A261088 -id:A261088 - 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

A261089 a(n) = least k such that A155043(k) = n; positions of records in A155043.

Original entry on oeis.org

0, 1, 3, 5, 7, 13, 17, 19, 23, 27, 29, 31, 35, 37, 41, 43, 51, 53, 57, 59, 61, 65, 67, 71, 73, 77, 79, 143, 149, 151, 155, 157, 161, 163, 173, 177, 179, 181, 185, 191, 193, 199, 203, 209, 211, 215, 219, 223, 231, 233, 237, 239, 241, 249, 251, 263, 267, 269, 271, 277, 285, 291, 293, 299, 303, 315, 317, 321, 327, 331, 335, 337, 341, 347, 349, 357, 359, 369, 515

Offset: 0

Antti Karttunen, Sep 23 2015

nonn

Note that there are even terms besides 0, and they all seem to be squares: a(915) = 7744 (= 88^2), a(41844) = 611524 (= 782^2), a(58264) = 872356 (= 934^2), a(66936) = 1020100 (= 1010^2), a(95309) = 1503076 (= 1226^2), a(105456) = 1653796 (= 1286^2), ...

Antti Karttunen, Table of n, a(n) for n = 0..110880

Cf. A000005, A155043, A049820, A259934, A261103.
Cf. A262503 (the last occurrence of n in A155043).
Cf. A262505 (difference between the last and the first occurrence).
Cf. A262507 (the number of occurrences of n in A155043).
Cf also A261085, A261088.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a261089 = fromJust . (`elemIndex` a155043_list)
-- Reinhard Zumkeller, Nov 27 2015

lim = 80; a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; t = Table[a@ n, {n, 0, 12 lim}]; Table[First@ Flatten@ Position[t, n] - 1, {n, 0, lim}] (* Michael De Vlieger, Sep 29 2015 *)

allocatemem(123456789);
uplim = 2162160; \\ = A002182(41).
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]);
n=0; k=0; while(k <= 10000, if(A155043(n)==k, write("b261089.txt", k, " ", n); k++); n++;);

;; With Antti Karttunen's IntSeq-library, two variants.
(definec (A261089 n) (let loop ((k 0)) (if (= n (A155043 k)) k (loop (+ 1 k)))))
(define A261089 (RECORD-POS 0 0 A155043))

Other identities. For all n >= 0:

A155043(a(n)) = n.

A262680 Number of squares encountered before zero is reached when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A010052(n) + a(A049820(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of perfect squares (A000290) encountered before zero is reached 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). This count includes n itself if it is a square, but excludes the final zero.
Also number of times the parity (of numbers encountered) changes until zero is reached when iterating A049820. This count includes also the last parity change 1 - d(1) -> 0 if coming to zero through 1.
There is a lower bound for this sequence that grows without limit if and only if either (1) A259934 is indeed the unique sequence (satisfying its given condition) and it contains an infinite number of squares (see A262514), or (2) more generally, if each one of all (hypothetically multiple) infinite branches of the tree (defined by parent-child relation A049820(child) = parent) contains an infinite number of squares. See also comments in A262509.

			For n=1, we subtract 1 - A000005(1) = 0, thus we reach zero in one step, and the starting value 1 is a square, thus a(1) = 1. Also, the parity changes once, from odd to even as we go from 1 to 0.
For n=24, when we start repeatedly subtracting the number of divisors (A000005), we obtain the following numbers: 24 - A000005(24) = 24 - 8 = 16, 16 - A000005(16) = 16 - 5 = 11, 11 - 2 = 9, 9 - 3 = 6, 6 - 4 = 2, 2 - 2 = 0. Of these numbers, 16 and 9 are squares larger than zero, thus a(24)=2. Also, we see that the parity changes twice: from even to odd at 16 and then back from odd to even at 9.

Antti Karttunen, Table of n, a(n) for n = 0..10201

Bisections: A262681, A262682.
Cf. A262687 (positions of records).
Cf. A000005, A000290, A010052, A049820, A155043, A259934, A261088, A262509, A262514, A262676, A262677.

