A262518 -id:A262518 - cp's OEIS frontend

A262520 a(n) = A262519(n) - A262518(n).

1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 2, 3, 0, 3, 4, 5, 4, 5, 3, 5, 6, 7, 5, 6, 1, 6, 7, 7, 8, 8, 10, 7, 2, 10, 9, 10, 13, 9, 11, 12, 1, 1, 4, 1, 3, 3, 2, 3, 7, 2, 2, 5, 7, 4, 9, 5, 6, 5, 5, 5, 6, 5, 1, 3, 7, 2, 8, 1, 8, 3, 9, 3, 3, 2, 3, 5, 3, 4, 6, 4, 6, 7, 4, 6, 2, 6, 6, 1, 7, 7, 10, 8, 9, 8, 8, 9, 10, 8, 1, 10, 10, 10, 11, 9, 11, 12, 10, 12, 13, 12, 13, 13, -2, -1, 2, 13, 13, 14, 14, 15

Offset: 0

Antti Karttunen, Oct 02 2015

sign

a(n) = How many steps more are needed to reach zero when starting from k = 2*n + 1 than when starting from k = 2*n and repeatedly applying the map that replaces k by k - d(k)? [Here d(k) is the number of divisors of k (A000005)]. If it takes more steps when starting from 2n than from 2n+1, then a(n) is negative.

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

Cf. A000005, A049820, A155043, A262518, A262519, A262521 (positions of negative values).

(define (A262520 n) (- (A262519 n) (A262518 n)))

a(n) = A262519(n) - A262518(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

A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.

Original entry on oeis.org

2, 5, 9, 13, 17, 23, 27, 32, 38, 44, 48, 56, 60, 66, 74, 80, 84, 93, 97, 105, 113, 119, 123, 133, 139, 145, 153, 161, 165, 177, 181, 188, 196, 202, 210, 222, 226, 232, 240, 250, 254, 266, 270, 278, 290, 296, 300, 312, 318, 327, 335, 343, 347, 359, 367, 377, 385, 391, 395, 411, 415, 421, 433, 441, 449, 461, 465, 473, 481

Offset: 1

Antti Karttunen, Oct 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10082
A. Karttunen, Ratio a(n)/A263085(n) drawn with OEIS Plot2-script

Cf. A000005, A006218, A099777, A263084, A263085, A060831.

Cf. also A262518, A262519.

with(numtheory): seq(add(tau(2*k), k=1..n), n= 1..60); # Ridouane Oudra, Aug 24 2019

Accumulate[DivisorSigma[0, 2 Range@ 69]] (* Michael De Vlieger, Oct 13 2015 *)

a(n) = sum(k=1, n, numdiv(2*k)); \\ Michel Marcus, Aug 25 2019

from math import isqrt
def A263086(n): return (t:=isqrt(m:=n>>1))**2-((s:=isqrt(n))**2<<1)+((sum(n//k for k in range(1,s+1))<<1)-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Oct 23 2023

a(1) = 2; for n > 1, a(n) = A000005(2*n) + a(n-1) [where A000005(k) gives the number of divisors of k].
Other identities. For all n >= 1:
a(n) = A263084(n) + A263085(n).
a(n) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019
From Ridouane Oudra, Aug 24 2019: (Start)
a(n) = Sum_{k=1..n} A000005(2*k)
a(n) = A006218(n) + A060831(n). (End)

A262519 Odd bisection of A155043.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 5, 5, 6, 7, 7, 8, 6, 9, 10, 11, 11, 12, 13, 13, 14, 15, 14, 15, 10, 16, 17, 17, 18, 19, 20, 19, 21, 22, 22, 23, 24, 23, 25, 26, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 22, 21, 22, 23, 23, 24, 24, 24, 25, 21, 26, 22, 23, 23, 24, 24, 24, 25, 26, 26, 27, 27, 27, 28, 29, 28, 30, 31, 31, 32, 33, 32, 33, 28, 33, 34, 29, 35, 36, 37, 37

Offset: 0

Antti Karttunen, Oct 02 2015

nonn

Number of steps needed to reach zero when starting from k = 2*n + 1 and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005).

Antti Karttunen, Table of n, a(n) for n = 0..10000
A. Karttunen, Ratio a(n)/A262518(n) drawn with the help of OEIS Plot2-script

Cf. A000005, A049820, A155043, A262518, A262520, A262521, A262681.

Table[Length[NestWhileList[#-DivisorSigma[0,#]&,n,#!=0&]]-1,{n,1,200,2}] (* Harvey P. Dale, Aug 31 2017 *)

(define (A262519 n) (A155043 (+ n n 1)))

a(n) = A155043(2*n + 1).

A262521 Numbers where A262520 takes a negative value; numbers n for which A155043(2n) > A155043(2n + 1).

Original entry on oeis.org

112, 113, 544, 545, 684, 760, 930, 1306, 1514, 1522, 1624, 1625, 2048, 2386, 2389, 2393, 2399, 2402, 2818, 2827, 2966, 2969, 3280, 3281, 3791, 3797, 3878, 4324, 4326, 4328, 4330, 4331, 4333, 4334, 4336, 4340, 4342, 4346, 4352, 5513, 5515, 5519, 5521, 5527, 5531, 5724, 5726, 6050, 6161, 6165, 6167, 6168, 6169, 6172, 6173, 6176, 6177, 6179, 6181, 6183, 6184, 6185, 6272

Offset: 1

Antti Karttunen, Oct 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1004

Cf. A000005, A049820, A155043, A262518, A262519, A262520.

A263084 a(n) = A263086(n) - A263085(n).

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 13, 14, 18, 22, 22, 28, 29, 31, 37, 41, 41, 46, 48, 52, 58, 62, 60, 68, 71, 73, 79, 83, 83, 93, 95, 96, 100, 104, 108, 118, 120, 120, 124, 132, 131, 141, 141, 145, 155, 157, 157, 165, 169, 172, 178, 184, 180, 190, 196, 202, 208, 210, 208, 220, 221, 223, 231, 237, 241, 251, 251, 251, 257, 267, 267, 278

Offset: 1

Antti Karttunen, Oct 12 2015

nonn

cp's OEIS Frontend

A262520 a(n) = A262519(n) - A262518(n).

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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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A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.

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A262519 Odd bisection of A155043.

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A262521 Numbers where A262520 takes a negative value; numbers n for which A155043(2n) > A155043(2n + 1).

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A263084 a(n) = A263086(n) - A263085(n).

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