A332895 -id:A332895 - cp's OEIS frontend

A332995 a(n) = A332895(A332817(n)).

0, 1, 2, 2, 4, 5, 4, 5, 8, 8, 10, 11, 8, 8, 10, 10, 16, 17, 16, 17, 20, 20, 22, 23, 16, 17, 16, 16, 20, 21, 20, 21, 32, 32, 34, 35, 32, 32, 34, 34, 40, 41, 40, 41, 44, 44, 46, 47, 32, 32, 34, 34, 32, 33, 32, 32, 40, 40, 42, 43, 40, 40, 42, 42, 64, 65, 64, 65, 68, 68, 70, 71, 64, 65, 64, 64, 68, 69, 68, 69, 80, 80, 82, 83, 80, 80, 82, 82, 88, 89, 88, 89, 92

Offset: 0

Antti Karttunen, Mar 05 2020

nonn

In contrast to similarly constructed A292271, 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. A292271, A332817, A332895, A332996, A332997.

Differs from a similarly constructed A292593 for the first time at n=511, where a(511) = 341, while A292593(511) = 340.

PARI
```
A332995(n) = A332895(A332817(n));
```

a(n) = A332895(A332817(n)).
a(n) = n - A332996(n) = n XOR A332996(n).
A000120(a(n)) = A332997(n).

A332896 a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 5, 0, 0, 4, 21, 4, 10, 10, 5, 0, 42, 0, 85, 8, 8, 42, 341, 8, 0, 20, 1, 20, 170, 10, 1365, 0, 40, 84, 11, 0, 682, 170, 21, 16, 2730, 16, 5461, 84, 8, 682, 21845, 16, 0, 0, 85, 40, 10922, 2, 43, 40, 168, 340, 87381, 20, 43690, 2730, 17, 0, 16, 80, 349525, 168, 680, 22, 1398101, 0, 174762, 1364, 1, 340, 32, 42, 5592405, 32, 0, 5460

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A332893(x) is iterated down to 1, starting from x=n. See the binary tree illustrated in A332815.

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

Cf. A000975, A108546, A332815, A332893, A332895,

Cf. also A292383.

A332896(n) = if(1==n,n-1,2*A332896(A332893(n)) + (3==(n%4)));

a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].
Other identities. For n >= 1:
a(2n) = 2*a(n).
a(A108546(n)) = A000975(n-1).

A332897 a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 2, 1, 1, 4, 2, 4, 2, 3, 3, 5, 1, 3, 3, 1, 2, 5, 1, 6, 1, 4, 4, 1, 2, 6, 4, 2, 2, 7, 3, 7, 3, 2, 5, 8, 1, 4, 3, 3, 3, 8, 1, 2, 2, 5, 5, 9, 1, 9, 6, 2, 1, 4, 4, 10, 4, 6, 1, 11, 2, 10, 6, 2, 4, 5, 2, 12, 2, 2, 7, 13, 3, 5, 7, 4, 3, 11, 2, 1, 5, 7, 8, 3, 1, 12, 4, 3, 3, 13, 3, 14, 3, 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 their prime factorization to 4k+3 prime and vice versa (except 3 -> 2), like in A332819. a(n) counts the numbers of the form 4k+1 encountered until 1 has been reached, which is also included in the count when n > 1. This count includes also n itself when it is of the form 4k+1 (A016813) and larger than 1.

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

Cf. A016813, A332893, A332895, A332898, A332899.

Cf. also A292375.

A332897(n) = if(n<=2,n-1,A332897(A332893(n)) + (1==(n%4)));

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

a(n) = A000120(A332895(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).

cp's OEIS Frontend

A332995 a(n) = A332895(A332817(n)).

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A332896 a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].

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PARI

Formula

A332897 a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

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PARI

Formula

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

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