A264970 -id:A264970 - cp's OEIS frontend

A263271 Square array A(row,col): A(row,0) = row and for col >= 1, if A262686(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A262686(row),col-1).

0, 2, 1, 6, 4, 2, 12, 8, 6, 3, 18, 0, 12, 5, 4, 22, 0, 18, 7, 8, 5, 30, 0, 22, 0, 0, 7, 6, 34, 0, 30, 0, 0, 0, 12, 7, 42, 0, 34, 0, 0, 0, 18, 0, 8, 46, 0, 42, 0, 0, 0, 22, 0, 0, 9, 54, 0, 46, 0, 0, 0, 30, 0, 0, 11, 10, 58, 0, 54, 0, 0, 0, 34, 0, 0, 16, 14, 11

Offset: 0

Antti Karttunen, Nov 29 2015

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...

Each row n lists all the nodes in A263267-tree that one encounters when one starts from node n and always chooses the largest possible child of it (A262686), and then the largest possible child of that child, etc, until a leaf-child (one of the terms of A045765) is encountered, after which the rest of the row contains only zeros.

			The top left corner of the array:
   0,  2,  6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 66,  0,  0,  0,  0
   1,  4,  8,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   2,  6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 66,  0,  0,  0,  0,  0
   3,  5,  7,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   4,  8,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   5,  7,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 66,  0,  0,  0,  0,  0,  0
   7,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   8,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   9, 11, 16, 24,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  10, 14, 20,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  11, 16, 24,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  12, 18, 22, 30, 34, 42, 46, 54, 58, 66,  0,  0,  0,  0,  0,  0,  0
  13,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  14, 20,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  15, 17, 21, 23, 27, 36,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  ...

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals

Column 0: A001477, Column 1: A262686.
Cf. A264971 (number of significant terms on each row, position where the first trailing zero-term occurs).
Cf. A264970.
Cf. also A000005, A045765, A263267.
See also array A265751 constructed in the same way, but obtained by following always the smallest child A082284, instead of the largest child A262686.

(define (A263271 n) (A263271bi (A002262 n) (A025581 n)))
(define (A263271bi row col) (cond ((zero? col) row) ((A262686 row) => (lambda (lad) (if (zero? lad) lad (A263271bi lad (- col 1)))))))
;; An alternative implementation, reflecting the new recurrence:
(define (A263271bi row col) (cond ((zero? col) row) ((and (zero? row) (= 1 col)) 2) ((zero? (A263271bi row (- col 1))) 0) (else (A262686 (A263271bi row (- col 1))))))

A(row,0) = row and for col >= 1, if A262686(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A262686(row),col-1).

A(0,0) = 0, A(0,1) = 2; if col = 0, A(row,0) = row; and for col > 0, if A(row,col-1) = 0, then A(row,col) = 0, otherwise A(row,col) = A262686(A(row,col-1)). [Another, perhaps more intuitive recurrence for this array.] - Antti Karttunen, Dec 21 2015

A264971 If A262686(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

13, 3, 12, 3, 2, 2, 11, 1, 1, 4, 3, 3, 10, 1, 2, 6, 2, 5, 9, 1, 1, 4, 8, 3, 1, 1, 3, 2, 1, 2, 7, 7, 2, 1, 6, 6, 1, 1, 7, 5, 1, 2, 5, 1, 2, 4, 4, 3, 6, 1, 1, 2, 1, 3, 3, 1, 1, 2, 2, 5, 5, 4, 4, 1, 1, 3, 1, 1, 1, 2, 3, 4, 4, 3, 1, 1, 3, 2, 5, 1, 2, 2, 4, 4, 3, 1, 3, 3, 1, 5, 4, 2, 2, 4, 3, 6, 2, 5, 1, 3, 1

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

See comments at A264970.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A049820, A060990, A262686.
One more than A264970.
Number of significant terms on row n of A263271.

If A060990(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A262686(n)).
Other identities. For all n >= 0:
a(n) = 1 + A264970(n).

A266110 If A082284(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

4, 3, 4, 2, 1, 1, 3, 0, 0, 2, 2, 1, 3, 0, 1, 2, 1, 1, 2, 0, 0, 6, 1, 5, 0, 0, 2, 4, 0, 3, 4, 2, 1, 0, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 1, 8, 1, 7, 1, 0, 0, 6, 0, 5, 3, 0, 0, 4, 2, 3, 7, 2, 1, 0, 0, 1, 0, 0, 0, 4, 1, 3, 6, 2, 0, 0, 8, 1, 4, 0, 1, 7, 3, 6, 5, 0, 2, 5, 0, 4, 12, 5, 1, 3, 11, 4, 4, 3, 0, 2, 0, 1, 10, 0, 3, 2, 9, 1, 0, 0, 1, 1, 0, 0, 8, 2, 0, 0, 7, 1, 2, 6

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

Starting from n, search for a smallest number k such that k - d(k) = n, and if found such a number, replace n with k and repeat the procedure. When eventually such k is no longer found, then (new) n must be one of the terms of A045765. The number of times the procedure can be repeated before that happens is the value of a(n). Sequence A266116 gives the stopping value of n.

			Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus a(21) = 6.

Antti Karttunen, Table of n, a(n) for n = 0..124340

One less than A266111.
Cf. A000005, A060990, A082284, A266116.
Cf. A045765 (positions of zeros).
Cf. tree A263267 (and its illustration).
Cf. also A264970.

cp's OEIS Frontend

A263271 Square array A(row,col): A(row,0) = row and for col >= 1, if A262686(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A262686(row),col-1).

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A264971 If A262686(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

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Formula

A266110 If A082284(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

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