A263259 -id:A263259 - cp's OEIS frontend

A263266 Inverse permutation to A263265: a(0) = 0; for n >= 1, a(n) = A263259(n) + A263260(A155043(n)-1) - 1.

0, 1, 2, 3, 4, 6, 5, 11, 7, 8, 9, 12, 10, 15, 13, 16, 17, 20, 14, 26, 18, 27, 19, 30, 21, 22, 23, 34, 24, 38, 25, 46, 28, 47, 29, 50, 39, 54, 31, 55, 32, 59, 33, 67, 35, 60, 36, 68, 37, 40, 41, 74, 42, 81, 43, 82, 44, 88, 48, 95, 45, 103, 51, 96, 97, 108, 52, 114, 56, 115, 57, 120, 49, 128, 61, 121, 62, 138, 63, 145, 64, 69, 70, 75, 53

Offset: 0

Antti Karttunen, Nov 24 2015

Antti Karttunen, Table of n, a(n) for n = 0..131071
A. Karttunen, Graph plotted with OEIS Plot script up to n=10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A263265.
Cf. A263281 (fixed points).
Cf. A155043, A263259, A263260.
Differs from A263268 for the first time at n=38, where a(38) = 31, while A263268(38) = 32.

(define (A263266 n) (if (zero? n) n (+ -1 (A263259 n) (A263260 (- (A155043 n) 1)))))

a(0) = 0; for n >= 1, a(n) = A263259(n) + A263260(A155043(n)-1) - 1.

A263279 a(n) = A263259(A259934(n)); one-based position of A259934(n) at row n of table A263265.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

a(n) gives the number of integers k <= A259934(n) for which A155043(k) = n = A155043(A259934(n)).

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

Cf. A155043, A259934, A263259, A263265, A263280.

Cf. also A261103.

(define (A263267 n) (A263259 (A259934 n)))

a(n) = A263259(A259934(n)).

A155043 a(0)=0; for n >= 1, a(n) = 1 + a(n-d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 4, 3, 3, 3, 4, 3, 5, 4, 5, 5, 6, 4, 7, 5, 7, 5, 8, 6, 6, 6, 9, 6, 10, 6, 11, 7, 11, 7, 12, 10, 13, 8, 13, 8, 14, 8, 15, 9, 14, 9, 15, 9, 10, 10, 16, 10, 17, 10, 17, 10, 18, 11, 19, 10, 20, 12, 19, 19, 21, 12, 22, 13, 22, 13, 23, 11, 24, 14, 23, 14, 25, 14, 26, 14, 15, 15

Offset: 0

Ctibor O. Zizka, Jan 19 2009

From Antti Karttunen, Sep 23 2015: (Start)
Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005).
The original name was: a(n) = 1 + a(n-sigma_0(n)), a(0)=0, sigma_0(n) number of divisors of n.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..124340
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Antti Karttunen, Graph plotted with OEIS Plot script up to n=10000
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.

Cf. A000005, A049820, A060990, A259934.
Sum of A262676 and A262677.
Cf. A261089 (positions of records, i.e., the first occurrence of n), A262503 (the last occurrence), A262505 (their difference), A263082.
Cf. A262518, A262519 (bisections, compare their scatter plots), A262521 (where the latter is less than the former).
Cf. A261085 (computed for primes), A261088 (for squares).
Cf. A262507 (number of times n occurs in total), A262508 (values occurring only once), A262509 (their indices).
Cf. A263265 (nonnegative integers arranged by the magnitude of a(n)).
Cf. also A263077, A263078, A263079, A263080.
Cf. also A261104, A262680, A262904, A263254, A263259, A263260, A263266, A263270.
Cf. also A004001, A005185.
Cf. A264893 (first differences), A264898 (where repeating values occur).

import Data.List (genericIndex)
a155043 n = genericIndex a155043_list n
a155043_list = 0 : map ((+ 1) . a155043) a049820_list
-- Reinhard Zumkeller, Nov 27 2015

with(numtheory): a := proc (n) if n = 0 then 0 else 1+a(n-tau(n)) end if end proc: seq(a(n), n = 0 .. 90); # Emeric Deutsch, Jan 26 2009

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[a@n, {n, 0, 82}] (* Michael De Vlieger, Sep 24 2015 *)

uplim = 110880; \\ = A002182(30).
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
A155043 = n -> if(!n,n,v155043[n]);
for(n=0, uplim, write("b155043.txt", n, " ", A155043(n)));
\\ Antti Karttunen, Sep 23 2015

from sympy import divisor_count as d
def a(n): return 0 if n==0 else 1 + a(n - d(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 03 2017

(definec (A155043 n) (if (zero? n) n (+ 1 (A155043 (A049820 n)))))
;; Antti Karttunen, Sep 23 2015

From Antti Karttunen, Sep 23 2015 & Nov 26 2015: (Start)
a(0) = 0; for n >= 1, a(n) = 1 + a(A049820(n)).
a(n) = A262676(n) + A262677(n). - Oct 03 2015.
Other identities. For all n >= 0:
a(A259934(n)) = a(A261089(n)) = a(A262503(n)) = n. [The sequence works as a left inverse for sequences A259934, A261089 and A262503.]
a(n) = A262904(n) + A263254(n).
a(n) = A263270(A263266(n)).
A263265(a(n), A263259(n)) = n.
(End)

Extended by Emeric Deutsch, Jan 26 2009

Name edited by Antti Karttunen, Sep 23 2015

A263265 Irregular triangle T(n,k), n >= 0, k = 1 .. A262507(n), read by rows, where each row n lists in ascending order all integers x for which A155043(x) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 7, 11, 14, 18, 13, 15, 16, 20, 22, 17, 24, 25, 26, 28, 30, 19, 21, 32, 34, 23, 38, 40, 42, 27, 44, 46, 48, 29, 36, 49, 50, 52, 54, 56, 60, 31, 33, 58, 72, 35, 62, 66, 84, 37, 39, 68, 70, 96, 41, 45, 74, 76, 78, 80, 104, 108, 43, 47, 81, 82, 88, 90, 120, 51, 83, 85, 86, 94, 128, 132, 53, 55, 87, 92, 102, 136, 140

Offset: 0

Antti Karttunen, Nov 24 2015

			Rows 0 - 8 of the triangle:
0;
1, 2;
3, 4, 6;
5, 8, 9, 10, 12;
7, 11, 14, 18;
13, 15, 16, 20, 22;
17, 24, 25, 26, 28, 30;
19, 21, 32, 34;
23, 38, 40, 42;
Row n contains A262507(n) terms, the first of which is A261089(n) and the last of which is A262503(n). For all terms on row n, A155043(n) = n.

Inverse: A263266.
Cf. A261089 (left edge), A262503 (right edge), A262507 (number of terms on each row).
Cf. A263279 (gives the positions of terms of A259934 on each row), A263280 (and their distance from the right edge).
Cf. A155043, A263259, A263270.
Cf. also permutations A263267 & A263268 and A263255 & A263256.
Differs from A263267 for the first time at n=31, where a(31) = 38, while A263267(31) = 40.

Other identities. For all n >= 0:

A155043(a(n)) = A263270(n).

cp's OEIS Frontend

A263266 Inverse permutation to A263265: a(0) = 0; for n >= 1, a(n) = A263259(n) + A263260(A155043(n)-1) - 1.

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A263279 a(n) = A263259(A259934(n)); one-based position of A259934(n) at row n of table A263265.

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A155043 a(0)=0; for n >= 1, a(n) = 1 + a(n-d(n)), where d(n) is the number of divisors of n (A000005).

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A263265 Irregular triangle T(n,k), n >= 0, k = 1 .. A262507(n), read by rows, where each row n lists in ascending order all integers x for which A155043(x) = n.

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A263089 Least monotonic left inverse for A261089; a(n) = largest k for which A261089(k) <= n.

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