A261085 Number of steps needed to reach zero when starting from the n-th prime [i.e., setting k to A000040(n)] and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 15, 17, 19, 20, 22, 23, 24, 26, 16, 18, 20, 21, 22, 22, 23, 24, 23, 24, 25, 26, 28, 29, 31, 33, 33, 34, 36, 37, 39, 40, 40, 41, 44, 47, 34, 35, 49, 51, 52, 54, 54, 55, 57, 58, 59, 58, 59, 62, 48, 49, 50, 66, 69, 71, 73, 74, 74, 76, 55, 57, 59, 60, 61, 63, 63, 65, 68, 69, 71, 72, 74, 62, 64, 65, 66, 67, 67, 70, 72, 73, 75, 76, 77, 80, 81, 75, 77, 79, 79, 81

Offset: 1

Antti Karttunen, Sep 23 2015

nonn

			For n=4 we have prime(4) = 7, from which we start subtracting the number of divisors, to get the following path: 7 - 2 = 5, 5 - 2 = 3, 3 - 2 = 1, 1 - 1 = 0, and we have reached zero in four steps, thus a(4) = 4.
For n=5 we have prime(5) = 11, for which the similar process results: 11 - 2 = 9, 9 - 3 = 6, 6 - 4 = 2, 2 - 2 = 0, and again we have reached zero in four steps, thus also a(5) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..6543

Cf. A000005, A000040, A049820, A155043, A259934, A261088.

Cf. A261086 (gives the positions of drops, i.e., where nonmonotonic) and A261087 (the corresponding primes).

mpr[p_]:=Length[NestWhileList[#-DivisorSigma[0,#]&,p,#>0&]]-1; mpr/@Prime[ Range[ 120]] (* Harvey P. Dale, Aug 18 2022 *)

uplim = 65537;
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]);
n=0; forprime(p=2, uplim, n++; write("b261085.txt", n, " ", A155043(p)));

(define (A261085 n) (A155043 (A000040 n)))

a(n) = A155043(A000040(n)).

A263087 a(n) = A060990(n^2); number of solutions to x - d(x) = n^2, where d(x) is the number of divisors of x (A000005).

Original entry on oeis.org

2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 3, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0

Offset: 0

Antti Karttunen, Oct 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10082

Cf. A000005, A000290, A002182, A002183, A060990.
Cf. A263093 (positions of zeros), A263092 (nonzeros).
Cf. A263250, A263251 (bisections) and A263252, A263253 (their partial sums).
Cf. also A261088, A263088.

A060990(n) = { my(k = n + 2400, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit A002183(77)=2400 good for at least up to A002182(77) = 10475665200.
A263087(n) = A060990(n^2);
for(n=0, 10082, write("b263087.txt", n, " ", A263087(n)));

(define (A263087 n) (A060990 (A000290 n)))

a(n) = A060990(n^2) = A060990(A000290(n)).

A262687 a(n) = index of first n in A262680; positions of records in A262680.

Original entry on oeis.org

0, 1, 4, 169, 324, 1521, 125316, 126025, 425104, 1713481, 1716100, 4959529, 24760576

Offset: 0

Antti Karttunen, Oct 03 2015

All terms are squares (A262688 gives the square roots).

The even and odd terms alternate.

Cf. A262680, A261088, A262514.

Cf. A262688 (square roots of these terms).

allocatemem((2^31)+(2^30));
uplim = 2^25;
v262680 = vector(uplim);
v262680[1] = 1; v262680[2] = 0;
for(i=3, uplim, v262680[i] = issquare(i) + v262680[i-numdiv(i)];
if(!(i%65536),print1(i,", ")););
A262680 = n -> if(!n,n,v262680[n]);
n=0; k=0; while(n*n <= uplim, if(A262680(n*n)==k, write("b262687.txt", k, " ", n*n); k++); n++;);

;; With Antti Karttunen's IntSeq-library.
(define A262687 (RECORD-POS 0 0 A262680))

A263088 a(n) = A262697(n^2).

