A356867 -id:A356867 - cp's OEIS frontend

A365462 a(n) = A356867(n) - n.

0, 0, 0, 1, -1, 0, 3, 0, 0, -3, 3, 3, 12, 6, -3, 34, -1, 0, 16, 8, 9, 103, 17, 0, 75, 6, 0, -17, -7, -9, 24, 12, 9, 36, 21, 9, 12, 60, 36, 135, 99, 18, 207, 36, -9, 199, 149, 102, 576, 150, -3, 448, 11, 0, 22, 54, 48, 217, 29, 24, 289, 50, 27, 279, 425, 309, 808, 212, 51, 1180, 89, 0, 1152, 318, 225, 3049, 323, 18

Offset: 1

Antti Karttunen, Sep 15 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A356867, A364958 (positions of 0's), A365463.

Cf. also A364499.

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];
A365462(n) = (A356867(n)-n);

A365463 a(n) = gcd(n, A356867(n)), where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 2, 3, 1, 1, 6, 1, 8, 9, 1, 1, 3, 1, 2, 3, 2, 1, 18, 1, 4, 3, 1, 1, 24, 25, 2, 27, 1, 1, 3, 1, 4, 3, 2, 7, 9, 1, 2, 3, 5, 1, 6, 1, 4, 9, 1, 1, 6, 1, 50, 3, 4, 1, 54, 11, 2, 3, 1, 1, 12, 1, 2, 9, 1, 5, 3, 1, 4, 3, 10, 1, 72, 1, 2, 75, 1, 1, 6, 1, 16, 81, 1, 1, 3, 5, 2, 3, 2, 1, 9, 91, 2, 3, 1, 5, 12, 1, 2, 9, 5, 1, 6, 1, 8, 21

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A007949, A356867, A364957 (Dirichlet inverse), A365462, A365464, A365465.

Cf. also A364500.

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];
A365463(n) = gcd(n, A356867(n));

a(n) = gcd(n, A365462(n)) = gcd(A356867(n), A365462(n)).
a(n) = n / A365464(n) = A356867(n) / A365465(n).
For all n >= 1, A007949(a(n)) = A007949(n), A011655(a(n)) = A011655(n).

A365719 The least number with the same prime signature as A356867(1+n), where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 6, 8, 4, 2, 6, 6, 4, 12, 12, 12, 16, 12, 6, 12, 30, 8, 24, 24, 36, 32, 8, 2, 6, 6, 6, 12, 30, 30, 24, 12, 4, 12, 12, 12, 60, 60, 24, 48, 36, 12, 36, 60, 16, 72, 48, 72, 64, 24, 6, 30, 30, 12, 24, 60, 60, 48, 60, 8, 60, 24, 24, 120, 120, 48, 96, 72, 36, 72, 180, 32, 144, 96, 216, 128, 16, 2, 6, 6, 6, 12

Offset: 0

Antti Karttunen, Sep 17 2023

nonn

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

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

Cf. also A278222.

\\ Needs also code from A356867:
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A365719(n) = A046523(A356867(1+n));

a(n) = A046523(A356867(1+n)).

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).

A365465 a(n) = A356867(n) / gcd(n, A356867(n)).

Original entry on oeis.org

1, 1, 1, 5, 4, 1, 10, 1, 1, 7, 14, 5, 25, 10, 4, 25, 16, 1, 35, 7, 10, 125, 40, 1, 4, 16, 1, 11, 22, 7, 55, 11, 14, 35, 8, 5, 49, 49, 25, 35, 140, 10, 250, 20, 4, 245, 196, 25, 625, 4, 16, 125, 64, 1, 7, 55, 35, 275, 88, 7, 350, 56, 10, 343, 98, 125, 875, 70, 40, 125, 160, 1, 1225, 196, 4, 3125, 400, 16, 1000, 8, 1, 13, 26

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Denominator of n / A356867(n).

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A356867, A365462, A365463, A365464 (numerators).

Cf. also A364502.

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];
A365465(n) = (A356867(n)/gcd(n, A356867(n)));

a(n) = A356867(n) / A365463(n).

A365718 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365717(i) = A365717(j) for all i, j >= 0, where A365717(n) = A348717(A356867(1+n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 5, 6, 3, 2, 7, 4, 3, 8, 9, 10, 11, 12, 4, 13, 14, 6, 15, 16, 17, 18, 6, 2, 19, 5, 5, 20, 21, 22, 23, 9, 3, 24, 12, 9, 25, 26, 27, 28, 29, 12, 30, 31, 11, 32, 33, 34, 35, 36, 4, 37, 14, 8, 38, 39, 40, 41, 42, 6, 43, 36, 16, 44, 45, 46, 47, 48, 29, 49, 50, 18, 51, 52, 53, 54, 11, 2, 55, 7, 7, 56

Offset: 0

Antti Karttunen, Sep 17 2023

Restricted growth sequence transform of A365717.
For all i, j >= 0: a(i) = a(j) => A365720(i) = A365720(j).
In contrast to austere A103391, which is easily computed from n's binary expansion, the scatter plot here with its slender seaweed-like branchings suggests that this sequence is not just a simple derivation of base-3 expansion of n.

