cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A245613 Permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = A244991(a(k)), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = A244990(1+a(k)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 9, 7, 11, 10, 32, 18, 13, 12, 17, 15, 22, 20, 35, 19, 66, 14, 24, 21, 34, 25, 23, 33, 31, 45, 63, 37, 27, 26, 41, 36, 29, 43, 69, 40, 134, 30, 47, 39, 44, 68, 71, 50, 38, 46, 67, 131, 28, 49, 42, 70, 64, 52, 92, 48, 127, 65, 61, 75, 55, 51, 89, 83, 73, 60
Offset: 1

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Author

Antti Karttunen, Jul 27 2014

Keywords

Comments

This shares with the permutation A122111 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

Links

Crossrefs

Inverse: A245614.
Cf. A026424, A028260, A055037, A055038, A066829, A244990, A244991, A244992, A122111.
Similar entanglement permutations: A243287, A243343, A243345, A244321-A244322, A245603, A135141, A237427, A245605.

Formula

a(1) = 1, and for n > 1, if A066829(n) = 1, a(n) = A244991(a(A055038(n))), otherwise a(n) = A244990(1+a(A055037(n)-1)).
As a composition of related permutations:
a(n) = A244322(A245603(n)).
For all n >= 1, A066829(n) = A244992(a(n)).

A285111 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(A005117(1+n)) = 2*a(n), a(A065642(n)) = 1 + 2*a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 13, 14, 10, 24, 15, 32, 27, 26, 25, 28, 20, 48, 55, 9, 30, 11, 21, 64, 54, 52, 31, 50, 56, 40, 111, 96, 110, 18, 51, 60, 22, 42, 41, 49, 128, 108, 223, 17, 103, 104, 61, 62, 447, 100, 43, 112, 80, 222, 109, 192, 220, 57, 63, 36, 102, 120, 113, 44, 84, 82, 895, 98, 256, 99, 221, 216, 446, 34, 207, 23
Offset: 1

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Comments

Note the indexing: the domain starts from 1, while the range includes also zero.

Links

Crossrefs

Inverse: A285112.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A284571.

Programs

  • Python
    from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum([1 for i in range(1, n) if core(i) == i])
def a(n):
    if n<3: return n - 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n))
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Apr 17 2017

Formula

a(1) = 0, a(2) = 1, and for n > 2, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)).

A284571 Permutation of natural numbers: a(1) = 1, a(A005117(1+n)) = 2*a(n), a(A065642(1+n)) = 1 + 2*a(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 9, 5, 12, 32, 17, 18, 10, 24, 33, 64, 65, 34, 11, 36, 20, 48, 129, 7, 66, 19, 37, 128, 130, 68, 49, 22, 72, 40, 97, 96, 258, 14, 69, 132, 38, 74, 73, 21, 256, 260, 81, 13, 29, 136, 15, 98, 521, 44, 39, 144, 80, 194, 257, 192, 516, 23, 137, 28, 138, 264, 45, 76, 148, 146, 197, 42, 512, 147, 193, 520, 162, 26, 27
Offset: 1

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Links

Crossrefs

Inverse: A284572.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A285111.

Programs

  • Python
    from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
def a(n):
    if n==1: return 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n) - 1)
[a(n) for n in range(1, 121)] # Indranil Ghosh, Apr 17 2017

Formula

a(1) = 1, for n > 1, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)-1).

A284584 a(1) = 0; for n > 1, if n is not squarefree, then a(n) = A057627(n), otherwise a(n) = A013928(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 5, 2, 3, 6, 7, 4, 8, 9, 10, 5, 11, 6, 12, 7, 13, 14, 15, 8, 9, 16, 10, 11, 17, 18, 19, 12, 20, 21, 22, 13, 23, 24, 25, 14, 26, 27, 28, 15, 16, 29, 30, 17, 18, 19, 31, 20, 32, 21, 33, 22, 34, 35, 36, 23, 37, 38, 24, 25, 39, 40, 41, 26, 42, 43, 44, 27, 45, 46, 28, 29, 47, 48, 49, 30, 31, 50, 51, 32, 52, 53, 54, 33, 55, 34, 56, 35, 57, 58, 59, 36
Offset: 1

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Comments

Each number n > 0 occurs exactly twice in this sequence, at the positions A005117(1+n) and A013929(n).

Links

Crossrefs

Cf. A005117, A008683, A013928, A013929, A057627, A243343, A243345, A243347, A243352.
Cf. A066136 (a similar sequence).

Programs

  • Python
    from sympy import mobius
from sympy.ntheory.factor_ import core
def a057627(n): return n - sum([mobius(k)**2 for k in range(1, n + 1)])
def a013928(n): return sum([1 for i in range(1, n) if core(i) == i])
def a(n):
    if n==1: return 0
    if core(n)==n: return a013928(n)
    else: return a057627(n)
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Apr 17 2017
  • Scheme
    (define (A284584 n) (cond ((= 1 n) 0) ((zero? (A008683 n)) (A057627 n)) (else (A013928 n))))

Formula

a(1) = 0; for n > 1, if A008683(n) is 0 [when n is not squarefree], then a(n) = A057627(n), otherwise a(n) = A013928(n).
Previous Showing 11-14 of 14 results.

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