Original entry on oeis.org

0, 6, 2, 38, 2, 1, 1, 1, 1, 22, 1, 0, 0, 2, 1, 3, 1, 9, 1, 39, 1, 47, 1, 51, 4, 114, 1, 1, 1, 529, 2, 6, 2, 3, 1, 1, 22, 1, 11, 3, 2, 4, 7, 93, 7, 967, 1, 1, 3, 4, 1, 3, 2, 4, 1, 3, 1, 3, 1, 1, 1, 2, 1, 139, 2, 265, 2, 1, 6, 464, 12, 4, 22, 1, 2, 1503, 2, 6, 1, 5, 2, 2, 1, 2, 5, 1, 2, 4, 2, 1, 1, 6, 3, 386, 1, 1, 3, 800, 1, 2, 1, 7, 5, 1, 1, 3353, 1, 2, 21, 3, 1, 17, 3, 3, 1, 4, 1, 5, 1, 3, 9, 2

Offset: 0

Antti Karttunen, Oct 12 2015

nonn

a(n)=0 if n^2 is in A259934, otherwise number of nodes in that finite subtree whose root is n^2 and edge-relation is defined by A049820(child) = parent. This count includes also leaves and n^2 itself.

Antti Karttunen, Table of n, a(n) for n = 0..222

Cf. A000290, A049820, A259934, A262697.
Cf. also A261088, A263087.
Cf. A262515 (positions of zeros), A263093 (positions of ones).

(define (A263088 n) (A262697 (A000290 n)))

a(n) = A262697(A000290(n)) = A262697(n^2).
Other identities. For all n >= 0:
If A263087(n) = 0, a(n) = 1.

A330877 Number of steps needed to reach zero or a cycle when starting from k = n and repeatedly applying the map that replaces k by k - d(k) if k is even, by k + d(k) if k is odd, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 2, 1, 7, 3, 6, 2, 5, 4, 4, 3, 12, 3, 11, 4, 10, 13, 10, 4, 9, 5, 8, 5, 8, 14, 7, 6, 32, 6, 32, 6, 31, 7, 30, 7, 29, 33, 29, 8, 28, 8, 28, 8, 27, 9, 26, 9, 12, 9, 11, 10, 25, 10, 25, 10, 24, 10, 23, 11, 23, 10, 22, 12, 21, 24, 21, 12, 21, 13

Offset: 0

Ctibor O. Zizka, Apr 29 2020

nonn

First cycle we see for n = 83. The length of the cycle is 38 steps. To reach a cycle means the time to first step into the loop.

			n = 1, mapping steps are 1 + 1 = 2, 2 - 2 = 0, a(1) = 2;
n = 2, mapping steps are 2 - 2 = 0, a(2) = 1;
n = 3, mapping steps are 3 + 2 = 5, 5 + 2 = 7, 7 + 2 = 9, 9 + 3 = 12, 12 - 6 = 6, 6 - 4 = 2, 2 - 2 = 0, a(3) = 7;
n = 4, mapping steps are 4 - 3 = 1, 1 + 1 = 2, 2 - 2 = 0, a(4) = 3;
n = 5, mapping steps are 5 + 2 = 7, 7 + 2 = 9, 9 + 3 = 12, 12 - 6 = 6, 6 - 4 = 2, 2 - 2 = 0, a(5) = 6.

Cf. A000005, A155043, A261085, A261088.

Cf. A049820, A062249.

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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A261089 a(n) = least k such that A155043(k) = n; positions of records in A155043.

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A262680 Number of squares encountered before zero is reached when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A010052(n) + a(A049820(n)).

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A261085 Number of steps needed to reach zero when starting from the n-th prime [i.e., setting k to A000040(n)] and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

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A263087 a(n) = A060990(n^2); number of solutions to x - d(x) = n^2, where d(x) is the number of divisors of x (A000005).

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A262687 a(n) = index of first n in A262680; positions of records in A262680.

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A263088 a(n) = A262697(n^2).

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A330877 Number of steps needed to reach zero or a cycle when starting from k = n and repeatedly applying the map that replaces k by k - d(k) if k is even, by k + d(k) if k is odd, where d(k) is the number of divisors of k (A000005).