A263079 -id:A263079 - cp's OEIS frontend

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).

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

A263077 a(n) = greatest k where A155043(k) < A155043(n).

Original entry on oeis.org

0, 0, 2, 2, 6, 2, 12, 6, 6, 6, 12, 6, 18, 12, 18, 18, 22, 12, 30, 18, 30, 18, 34, 22, 22, 22, 42, 22, 48, 22, 60, 30, 60, 30, 72, 48, 84, 34, 84, 34, 96, 34, 108, 42, 96, 42, 108, 42, 48, 48, 120, 48, 132, 48, 132, 48, 140, 60, 140, 48, 140, 72, 140, 140, 140, 72, 140, 84, 140, 84, 140, 60, 140, 96, 140, 96, 150, 96, 156, 96, 108, 108, 120, 72, 120, 120, 132, 108, 140, 108, 140, 132, 140, 120, 140, 84

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..124340

Cf. A155043, A261089, A262503, A263078, A263079, A263080, A263082.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k, {n, 96}] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem((2^31)+(2^30));
uplim1 = 36756720 + 640; \\ = A002182(53) + A002183(53).
uplim2 = 36756720; \\ = A002182(53).
uplim3 = 32432400; \\ = A002182(52). Really just some Ad Hoc value smaller than above.
v155043 = vector(uplim1);
vother = vector(uplim3); \\ Contains A262503 and A263082 in succession.
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim1, v155043[i] = 1 + v155043[i-numdiv(i)]; if(!(i%1048576),print1(i,", ")));
A155043 = n -> if(!n,n,v155043[n]);
maxlen = 0; for(i=1, uplim2, len = v155043[i]; vother[len] = i; maxlen = max(maxlen,len); if(!(i%1048576),print1(i,", "))); \\ First it will be A262503.
print("uplim2=", uplim2, " uplim3=", uplim3, " maxlen=", maxlen);
\\ Then we convert it to A263082:
m = 0; for(i=1, maxlen, m = max(m, vother[i]); vother[i] = m; if(!(i%1048576),print1(i,", ")));
A263082 = n -> if(!n,n,vother[n]);
A263077 = n -> A263082(A155043(n)-1);
\\ Finally we can compute A263077:
for(i=1, uplim3, write("b263077.txt", i, " ", A263077(i)); );

a(n) = A263082(A155043(n)-1).

A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

Original entry on oeis.org

-1, -2, -1, -2, 1, -4, 5, -2, -3, -4, 1, -6, 5, -2, 3, 2, 5, -6, 11, -2, 9, -4, 11, -2, -3, -4, 15, -6, 19, -8, 29, -2, 27, -4, 37, 12, 47, -4, 45, -6, 55, -8, 65, -2, 51, -4, 61, -6, -1, -2, 69, -4, 79, -6, 77, -8, 83, 2, 81, -12, 79, 10, 77, 76, 75, 6, 73, 16, 71, 14, 69, -12, 67, 22, 65, 20, 73, 18, 77, 16, 27, 26, 37, -12, 35, 34, 45, 20, 51, 18, 49, 40, 47, 26, 45, -12, 43, 42, 41, 40, 39, 30

Offset: 1

Antti Karttunen, Oct 09 2015

sign

			For n=1 we have A049820(1) = 0, thus A155043(1) = 1, and 0 is the only (and thus the largest) number from which zero can be reached with less steps (namely in zero steps, A155043(0) = 0), thus a(1) = 0 - 1 = -1.
For n=7, we have A155043(7) = 4 [as A049820(7) = 5, A049820(5) = 3, A049820(3) = 1, A049820(1) = 0], but there exists x=12 for which we have A049820(12) = 6, A049820(6) = 2, A049820(2) = 0, and this is the largest x such that A155043(x) < A155043(7), thus a(7) = 12 - 7 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..124340

Cf. A155043, A261089, A262503, A262909, A263077.

Cf. A263079 (indices of the negative terms), A263080 (of the positive terms).

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k - n, {n, 102}] (* Michael De Vlieger, Oct 13 2015 *)

A263078 = n -> A263077(n) - n;
for(n=1,124340,write("b263078.txt",n," ",A263078(n)));
\\ Other code as in A263077

a(n) = A263077(n)-n.

A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

Original entry on oeis.org

5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

			5 is present, because if we start iterating A049820 from it as: A049820(5) = 3, A049820(3) = 1, A049820(1) = 0, we get a path of length 3 to reach zero, thus A155043(5) = 3. On the other hand, if we start from 6, the path is one step shorter: A049820(6) = 2, A049820(2) = 0 [i.e., A155043(6) = 2], thus there exists a number larger than 5 which results a shorter path to zero.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A155043, A261089, A262503, A263077, A263078.

Complement: A263079.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Position[Table[k = 3 n; While[a@ k >= a@ n, k--]; k - n, {n, 121}], Integer?Positive] // Flatten (* _Michael De Vlieger, Oct 13 2015 *)

n=0; i=0; while(i < 10000, n++; if((A263077(n) > n), i++; write("b263080.txt",i," ",n)));
\\ Other code as in A263077.

;; With Antti Karttunen's IntSeq-library.
(define A263080 (MATCHING-POS 1 1 (COMPOSE positive? A263078)))

A263082 a(n) = Max( A262503(k) : k=0,1,2,3,...,n ), where A262503(k) = largest x such that A155043(x) = k.

Original entry on oeis.org

0, 2, 6, 12, 18, 22, 30, 34, 42, 48, 60, 72, 84, 96, 108, 120, 132, 140, 140, 140, 140, 140, 140, 140, 150, 156, 168, 180, 180, 184, 192, 204, 216, 228, 240, 248, 264, 280, 280, 280, 280, 288, 296, 312, 312, 320, 328, 340, 352, 364, 372, 372, 372, 372, 384, 396, 420, 420, 420, 420, 432, 450, 468, 480, 504, 520, 540, 560, 572, 580, 594, 612, 612, 618, 622, 628, 648, 672, 672, 672, 672, 672

Offset: 0

Antti Karttunen, Oct 09 2015

nonn

From position a(n)+1 onward only terms > n will occur in A155043.

Antti Karttunen, Table of n, a(n) for n = 0..10005

Cf. A155043, A262503, A263077, A263079.

a(0) = 0; for n >= 1, a(n) = max(A262503(n),a(n-1)).
Other identities and observations:
For all n >= 0 and for any k > a(n): A155043(k) > n. [See the comment above.]
For all n >= 0: A155043(a(n)) <= n.

cp's OEIS Frontend

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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A263077 a(n) = greatest k where A155043(k) < A155043(n).

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A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

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A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

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A263082 a(n) = Max( A262503(k) : k=0,1,2,3,...,n ), where A262503(k) = largest x such that A155043(x) = k.

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