A005245 -id:A005245 - cp's OEIS frontend

A350723 Integer defect of n: a(n) = A005245(n) - A349983(n).

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

Offset: 1

Harry Altman, Feb 06 2022

nonn

Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217.

Harry Altman, Integer Complexity: The Integer Defect, arXiv:1804.07446 [math.NT], 2018.

Difference between A005245 and A349983 (the latter being almost the same as A007600).

A189125 Primes p such that A005245(p) < A005245(p-1)+1.

Original entry on oeis.org

353942783, 1163385647, 1932319583, 2336924879, 4166809919, 4937621453, 5123340683, 5184740299, 5390865059, 5455982879, 5467766947, 6432033887, 6459553799, 6545574839, 6714582263, 6988649399, 7378517519, 7515851039, 7657182539, 7756383347

Offset: 1

Max Alekseyev, Apr 17 2011

Counterexamples to the conjecture from section F26 in UPINT.
a(1) was found by Martin Fuller; a(2) was found by Janis Iraids.
The prime terms of each of the sequences A189123 and A189124.

R. K. Guy, Unsolved Problems in Number Theory, Section F26.

Janis Iraids, Table of n, a(n) for n = 1..1000
Martin N. Fuller, C program for A005245

Cf. A005245, A189123, A189124.

More terms from Janis Iraids, Apr 20 2011.

A244743 Smallest number n with ||n-1||-||n|| = k where ||n||=A005245(n) denotes the complexity of n.

Original entry on oeis.org

6, 12, 24, 108, 720, 1440, 81648, 2041200, 612360000

Offset: 0

Juan Arias-de-Reyna, Jul 05 2014

The k-th term of this sequence is the least n with ||n-1||-||n|| = k if such an n exists.

It is conjectured that ||n-1||-||n|| is not bounded. But there is no proof that the sequence is infinite or is well defined.

			||719||=23, ||720||=19, therefore ||719||-||720||=4. This is the first time the difference 4 appears in this way. Hence a(4)=720.

J. Arias de Reyna and J. van de Lune, The question "How many 1's are needed?" revisited, arXiv preprint arXiv:1404.1850 [math.NT], 2014.

Cf. A005245, A252739 (see comments).

A117618 Least number with complexity height of n, under integer complexity A005245.

Original entry on oeis.org

1, 6, 7, 10, 22, 683

Offset: 1

Jonathan Vos Post, Apr 07 2006

Consider the recursion: A005245(n), A005245(A005245(n)), A005245(A005245(A005245(n))), ... which we know is finite before reaching a fixed point, as A005245(n) <= n. The number of steps needed to reach such a fixed point is the complexity height of n (with respect to the A005245 measure of complexity, there being others in the OEIS).

a(7) >= 872573642639 = A005520(89). - David A. Corneth, May 06 2024

			a(1) = 1 because the A005245 complexity of 1 is 1, already giving a fixed point.
a(2) = 6 because it is the smallest x such that A005245(x) =/= x and A005245(x) = A005245(A005245(x)).
a(3) = 7 because 7 is the least number x with complexity 6, thus taking a further step of recursion to reach a fixed point.
a(4) = 10 because 10 is the least number with complexity 7.
a(5) = 22 because 22 is the least number with complexity 10.
a(6) = 683 because 683 is the least number with complexity 22.
a(7) = the least number with complexity 683.

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971.
R. K. Guy, Unsolved Problems in Number Theory, Sect. F26.

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
E. Pegg, Jr., Integer Complexity.
Eric Weisstein's World of Mathematics, Integer Complexity.
Index to sequences related to the complexity of n

Cf. A003037, A003313, A005245, A005421, A005520, A025280, A061373, A064097, A076091, A076142.

a(n) = least k such that A005245^(n)(k) = A005245^(n-1)(k) but (if n>1) A005245^(n-1)(k) != A005245^(n-2)(k), where ^ denotes repeated application.

For n >= 3, a(n) = A005520(a(n-1)). - Max Alekseyev, May 06 2024

a(2)=6 inserted by Giovanni Resta, Jun 15 2016

Edited by Max Alekseyev, May 06 2024

A189123 Positive integers n such that A005245(n-i) + A005245(i) < A005245(n-1) + 1 for some i, 1 < i < n.

Original entry on oeis.org

21080618, 63241604, 67139098, 116385658, 117448688, 126483083, 152523860, 189724562, 212400458, 229762259, 318689258, 348330652, 353942783, 366873514, 373603732, 379448999, 385159320, 404764540, 409108300, 460759642, 465722100, 477258719, 498197068, 511069678, 516743639, 519835084, 538858312, 545438698, 545790940, 546853138, 574842670, 575550972, 581106238, 590785918

Offset: 1

Max Alekseyev, Apr 17 2011

nonn

Counterexamples to the first conjecture of David Wilson on A005245.
a(1) was found by Igor aka CD_Eater from Moscow in January, 2008
Prime terms are listed in A189125.

