A332896 -id:A332896 - cp's OEIS frontend

A332901 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A332896(i)) = A278222(A332896(j)) for all i, j.

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

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Restricted growth sequence transform of A278222(A332896(n)).
This is a variant of A292583: Instead of runs of numbers of the form 4k+3 encountered on trajectories of the standard Doudna-tree (A005940), this relates to the corresponding trajectories in A332815-tree. See comments in A292583.
For all i, j:
  a(i) = a(j) => A053866(i) = A053866(j),
  a(i) = a(j) => A332898(i) = A332898(j).

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

Cf. A053866, A278222, A332896, A332898, A332900.
Cf. A028982 (positions of ones).
Cf. also A292583.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v332901 = rgs_transform(vector(up_to,n,A278222(A332896(n))));
A332901(n) = v332901[n];

A332996 a(n) = A332896(A332817(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 2, 0, 1, 0, 0, 4, 5, 4, 5, 0, 0, 2, 2, 0, 1, 0, 0, 8, 8, 10, 11, 8, 8, 10, 10, 0, 1, 0, 0, 4, 5, 4, 5, 0, 0, 2, 2, 0, 1, 0, 0, 16, 17, 16, 17, 20, 20, 22, 23, 16, 17, 16, 16, 20, 21, 20, 21, 0, 0, 2, 2, 0, 1, 0, 0, 8, 8, 10, 11, 8, 8, 10, 10, 0, 1, 0, 0, 4, 5, 4, 5, 0, 0, 2, 2, 0, 1, 0, 0, 32, 32, 34, 35, 32, 32, 34, 34, 40, 41

Offset: 0

Antti Karttunen, Mar 05 2020

nonn

In contrast to similarly constructed A292274, this sequence can be computed directly from the binary expansion of n, without involving primes or their distribution at all.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A292274, A332817, A332896, A332995, A332998.

Differs from a similarly constructed A292592 for the first time at n=511, where a(511) = 170, while A292592(511) = 171.

PARI
```
A332996(n) = A332896(A332817(n));
```

a(n) = A332896(A332817(n)).
a(n) = n - A332995(n) = n XOR A332995(n).
A000120(a(n)) = A332998(n).

A332898 a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 3, 1, 2, 2, 2, 0, 3, 0, 4, 1, 1, 3, 5, 1, 0, 2, 1, 2, 4, 2, 6, 0, 2, 3, 3, 0, 5, 4, 3, 1, 6, 1, 7, 3, 1, 5, 8, 1, 0, 0, 4, 2, 7, 1, 4, 2, 3, 4, 9, 2, 8, 6, 2, 0, 1, 2, 10, 3, 4, 3, 11, 0, 9, 5, 1, 4, 1, 3, 12, 1, 0, 6, 13, 1, 2, 7, 5, 3, 10, 1, 4, 5, 5, 8, 5, 1, 11, 0, 3, 0, 12, 4, 14, 2, 2

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Starting from x=n, iterate the map x -> A332893(x) which divides even numbers by 2, and for odd n, changes every 4k+1 prime in the prime factorization to 4k+3 prime and vice versa (except 3 --> 2), like in A332819. a(n) counts the numbers of the form 4k+3 encountered until 1 has been reached. The count includes also n itself if it is of the form 4k+3 (A004767).

In other words, locate the node which contains n in binary tree A332815 and traverse from that node towards the root, counting all numbers of the form 4k+3 that occur on the path.

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

Cf. A000120, A004767, A332815, A332893, A332896, A332897, A332899.
Cf. A028982 (positions of zeros).
Cf. also A292377.

A332898(n) = if(1==n,0,A332898(A332893(n)) + (3==(n%4)));

a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + [n == 3 (mod 4)].

a(n) = A000120(A332896(n)).

A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 2, 2, 5, 4, 10, 4, 5, 10, 42, 8, 21, 20, 8, 8, 85, 10, 170, 20, 21, 84, 682, 16, 11, 42, 8, 40, 341, 16, 2730, 16, 85, 170, 16, 20, 1365, 340, 40, 40, 5461, 42, 10922, 168, 17, 1364, 43690, 32, 23, 22, 168, 84, 21845, 16, 80, 80, 341, 682, 174762, 32, 87381, 5460, 40, 32, 43, 170, 699050, 340, 1365, 32, 2796202

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000975, A108546, A332815, A332893, A332896, A332897.

Cf. also A292385.

A332895(n) = if(n<=2,n-1,2*A332895(A332893(n)) + (1==(n%4)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].
For n > 1, a(2n) = 2*a(n).
For n >= 1, a(A108546(n)) = A000975(n); A000120(a(n)) = A332897(n).

A332811 a(n) = A243071(A332808(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 63, 12, 31, 30, 13, 8, 127, 10, 255, 28, 29, 126, 1023, 24, 11, 62, 9, 60, 511, 26, 4095, 16, 125, 254, 27, 20, 2047, 510, 61, 56, 8191, 58, 16383, 252, 25, 2046, 65535, 48, 23, 22, 253, 124, 32767, 18, 123, 120, 509, 1022, 262143, 52, 131071, 8190, 57, 32, 59, 250, 1048575, 508, 2045, 54, 4194303, 40, 524287, 4094, 21

Offset: 1

Antti Karttunen, Mar 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A332817 (inverse permutation).
Cf. A054429, A243071, A332808, A332816, A332895, A332896, A332899.
Cf. also A332215.

up_to = 26927;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };
A332811(n) = A243071(A332808(n));

a(n) = A243071(A332808(n)).
For n > 1, a(n) = A054429(A332816(n)).
a(n) = A332895(n) + A332896(n).
a(n) = A332895(n) OR A332896(n) = A332895(n) XOR A332896(n).
A000120(a(n)) = A332899(n).

A332900 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 1 and n is a square or twice square, with f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 2, 7, 8, 9, 10, 11, 12, 2, 13, 2, 14, 15, 16, 17, 18, 19, 2, 20, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 2, 2, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 54, 55, 56, 57, 58, 59, 60, 2, 61, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 2, 85, 2

Offset: 1

Antti Karttunen, Mar 05 2020

nonn

For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A292383(i) = A292383(j) => A292583(i) = A292583(j),
  a(i) = a(j) => A332896(i) = A332896(j) => A332901(i) = A332901(j).

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

Cf. A292383, A292583, A324400, A332896, A332901.

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; };
A332900aux(n) = if((n>1)&&(issquare(n)||issquare(2*n)),0,n);
v332900 = rgs_transform(vector(up_to,n,A332900aux(n)));
A332900(n) = v332900[n];

cp's OEIS Frontend

A332901 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A332896(i)) = A278222(A332896(j)) for all i, j.

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PARI

A332996 a(n) = A332896(A332817(n)).

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Formula

A332898 a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + [n == 3 (mod 4)].

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Formula

A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

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Formula

A332811 a(n) = A243071(A332808(n)).

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A332900 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 1 and n is a square or twice square, with f(n) = n for all other numbers.

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