A348717 -id:A348717 - cp's OEIS frontend

A305897 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A348717, or equally, of A246277.
Filter sequence for prime factorization patterns, including also information about gaps between prime factors. - Original name, gives the motivation for this sequence. Here the "gaps" refers to differences between the indices of primes present, not the prime gaps as usually understood.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j).
  a(i) = a(j) => A243055(i) = A243055(j).

			a(10) = a(21) (= 6) because both have prime exponents [1, 1] and the difference between the prime indices is the same, as 10 = prime(1)*prime(3), and 21 = prime(2)*prime(4).
a(12) != a(18) because the prime exponents [2,1] and [1,2] do not occur in the same order.
a(140) = a(693) (= 71) because both numbers have prime exponents [2, 1, 1] (in this order) and the differences between the indices of the successive prime factors are same: 140 = prime(1)^2 * prime(3) * prime(4), 693 = prime(2)^2 * prime(4) * prime(5).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from indices in prime factorization

Cf. A077462, A101296, A246277, A300247, A305800, A348717.

Cf. also A286621.

up_to = 65537;
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; };
A246277(n) = if(1==n, 0, 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)/2);
v305897 = rgs_transform(vector(up_to,n,A246277(n)));
A305897(n) = v305897[n];

Name changed by Antti Karttunen, Apr 30 2022

A354186 Dirichlet inverse of A348717.

Original entry on oeis.org

1, -2, -2, 0, -2, 2, -2, 0, 0, -2, -2, 4, -2, -6, 2, 0, -2, -2, -2, 12, -2, -14, -2, 0, 0, -18, 0, 20, -2, 10, -2, 0, -6, -26, 2, 8, -2, -30, -14, 0, -2, 30, -2, 36, 4, -38, -2, 0, 0, -18, -18, 44, -2, -6, -2, 0, -26, -50, -2, -4, -2, -54, 12, 0, -6, 54, -2, 60, -30, 2, -2, -8, -2, -66, -2, 68, 2, 90, -2, 0, 0, -74

Offset: 1

Antti Karttunen, May 19 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A348717, A354187.

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));
memoA354186 = Map();
A354186(n) = if(1==n,1,my(v); if(mapisdefined(memoA354186,n,&v), v, v = -sumdiv(n,d,if(dA348717(n/d)*A354186(d),0)); mapput(memoA354186,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A348717(n/d) * a(d).

a(n) = A354187(n) - A348717(n).

A364297 a(n) = A348717(A163511(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 6, 2, 16, 8, 18, 4, 12, 6, 10, 2, 32, 16, 54, 8, 36, 18, 50, 4, 24, 12, 30, 6, 20, 10, 14, 2, 64, 32, 162, 16, 108, 54, 250, 8, 72, 36, 150, 18, 100, 50, 98, 4, 48, 24, 90, 12, 60, 30, 70, 6, 40, 20, 42, 10, 28, 14, 22, 2, 128, 64, 486, 32, 324, 162, 1250, 16, 216, 108, 750, 54, 500, 250, 686, 8, 144

Offset: 0

Antti Karttunen, Aug 15 2023

nonn

For all i, j: a(i) = a(j) => A278531(i) = A278531(j).
As the underlying sequence A163511 can be represented as a binary tree, so can this be also:
                                       1
                                       |
                    ...................2...................
                   4                                       2
         8......../ \........4                   6......../ \........2
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       8          18       4          12       6          10       2
  32  16   54 8       36  18   50 4       24  12   30 6       20  10   14 2
etc.
Each rightward leaning branch stays constant, because a(2n+1) = a(n).
Conjecture: Mersenne primes (A000668) gives all such odd numbers k for which a(k) = A348717(k). If true, then it immediately implies that map n -> A163511(n) [or equally: map n ->  A243071(n)] has no other fixed points than those given by A007283. But see also A364959. - Edited Sep 03 2023

Cf. A000265, A000668, A007283, A163511, A243071, A348717, A364949, A364956, A365423.

