A365719 -id:A365719 - cp's OEIS frontend

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)).

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];

A356867 For n >= 1, write n = 3^m + k, where m >= 0 is the greatest power of 3 <= n, and k is in the range 0 <= k < 3^(m+1) - 3^m, then for n such that k=0, a(n)=n, and for n such that k > 0, a(n) is the smallest prime multiple p*a(k), p != 3, that is not already a term.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 8, 9, 7, 14, 15, 25, 20, 12, 50, 16, 18, 35, 28, 30, 125, 40, 24, 100, 32, 27, 11, 22, 21, 55, 44, 42, 70, 56, 45, 49, 98, 75, 175, 140, 60, 250, 80, 36, 245, 196, 150, 625, 200, 48, 500, 64, 54, 77, 110, 105, 275, 88, 84, 350, 112, 90, 343

Offset: 1

David James Sycamore, Sep 01 2022

Any prime p may be used to generate a sequence D(p) of this kind. The present sequence is D(3), and D(2) is the Doudna sequence, A005940.
Conjectured to be a permutation of the positive integers in which the primes appear in order.
From Antti Karttunen, Sep 16 2023: (Start)
The conjecture is true: Sequence is a permutation of natural numbers. By definition it is injective, and the surjectivity is guaranteed by the fact that there are infinitely many such n > k encountered by the greedy algorithm that a(n) will be a multiple of a(k), and "the smallest prime multiple" condition guarantees that all multiples of a(k) will eventually appear. That the primes and A100484 appear in order follows from the formulas a(3^m + 1) = prime(m+2), and a(3^m + 2) = 2*prime(m+2).
If the base-3 representation of n-1 has the base-3 representation of k-1 as its suffix, then a(n) is a multiple of a(k). For example, A007089(16-1) = 120, and A007089(43-1) = 1120, thus the former is the suffix of the latter, and a(16) = 50 indeed divides a(43) = 250.
(End)

			n=1=3^0+0 so a(1)=1. n=2=3^0+1 so k=1 and a(2)=2. Similarly a(3)=3 and a(9)=9.
n=10=3^2+1, therefore k=1 and a(1)=1 so a(10)=1*7=7 (since 2 and 5 have already occurred).

Michael De Vlieger, Table of n, a(n) for n = 1..19683 (19683 = 3^9)
Michael De Vlieger, Fan style ternary tree showing a(n) for n = 1..3^9, with a heat map color function for level m where 3^m is blue, smaller values are bluer, and larger are yellow-green. The smallest value in level m is shown in purple and largest is shown in red.
Index entries for sequences that are permutations of the natural numbers

Cf. A007089, A007949, A011655, A048473, A100484, A053735, A364958 (fixed points), A365390 (inverse permutation), A365424, A365459, A365462 [= a(n)-n], A365463 [= gcd(a(n),n)], A365464, A365465, A365717 [= A348717(a(1+n))], A365719 [= A046523(a(1+n))], A365721 [= omega(a(1+n))], A365722 [= bigomega(a(1+n))].

Cf. also A005940, A364611, A364628 for variants D(2), D(5) and D(7).

Block[{a, c, i, j, k, m, t, nn}, nn = 64; m = 1; i = 2; p = Prime[i]; c[] = False; Do[Set[{m, k}, {1, n - p^Floor[Log[p, n]]}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], While[Set[t, Prime[m] a[k]]; Or[m == i, c[t]], m++]; Set[{a[n], c[t]}, {t, True}]], {n, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Sep 01 2022 *)

up_to = 19683;
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); };
v356867 = A356867list(up_to);
A356867(n) = v356867[n]; \\ Antti Karttunen, Sep 15 2023

from sympy import nextprime
from sympy.ntheory import digits
from itertools import count, islice
def b(n): return n - 3**(len(digits(n,3)) - 2)
def agen():
    aset, alst = set(), [None]
    for n in count(1):
        k = b(n)
        if k == 0: an = n
        else:
            ak, p = alst[k], 2
            while p == 3 or p*ak in aset: p = nextprime(p)
            an = p*ak
        yield an; aset.add(an); alst.append(an)
print(list(islice(agen(), 64))) # Michael S. Branicky, Sep 02 2022

