A235048 -id:A235048 - cp's OEIS frontend

A234744 Multiplicative permutation of integers: a(n) = A234840(A235199(n)).

0, 1, 3, 2, 9, 61, 6, 19, 27, 4, 183, 101, 18, 281, 57, 122, 81, 11, 12, 5, 549, 38, 303, 263, 54, 3721, 843, 8, 171, 157, 366, 31, 243, 202, 33, 1159, 36, 1811, 15, 562, 1647, 1091, 114, 29, 909, 244, 789, 131, 162, 361, 11163, 22, 2529, 541, 24, 6161, 513, 10, 471, 59, 1098, 7

Offset: 0

Antti Karttunen, Jan 04 2014

Please see the comments at A234743. The observations given there apply also here.

Antti Karttunen, Table of n, a(n) for n = 0..378
Index entries for sequences that are permutations of the natural numbers

Inverse: A234743. Similarly composed multiplicative permutations, with more tractable cycle structures: A235485/A235486, A235493/A235494.

Cf. A234840, A235199, A235048.

(define (A234744 n) (A234840 (A235199 n)))

a(n) = A234840(A235199(n)).

Multiplicative with a(p_i) = p_{A235048(i)} for primes with index i, except for cases i=3 and i=4, use p_18 and p_8 (61 and 19) instead of 19 and 61. For the composites, the value is determined as: a(u*v) = a(u) * a(v).

A281010 Triangle read by rows in which row 2n-1 lists the widths of the symmetric representation of sigma(n), and row 2n lists a finite sequence S together with -1, with the property that the partial sums of S give the row 2n-1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 12 2017

The row 2n-1 lists the widths of the terraces at the n-th level (starting from the top) of the pyramid described in A245092.
The sum of the areas of these terraces equals A000203(n): the sum of the divisors of n.
The k-th element of row 2n is associated to the k-th vertical cells at the n-th level of the pyramid.
The row 2n shows where the subparts (or subregions) of the terraces starting and ending, in accordance with the values 1 or -1.
The number of subparts in the n-th terrace equals A001227(n): the number of odd divisors of n.
If n is odd then the number of subparts in the n-th terrace is also A000005(n): the number of divisors of n.

			Triangle begins:
1;
1,-1;
1, 1, 1;
1, 0, 0,-1;
1, 1, 0, 1, 1;
1, 0,-1, 1, 0;-1;
1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0,-1;
1, 1, 1, 0, 0, 0, 1, 1, 1;
1, 0, 0,-1, 0, 0, 1, 0, 0,-1;
1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 1,-1, 0, 0, 0, 0,-1;
1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1;
1, 0, 0, 0,-1, 0, 0, 0, 0, 1, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0,-1, 0, 1, 0, 0,-1, 0, 1, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
...
Written as an isosceles triangle the sequence begins:
.
.                                        1;
.                                      1, -1;
.                                    1,  1,  1;
.                                  1,  0,  0, -1;
.                                1,  1,  0,  1,  1;
.                              1,  0, -1,  1,  0, -1;
.                            1,  1,  1,  1,  1,  1,  1;
.                          1,  0,  0,  0,  0,  0,  0, -1;
.                        1,  1,  1,  0,  0,  0,  1,  1,  1;
.                      1,  0,  0, -1,  0,  0,  1,  0,  0, -1;
.                    1,  1,  1,  1,  1,  2,  1,  1,  1,  1,  1;
.                  1,  0,  0,  0,  0,  1, -1,  0,  0,  0,  0, -1;
.                1,  1,  1,  1,  0,  0,  0,  0,  0,  1,  1,  1,  1;
.              1,  0,  0,  0, -1,  0,  0,  0,  0,  1,  0,  0,  0, -1;
.            1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1;
.          1,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
.        1,  1,  1,  1,  1,  0,  0,  1,  1,  1,  0,  0,  1,  1,  1,  1,  1;
.      1,  0,  0,  0,  0, -1,  0,  1,  0,  0, -1,  0,  1,  0,  0,  0,  0, -1;
.    1,  1,  1,  1,  1,  1,  1,  1,  1,  0,  1,  1,  1,  1,  1,  1,  1,  1,  1;
.  1,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1,  0,  0,  0,  0,  0,  0,  0,  0, -1;
...

Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

The sum of row 2n-1 is A000203(n).
The sum of row 2n is A000004(n) = 0.
The number of positive terms in row 2n is A001227(n).
The number of nonzero terms in row 2n is A054844(n).
Middle diagonal (or central column of the isosceles triangle) gives A067742.
Row 2n-1 is also the n-th row of A249351.
Row 2n is also the n-th row of A281011.
Row 2n-1 lists the partial sums of the terms, except the last term, of the row 2n.
Cf. A000005, A001227, A196020, A235048, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A279387, A279388, A279391.

A235047 Permutation of nonnegative integers: a(n) = A235199(A234840(n+1)-1).

Original entry on oeis.org

0, 2, 1, 8, 18, 7, 84, 26, 3, 40, 14, 11, 280, 130, 37, 112, 196, 17, 4, 154, 289, 32, 262, 53, 504, 842, 5, 932, 20, 181, 86, 578, 15, 302, 2022, 35, 1582, 10, 561, 512, 2674, 329, 156, 50, 147, 1228, 264, 115, 9912, 2386, 201, 4448, 756, 23, 208, 2762, 9, 58, 42, 1003, 6, 272, 243

Offset: 0

Antti Karttunen, Jan 10 2014

nonn

Let b(n)=a(n), but with instead of a(8)=3 and a(18)=4, define b(8)=4 and b(18)=3 (i.e. otherwise same, but the values in positions 8 and 18 are swapped). The sequence b is then a permutation induced when A234743 is restricted to primes, and the indices of the reordered primes are collected: We have A049084(A234743(A000040(n))) = b(n) for all n. Or in other words, the permutation b completely determines the permutation A234743, because the latter is multiplicative. See further comments there.

Antti Karttunen, Table of n, a(n) for n = 0..361
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A235048. Cf. A234840, A235199, A234743.

(define (A235047 n) (A235199 (- (A234840 (+ n 1)) 1)))

a(n) = A235199(A234840(n+1)-1).

cp's OEIS Frontend

A234744 Multiplicative permutation of integers: a(n) = A234840(A235199(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A281010 Triangle read by rows in which row 2n-1 lists the widths of the symmetric representation of sigma(n), and row 2n lists a finite sequence S together with -1, with the property that the partial sums of S give the row 2n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A235047 Permutation of nonnegative integers: a(n) = A235199(A234840(n+1)-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula