A277196 -id:A277196 - cp's OEIS frontend

A277006 a(n) = A005940(1+A003714(n)); permutation of squarefree numbers.

1, 2, 3, 5, 6, 7, 10, 15, 11, 14, 21, 35, 30, 13, 22, 33, 55, 42, 77, 70, 105, 17, 26, 39, 65, 66, 91, 110, 165, 143, 154, 231, 385, 210, 19, 34, 51, 85, 78, 119, 130, 195, 187, 182, 273, 455, 330, 221, 286, 429, 715, 462, 1001, 770, 1155, 23, 38, 57, 95, 102, 133, 170, 255, 209, 238, 357, 595, 390, 247, 374, 561, 935, 546, 1309, 910, 1365, 323

Offset: 0

Antti Karttunen, Oct 07 2016

nonn

Permutation of A005117 (squarefree numbers).

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A000040, A000045, A003714, A005117, A005940, A048675, A048679, A087808.

Cf. also A019565, A277010, A277196.

(define (A277006 n) (A005940 (+ 1 (A003714 n))))

a(n) = A005940(1+A003714(n)).
Other identities.
For n >= 0, A048675(a(n)) = A087808(A003714(n)) = A048679(n).
For n >= 1, a(A000045(n+1)) = A000040(n).

A277195 Permutation of nonnegative integers: a(n) = A022290(A277010(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 13, 9, 7, 21, 34, 10, 14, 55, 22, 89, 12, 144, 15, 35, 11, 233, 56, 23, 377, 17, 610, 90, 987, 36, 1597, 16, 57, 145, 2584, 4181, 234, 24, 25, 6765, 91, 19, 10946, 17711, 378, 18, 38, 28657, 611, 46368, 37, 988, 146, 75025, 26, 235, 1598, 58, 121393, 196418, 59, 317811, 20, 2585, 514229, 832040, 27, 379, 1346269, 93, 92

Offset: 1

Antti Karttunen, Oct 07 2016

nonn

Note the indexing: domain starts from 1, but the range includes also 0.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A277196.

Cf. A005117, A022290, A156552, A277010.

from math import isqrt
from sympy import fibonacci, mobius, primepi, factorint
def A277195(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return sum(fibonacci(primepi(p)+i) for i, p in enumerate(factorint(bisection(f), multiple=True),1)) # Chai Wah Wu, Aug 31 2024

(define (A277195 n) (A022290 (A277010 n)))

a(n) = A022290(A277010(n)) = A022290(A156552(A005117(n))).

cp's OEIS Frontend

A277006 a(n) = A005940(1+A003714(n)); permutation of squarefree numbers.

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