Cf. also A245611, A245612, A278531, A286531, A364959, A364963.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
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));
A364297(n) = A348717(A163511(n));

a(0) = 1, a(1) = 2, a(2n) = A163511(2n) = 2*A163511(n), and for n > 0, a(2n+1) = a(n).

A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 4, 7, 3, 8, 3, 9, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 4, 17, 18, 19, 3, 20, 3, 21, 22, 23, 10, 24, 3, 25, 26, 27, 3, 28, 3, 29, 8, 30, 3, 31, 4, 32, 33, 34, 3, 35, 14, 36, 37, 38, 3, 39, 3, 40, 41, 42, 43, 44, 3, 45, 46, 47, 3, 48, 3, 49, 50, 51, 10, 52, 3, 53, 11, 54, 3, 55, 26, 56, 57, 58, 3, 59, 14, 60, 61, 62, 33, 63, 3, 64, 65, 66

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].

Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

			a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.
a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.
a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.
a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.

Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.

Cf. also A305801, A305897, A355834, A355835.

up_to = 65537;
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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
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));
Aux355833(n) = [A342671(n), A348717(n)];
v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));
A355833(n) = v355833[n];

A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 5, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 25, 26, 3, 27, 28, 29, 3, 30, 3, 31, 32, 33, 3, 34, 17, 35, 36, 37, 3, 38, 39, 40, 41, 42, 3, 43, 3, 44, 45, 46, 47, 48, 3, 49, 50, 51, 3, 52, 3, 53, 54, 55, 56, 57, 3, 58, 59, 60, 3, 61, 62, 63, 64, 65, 3, 66, 67, 68, 69, 70, 71, 72, 3, 73, 74

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A355442(n)].
For all i, j: a(i) = a(j) => A355836(i) = A355836(j).
Terms that occur in positions given by A355822 may occur only a finite number of times in this sequence. Most of these seem to be in the singular equivalence classes, i.e., have unique values, apart from exceptions like pairs {6, 15}, {273, 1729}, (see the examples and the array A355926). In a coarser variant A355836 multiple such finite equivalence classes may coalesce together into several infinite equivalence classes.

			a(6) = a(15) [= 5 as allotted by the rgs-transform] because 15 = A003961(6) [i.e., 15 is in the same column in prime shift array A246278 as 6 is], and because A355442(6) = A355442(15) = 5.
a(138) = a(435) [= 103 as allotted by the rgs-transform] because 435 = A003961(138), and A355442(138) = A355442(435) = 5.
a(273) = a(1729) [= 205 as allotted by the rgs-transform] because 1729 = A003961(A003961(273)) [i.e., 273 and 1729 are in the same column of A246278], and A355442(273) = A355442(1729) = 11.

Cf. A003961, A276086, A348717, A355442, A355821, A355822, A355926.

Cf. also A246278, A305801, A355833, A355834.

up_to = 65537;
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));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
Aux355835(n) = [A348717(n), A355442(n)];
v355835 = rgs_transform(vector(up_to,n,Aux355835(n)));
A355835(n) = v355835[n];

A364949 a(n) = gcd(A348717(n), A348717(A163511(n))).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 2, 2, 12, 2, 2, 2, 16, 2, 18, 2, 4, 2, 2, 2, 24, 4, 2, 2, 4, 2, 2, 2, 32, 2, 2, 2, 36, 2, 2, 2, 8, 2, 6, 2, 4, 2, 2, 2, 48, 4, 10, 2, 4, 2, 2, 2, 8, 2, 2, 2, 4, 2, 2, 2, 64, 2, 6, 2, 4, 2, 10, 2, 72, 2, 2, 18, 4, 2, 2, 2, 16, 8, 2, 2, 12, 2, 2, 2, 8, 2, 6, 10, 4, 2, 2, 2, 96, 2, 2, 4, 20