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

Cf. A348717, A356867, A365720.

Cf. also A103391 (similar transformation applied to A005940) and A365715 (compare the scatter plot).

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; };
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));
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); };
v365718 = rgs_transform(apply(A348717,A356867list(1+up_to)));
A365718(n) = v365718[1+n];

A365390 Inverse permutation of A356867, where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 8, 9, 7, 28, 15, 82, 11, 12, 17, 244, 18, 730, 14, 30, 29, 2188, 24, 13, 83, 27, 20, 6562, 21, 19684, 26, 84, 245, 19, 45, 59050, 731, 246, 23, 177148, 33, 531442, 32, 36, 2189, 1594324, 51, 37, 16, 732, 86, 4782970, 54, 31, 35, 2190, 6563, 14348908, 42, 43046722, 19685, 90, 53, 85, 87, 129140164

Offset: 1

Antti Karttunen, Sep 15 2023

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A365389 (one less), A356867 (inverse), A364958 (fixed points).

Cf. also A005941.

up_to = 1+(3^15);
A365390list(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)); my(invs=List([]),x); for(i=1,oo,if(mapisdefined(met,i,&x), listput(invs,x), if(isprime(i)&&i>4, listput(invs, 1+(3^(primepi(i)-2))), return(Vec(invs))))); };
v365390 = A365390list(up_to);
A365390(n) = v365390[n];
for(n=1,#v365390,print1(A365390(n),", "));

PARI
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See Links section.
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For all n >= 1, A356867(a(n)) = n.

A365464 a(n) = n / gcd(n, A356867(n)).

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 7, 1, 1, 10, 11, 4, 13, 7, 5, 8, 17, 1, 19, 5, 7, 22, 23, 1, 1, 13, 1, 28, 29, 10, 31, 8, 11, 17, 5, 4, 37, 19, 13, 8, 41, 7, 43, 11, 5, 46, 47, 8, 49, 1, 17, 13, 53, 1, 5, 28, 19, 58, 59, 5, 61, 31, 7, 64, 13, 22, 67, 17, 23, 7, 71, 1, 73, 37, 1, 76, 77, 13, 79, 5, 1, 82, 83, 28, 17, 43, 29, 44, 89, 10

Offset: 1

Antti Karttunen, Sep 15 2023

Numerator of n / A356867(n).

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A356867, A365462, A365463, A365465 (denominators).

Cf. also A364501.

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];
A365464(n) = n/gcd(n, A356867(n));

a(n) = n / A365463(n).

A364958 Fixed points of A356867, where A356867 is Sycamore's Doudna variant D(3).

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 18, 24, 27, 54, 72, 81, 91, 162, 216, 243, 273, 486, 648, 729, 819, 1458, 1944, 2187, 2457, 4374, 5832, 6561, 7371, 13122, 17496, 19683, 22113, 39366, 52488, 59049, 66339, 118098, 157464, 177147, 199017, 354294, 472392, 531441, 597051, 1062882, 1417176, 1594323, 1791153, 3188646, 4251528, 4782969

Offset: 1

David James Sycamore and Antti Karttunen, Sep 15 2023

nonn

Conjecture: All terms are of the form k*3^n, where k = 1,2,8,91, and n >= 0. - David James Sycamore, Aug 16 2023

Fixed points of A356867 and of A365390, positions of 0's in A365462.

Block[{a, c, i, j, k, m, p, t, nn},
  nn = 3^12; m = 1; i = 2; p = Prime[i]; c[_] = False;
  Monitor[Reap[Do[Set[{m, k}, {1, n - p^Floor[Log[p, n]]}];
    If[k == 0, Sow[n]; Set[{a[n], c[n]}, {n, True}],
      While[Set[t, Prime[m] a[k]]; Or[m == i, c[t]], m++];
      If[t == n, Sow[n]]; Set[{a[n], c[t]}, {t, True}] ],
{n, nn}] ][[-1, 1]], n] ] (* Michael De Vlieger, Jul 02 2025 *)

up_to = 3^14;
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];
isA364958(n) = (A356867(n)==n);

{k | k==A356867(k)}.

cp's OEIS Frontend

A365462 a(n) = A356867(n) - n.

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A365463 a(n) = gcd(n, A356867(n)), where A356867 is Sycamore's Doudna variant D(3).

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A365719 The least number with the same prime signature as A356867(1+n), where A356867 is Sycamore's Doudna variant D(3).

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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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A365465 a(n) = A356867(n) / gcd(n, A356867(n)).

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A365718 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365717(i) = A365717(j) for all i, j >= 0, where A365717(n) = A348717(A356867(1+n)).

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A365390 Inverse permutation of A356867, where A356867 is Sycamore's Doudna variant D(3).

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PARI

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A365464 a(n) = n / gcd(n, A356867(n)).

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