A365465(j) for all i, j >= 1, where A365465(n) = A356867(n) / gcd(n, A356867(n)), and A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 1, 1, 5, 6, 2, 7, 4, 3, 7, 8, 1, 9, 5, 4, 10, 11, 1, 3, 8, 1, 12, 13, 5, 14, 12, 6, 9, 15, 2, 16, 16, 7, 9, 17, 4, 18, 19, 3, 20, 21, 7, 22, 3, 8, 10, 23, 1, 5, 14, 9, 24, 25, 5, 26, 27, 4, 28, 29, 10, 30, 31, 11, 10, 32, 1, 33, 21, 3, 34, 35, 8, 36, 15, 1, 37, 38, 12, 37, 38, 13, 39, 40, 5, 1

Offset: 1

Antti Karttunen, Sep 17 2023

Restricted growth sequence transform of A365465.

Compare to the scatter plots of A365431 (analogous sequence for Doudna variant D(2)), and also of A365393 and A365718.

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A356867, A365465.

Cf. also A365431, A365393, A365718.

\\ Needs also program from A356867:
up_to = 59049; \\ = 3^10.
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A365465(n) = (A356867(n)/gcd(n, A356867(n)));
v365715 = rgs_transform(vector(up_to,n,A365465(n)));
A365715(n) = v365715[n];

A365717 a(n) is the least k such that A003961^i(k) = A356867(1+n) for some i >= 0, where A003961^i denotes the i-th iterate of prime shift, and A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 10, 8, 4, 2, 14, 6, 4, 20, 12, 50, 16, 18, 6, 28, 30, 8, 40, 24, 100, 32, 8, 2, 22, 10, 10, 44, 42, 70, 56, 12, 4, 98, 18, 12, 140, 60, 250, 80, 36, 18, 196, 150, 16, 200, 48, 500, 64, 54, 6, 110, 30, 20, 88, 84, 350, 112, 90, 8, 490, 54, 24, 280, 120, 1250, 160, 72, 36, 392, 300, 32, 400, 96, 1000

Offset: 0

Antti Karttunen, Sep 17 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A348717, A356867, A365718 (rgs-transform), A365719, A365721, A365722.

A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A365717(n) = A348717(A356867(1+n)); \\ Needs also program from A356867.

a(n) = A348717(A356867(1+n)).

A365720 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365719(i) = A365719(j) for all i, j >= 0, where A365719(n) = A046523(A356867(1+n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 17 2023

Restricted growth sequence transform of A365719.
For all i, j >= 0:
  A365718(i) = A365718(j) => a(i) = a(j),
  a(i) = a(j) => A365721(i) = A365721(j),
  a(i) = a(j) => A365722(i) = A365722(j).

Antti Karttunen, Table of n, a(n) for n = 0..59049

Cf. A046523, A356867, A365718, A365720 (rgs-transform), A365721, A365722.

Cf. also A286622.

up_to = 59049; \\ = 3^10.
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v365720 = rgs_transform(apply(A046523,A356867list(1+up_to)));
A365720(n) = v365720[1+n];

cp's OEIS Frontend

A103391 "Even" fractal sequence for the natural numbers: Deleting every even-indexed term results in the same sequence.

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A365715 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365465(i) = A365465(j) for all i, j >= 1, where A365465(n) = A356867(n) / gcd(n, A356867(n)), and A356867 is Sycamore's Doudna variant D(3).

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A365717 a(n) is the least k such that A003961^i(k) = A356867(1+n) for some i >= 0, where A003961^i denotes the i-th iterate of prime shift, and A356867 is Sycamore's Doudna variant D(3).

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A365720 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365719(i) = A365719(j) for all i, j >= 0, where A365719(n) = A046523(A356867(1+n)).

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