Offset: 1

Antti Karttunen, Aug 16 2023

nonn

Cf. A163511, A348717, A364255, A364297.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
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));
A364949(n) = gcd(A348717(n),A348717(A163511(n)));

a(n) = gcd(A348717(n), A364297(n)).

a(2*n) = A364255(2*n) = 2*A364255(n). (Edited Sep 03 2023)

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];

A353267 The least number with the same prime factorization pattern (A348717) as A332449(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, 4, 4, 10, 4, 16, 4, 30, 10, 36, 4, 22, 4, 100, 16, 90, 4, 40, 4, 250, 36, 196, 4, 66, 10, 484, 30, 490, 4, 64, 4, 270, 100, 676, 16, 154, 4, 1156, 196, 750, 4, 144, 4, 1210, 22, 1444, 4, 198, 10, 84, 484, 1690, 4, 120, 36, 1470, 676, 2116, 4, 34, 4, 3364, 250, 810, 100, 400, 4, 2890, 1156, 324, 4, 462, 4, 3844

Offset: 1

Antti Karttunen, Apr 09 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A332449, A348717.

Cf. also A305897 (rgs-transform), A352892, A353268.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A353267(n) = A348717(A332449(n));

a(n) = A348717(A332449(n)) = A332449(A348717(n)).

A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 4, 9, 2, 10, 2, 11, 6, 12, 2, 13, 3, 14, 5, 15, 2, 16, 2, 17, 18, 19, 20, 21, 2, 22, 12, 23, 2, 24, 2, 25, 26, 27, 2, 28, 3, 29, 30, 31, 2, 32, 6, 33, 19, 34, 2, 35, 2, 36, 11, 37, 8, 38, 2, 39, 40, 41, 2, 42, 2, 43, 44, 45, 20, 46, 2, 47, 9, 48, 2, 49, 12, 50, 51, 52, 2, 53, 54, 55, 34, 56, 57, 58, 2, 59, 60, 61, 2, 62, 2, 63, 16

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).

			a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.

Cf. A000203, A000265, A009194, A246278, A348717, A355931.

Cf. also A300230, A305800, A305897, A355833.

up_to = 65537;
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; };
A000265(n) = (n>>valuation(n,2));
A009194(n) = gcd(n, sigma(n));
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));
Aux355834(n) = [A000265(A009194(n)), A348717(n)];
v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
A355834(n) = v355834[n];

A353268 The least number with the same prime factorization pattern (A348717) as A329603(n) = A005940(1+(1+(3*A156552(n)))).

Original entry on oeis.org

2, 2, 8, 6, 18, 2, 50, 12, 20, 8, 98, 14, 242, 18, 32, 24, 338, 6, 578, 54, 72, 50, 722, 28, 42, 98, 60, 150, 1058, 2, 1682, 48, 200, 242, 162, 70, 1922, 338, 392, 108, 2738, 8, 3362, 294, 44, 578, 3698, 56, 110, 20, 968, 726, 4418, 12, 450, 300, 1352, 722, 5618, 26, 6962, 1058, 500, 96, 882, 18, 7442, 1014, 2312

Offset: 1

Antti Karttunen, Apr 09 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A348717, A353267.

Coincides with A352892 on even n, and with A329603 on odd n.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A353268(n) = A348717(A329603(n));

a(n) = A348717(A329603(n)).

For all n >= 1, a(2n) = A352892(2n), a(2n-1) = A329603(2n-1).

cp's OEIS Frontend

A305897 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j), for all i, j >= 1.

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A354186 Dirichlet inverse of A348717.

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Formula

A364297 a(n) = A348717(A163511(n)).

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A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

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A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

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A364949 a(n) = gcd(A348717(n), A348717(A163511(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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A353267 The least number with the same prime factorization pattern (A348717) as A332449(n) = A005940(1+(3*A156552(n))).

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A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

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