a(3^m + 1) = prime(m+2) for m >= 1.
Conjectures from Jianing Song, Nov 23 2022: (Start)
(1) a(3^m+2) = 2*prime(m+2) for m >= 2. - [The conjecture is true because a(2) = 2 and 3^m + 2 < 3^(1+m) + (3^m) + 1 for all m - Antti Karttunen, Sep 16 2023]
(2) For n > m >= 1, a(3^n+3^m+1) = prime(m+2)^2 for n = m+1; prime(n+2)*prime(m+2)^2 for n >= m+2.
(3) For n > m >= 1, a(3^n+3^m+2) = 4*prime(n+2) for n >= 3, m = 1; 2*prime(m+2)^2 for n = m+1, m >= 2; 2*prime(m+2)*prime(m+3) for n = m+2, m >= 2; 2*prime(n+2)*prime(m+2)^2 for n >= m+3, m >= 2. (End)
From Antti Karttunen, Sep 17 2023: (Start)
If A053735(n) = 1, then a(n) = n, otherwise a(n) = A365424(n) * a(A365459(n)).
For all n >= 1, A007949(a(n)) = A007949(n) and a(3*n) = 3*a(n).
For n >= 1, a(3^n - 1) = 2^(2n - 1), a(A048473(n)) = 2^(2*(n-1)).
These are conjectures so far:
For n >= 1, a(3^n - 2) = 10^(n-1).
For n >= 2, a(3^n - 3) = A002023(n-2) = 6*4^(n-2).
(End)

More terms from Michael De Vlieger, Sep 01 2022

A365721 The number of distinct prime factors in A356867(1+n), where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 17 2023

nonn

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

Cf. A001221, A356867, A365719, A365722.

Cf. also A069010.

A365721(n) = omega(A356867(1+n)); \\ Uses also the program given in A356867.

a(n) = A001221(A356867(1+n)) = A001221(A365719(n)).

a(n) <= A365722(n).

A365722 The number of prime factors (with multiplicity) in A356867(1+n), where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 3, 4, 3, 2, 3, 3, 3, 4, 4, 4, 5, 3, 1, 2, 2, 2, 3, 3, 3, 4, 3, 2, 3, 3, 3, 4, 4, 4, 5, 4, 3, 4, 4, 4, 5, 5, 5, 6, 4, 2, 3, 3, 3, 4, 4, 4, 5, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 4, 5, 5, 5, 6, 6, 6, 7, 4, 1, 2, 2, 2, 3, 3, 3, 4, 3, 2, 3, 3, 3, 4, 4, 4, 5, 4, 3, 4, 4, 4, 5, 5, 5

Offset: 0

Antti Karttunen, Sep 17 2023

Sum of digits minus the number of trailing 2's in the base-3 representation of n (A007089).

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

Cf. A001222, A007089, A007949, A053735, A356867, A365719, A365721.

Cf. also A000120.

A365722(n) = (sumdigits(n,3)-valuation(1+n,3));

A365722(n) = bigomega(A356867(1+n)); \\ Uses also the program given in A356867.

a(n) = A001222(A356867(1+n)) = A001222(A365719(n)).
a(n) = A053735(n) - A007949(1+n).
a(n) >= A365721(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)).

cp's OEIS Frontend

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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A356867 For n >= 1, write n = 3^m + k, where m >= 0 is the greatest power of 3 <= n, and k is in the range 0 <= k < 3^(m+1) - 3^m, then for n such that k=0, a(n)=n, and for n such that k > 0, a(n) is the smallest prime multiple p*a(k), p != 3, that is not already a term.

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A365721 The number of distinct prime factors in A356867(1+n), where A356867 is Sycamore's Doudna variant D(3).

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A365722 The number of prime factors (with multiplicity) in A356867(1+n), where 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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