A086956 -id:A086956 - cp's OEIS frontend

A086962 a(A086957(n))=A086957(n), a(A086958(n))=A086959(n), a(A086959(n))=A086958(n), see A086956.

1, 3, 2, 4, 5, 8, 7, 6, 9, 25, 11, 15, 13, 21, 12, 16, 17, 18, 19, 24, 14, 33, 23, 20, 10, 39, 27, 28, 29, 36, 31, 32, 22, 51, 35, 30, 37, 57, 26, 48, 41, 56, 43, 44, 54, 69, 47, 40, 49, 50, 34, 52, 53, 45, 55, 42, 38, 87, 59, 70, 61, 93, 63, 64, 65, 88, 67, 68, 46

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

A self-inverse permutation of natural numbers: a(a(n))=n;

a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Index entries for sequences that are permutations of the natural numbers

Cf. A086572, A086571.

A086963 a(A086957(n))=A086959(n), a(A086958(n))=A086958(n), a(A086959(n))=A086957(n), see A086956.

Original entry on oeis.org

3, 2, 1, 8, 25, 6, 21, 4, 15, 10, 33, 12, 39, 14, 9, 24, 51, 36, 57, 20, 7, 22, 69, 16, 5, 26, 54, 56, 87, 30, 93, 48, 11, 34, 280, 18, 111, 38, 13, 40, 123, 42, 129, 88, 45, 46, 141, 32, 343, 70, 17, 104, 159, 27, 275, 28, 19, 58, 177, 60, 183, 62, 168, 112, 325

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

A self-inverse permutation of natural numbers: a(a(n))=n;

a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Index entries for sequences that are permutations of the natural numbers

Cf. A086572, A086571.

A086957 Smallest m with A086956(m)=n.

Original entry on oeis.org

1, 4, 9, 16, 5, 18, 7, 32, 27, 50, 11, 72, 13, 28, 75, 64, 17, 108, 19, 100, 63, 44, 23, 192, 125, 52, 81, 196, 29, 180, 31, 128, 99, 68, 35, 288, 37, 76, 117, 200, 41, 294, 43, 176, 225, 92, 47, 384, 49, 250, 153, 208, 53, 486, 55, 392, 171, 116, 59, 420, 61

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

a(n) < A086958(n) < A086959(n), A086956(a(n))=A086956(A086958(n))=A086956(A086959(n));
n and A086956(n) are divisors of a(n);
A086960(a(n)) = A086958(n), A086960(A086959(n)) = a(n);
A086961(a(n)) = A086959(n), A086961(A086958(n)) = a(n);
A086962(a(n)) = a(n);
A086963(a(n)) = A086959(n), A086963(A086959(n)) = a(n);
A086964(a(n)) = A086958(n), A086964(A086958(n)) = a(n).

Cf. A086938, A086939.

A086958 m such that A086956(m)=n and A086956(k)=n for exactly one kA086957(n)).

Original entry on oeis.org

2, 6, 12, 20, 10, 30, 14, 40, 45, 60, 22, 84, 26, 42, 90, 80, 34, 126, 38, 120, 147, 66, 46, 216, 150, 78, 135, 224, 58, 210, 62, 160, 132, 102, 245, 324, 74, 114, 156, 320, 82, 336, 86, 220, 315, 138, 94, 432, 98, 300, 204, 260, 106, 540, 110, 448, 228, 174

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

A086957(n) < a(n) < A086959(n), A086956(a(n))=A086956(A086957(n))=A086956(A086959(n));
n and A086956(n) are divisors of a(n);
A086960(A086957(n)) = a(n), A086960(a(n)) = A086959(n);
A086961(a(n)) = A086957(n), A086961(A086959(n)) = a(n);
A086962(a(n)) = A086959(n), A086962(A086959(n)) = a(n);
A086963(a(n)) = a(n);
A086964(A086957(n)) = a(n), A086964(a(n)) = A086957(n).

Cf. A086938, A086939.

A086959 Greatest m with A086956(m)=n.

Original entry on oeis.org

3, 8, 15, 24, 25, 36, 21, 48, 54, 70, 33, 96, 39, 56, 105, 112, 51, 144, 57, 140, 168, 88, 69, 240, 175, 104, 162, 252, 87, 270, 93, 256, 165, 136, 280, 360, 111, 152, 195, 400, 123, 378, 129, 264, 405, 184, 141, 480, 343, 350, 255, 312, 159, 594, 275, 504

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

A086957(n) < A086958(n) < a(n), A086956(a(n))=A086956(A086957(n))=A086956(A086958(n));
n and A086956(n) are divisors of a(n);
A086960(A086958(n)) = a(n), A086960(a(n)) = A086957(n);
A086961(A086957(n)) = a(n), A086961(a(n)) = A086958(n);
A086962(A086958(n)) = a(n), A086962(a(n)) = A086958(n);
A086963(A086957(n)) = a(n), A086963(a(n)) = A086957(n);
A086964(a(n)) = a(n).

Cf. A086938, A086939.

A086960 a(A086957(n))=A086958(n), a(A086958(n))=A086959(n), a(A086959(n))=A086957(n), see A086956.

Original entry on oeis.org

2, 3, 1, 6, 10, 8, 14, 4, 12, 25, 22, 15, 26, 21, 9, 20, 34, 30, 38, 24, 7, 33, 46, 16, 5, 39, 45, 42, 58, 36, 62, 40, 11, 51, 245, 18, 74, 57, 13, 48, 82, 56, 86, 66, 54, 69, 94, 32, 98, 60, 17, 78, 106, 27, 110, 28, 19, 87, 118, 70, 122, 93, 147, 80, 130, 88, 134

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

permutation of natural numbers with inverse A086961: a(A086961(n)) = A086961(a(n)) = n;

a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Index entries for sequences that are permutations of the natural numbers

Cf. A086572, A086571.

A086961 a(A086957(n))=A086959(n), a(A086958(n))=A086957(n), a(A086959(n))=A086958(n), see A086956.

Original entry on oeis.org

3, 1, 2, 8, 25, 4, 21, 6, 15, 5, 33, 9, 39, 7, 12, 24, 51, 36, 57, 16, 14, 11, 69, 20, 10, 13, 54, 56, 87, 18, 93, 48, 22, 17, 280, 30, 111, 19, 26, 32, 123, 28, 129, 88, 27, 23, 141, 40, 343, 70, 34, 104, 159, 45, 275, 42, 38, 29, 177, 50, 183, 31, 168, 112, 325

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

permutation of natural numbers with inverse A086960: a(A086960(n)) = A086960(a(n)) = n;

a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Index entries for sequences that are permutations of the natural numbers

Cf. A086572, A086571.

A086964 a(A086957(n))=A086958(n), a(A086958(n))=A086957(n), a(A086959(n))=A086959(n), see A086956.

Original entry on oeis.org

2, 1, 3, 6, 10, 4, 14, 8, 12, 5, 22, 9, 26, 7, 15, 20, 34, 30, 38, 16, 21, 11, 46, 24, 25, 13, 45, 42, 58, 18, 62, 40, 33, 17, 245, 36, 74, 19, 39, 32, 82, 28, 86, 66, 27, 23, 94, 48, 98, 60, 51, 78, 106, 54, 110, 56, 57, 29, 118, 50, 122, 31, 147, 80, 130, 44, 134

Offset: 1

Reinhard Zumkeller, Jul 25 2003

nonn

A self-inverse permutation of natural numbers: a(a(n))=n;

a(n) = A086962(A086963(n)) = A086963(A086962(n)).

Index entries for sequences that are permutations of the natural numbers

Cf. A086572, A086571.

A068101 a(n) = Sum_{k|n, k<=sqrt(n)} mu(k) where mu(k) is the Moebius function and the sum is over the positive divisors k of n with k <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 1, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, -1, 1, 0, -1, 0, 0, 0, 1, 0, 0, -2, 1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, 0, 1

Offset: 1

Leroy Quet, Mar 22 2002

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A086956.

a068101 = sum . map a008683 . a161906_row
-- Reinhard Zumkeller, Jul 30 2013

Table[DivisorSum[n, MoebiusMu, # <= Sqrt[n] &], {n, 103}] (* Michael De Vlieger, Sep 24 2017 *)

a(n) = sumdiv(n, k, (k<=sqrt(n))*moebius(k)); \\ Michel Marcus, Jan 03 2017

a(n) = Sum_{k=1..A038548(n)} A008683(A161906(n,k)). - Reinhard Zumkeller, Jul 30 2013

G.f.: Sum_{k>=1} mu(k)*x^(k^2)/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017

cp's OEIS Frontend

A086962 a(A086957(n))=A086957(n), a(A086958(n))=A086959(n), a(A086959(n))=A086958(n), see A086956.

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A086963 a(A086957(n))=A086959(n), a(A086958(n))=A086958(n), a(A086959(n))=A086957(n), see A086956.

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A086957 Smallest m with A086956(m)=n.

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A086958 m such that A086956(m)=n and A086956(k)=n for exactly one kA086957(n)).

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A086959 Greatest m with A086956(m)=n.

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A086960 a(A086957(n))=A086958(n), a(A086958(n))=A086959(n), a(A086959(n))=A086957(n), see A086956.

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A086961 a(A086957(n))=A086959(n), a(A086958(n))=A086957(n), a(A086959(n))=A086958(n), see A086956.

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A086964 a(A086957(n))=A086958(n), a(A086958(n))=A086957(n), a(A086959(n))=A086959(n), see A086956.

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A068101 a(n) = Sum_{k|n, k<=sqrt(n)} mu(k) where mu(k) is the Moebius function and the sum is over the positive divisors k of n with k <= sqrt(n).

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