Discussion at dxdy.ru (in Russian)
Janis Iraids, Values of a(n) below 1.5*10^9
Martin N. Fuller, C program for A005245

Cf. A005245, A189124, A189125

A189124 Positive integers n such that A005245(n) < min{ A005245(n-1) + 1, A005245(x) + A005245(y) } where x,y range over positive integers with x*y = n.

Original entry on oeis.org

353942783, 516743639, 1163385647, 1542243239, 1932319583, 2336924879, 3113713259, 3444631199, 3878989487, 4103787551, 4166809919, 4937621453, 5123340683, 5170931639, 5184740299, 5200683263, 5390865059, 5455982879, 5467766947, 5570566315

Offset: 1

Max Alekseyev, Apr 17 2011

Counterexamples to the second conjecture of David Wilson on A005245.
a(1), a(2) were found by Martin Fuller; a(3) was found by Janis Iraids.
Prime terms are given by A189125.

Janis Iraids, Table of n, a(n) for n = 1..1000
Martin N. Fuller, C program for A005245

Cf. A005245, A189123, A189125.

More terms from Janis Iraids, Apr 20 2011

A195101 Solid numbers. Numbers m such that A005245(m) < A005245(k) + A005245(m-k) for 1<= k < m.

Original entry on oeis.org

1, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 26, 27, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111

Offset: 1

Juan Arias-de-Reyna, Sep 09 2011

nonn

It is useful for computing  A005245(m). To compute min_k  A005245(k) + A005245(m-k) we only need to check the cases in which k is a solid number.
The solid numbers  <= x appear to be  <= 0.6 * x.
We find many values where the minimum of A005245(k) + A005245(m-k) is not taken for k = 1. This is sequence A189123.
The first value of m needing k = 6 is 21080618, the first k = 8 is 385159320, the first with k = 9 is 3679353584.
Conjecture that for every solid number m > 1 there is some number n such that  A005245(n) = A005245(m)+A005245(n-m) and such that  for every representation as a product n = u*v with u, v >= 2 or every 1 < = k < m, we have A005245(n) < A005245(u)+A005245(v) and A005245(n) < A005245(k) + A005245(n-k).
The solid numbers are infinite. Proof by H. Altman, mentioned in link. For n>1, 3^n is a solid number.  If 3^n=a+b with 3n=||a||+||b||, then 3log_3(a)+3log_3(b)<=3n, and so ab<=3^n=a+b.  So either a=b=2 (impossible), or a=1 or b=1.  So suppose a=1.  Then b=3^n-1.  But since n>1 we have 3^n-1>(3/4)3^n>=E(3n-1) and thus ||3^n-1||>=3n, ||a||+||b||>=3n+1, contradiction. - Juan Arias-de-Reyna, Jan 09 2014

			m = 8 is a term of the sequence because
A005245(8) = 6 < A005245(7) + A005245(1)=6+1; A005245(8) < A005245(6) + A005245(2)=5+2; A005245(8)  < A005245(5) + A005245(3)=5+3;
A005245(8) < A005245(4) + A005245(4)=4+4.
m = 7 is not a term of the sequence because A005245(7) = 6 = A005245(6) + A005245(1) = 5 + 1.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
H. Altman and J. Zelinsky, Numbers with integer complexity close to the lower bound, Integers, 12 (2012) 1093-1125 (article where sequence is first introduced).
Juan Arias de Reyna, Complexity of natural numbers and arithmetic compact sets, arXiv:2302.06224 [math.NT], 2023.
Juan Arias de Reyna, Arithmetical Self-Similar Compact Sets, Integers (2024) Vol. 24, A21. See p. 13.

Cf. A005245, A189123.

nn = 200; a5245[n_] :=  a5245[n] = If[n == 1, 1, Min[Sequence @@ Table[a5245[i] + a5245[n - i], {i, 1, n/2}], Sequence @@ Table[a5245[d] + a5245[n/d], {d, Divisors[n]~Complement~{1, n}}]]]; t = Table[a5245[n], {n, nn}]; Select[Range[nn], And @@ Table[t[[#]] < t[[k]] + t[[# - k]], {k, # - 1}] &] (* T. D. Noe, Apr 09 2014 *)

Name and comments change using "solid numbers" notation by Juan Arias-de-Reyna, Jan 09 2014

A354023 Terms m of A351467 such that A005245(m) == 0 (mod 3).

Original entry on oeis.org

3, 7, 8, 9, 19, 20, 21, 24, 27, 55, 56, 57, 60, 63, 64, 72, 81, 163, 164, 165, 168, 171, 180, 189, 192, 216, 243, 487, 488, 489, 492, 495, 504, 512, 513, 540, 567, 576, 648, 729, 1459, 1460, 1461, 1464, 1467, 1476, 1485, 1512, 1536, 1539

Offset: 1

Harry Altman, May 14 2022

m appears in this list if and only if m>1 and it can be written as 2^p*3^r for p in {0,3,6,9} or as 2(3^r+1)3^s (r>0) or (2*3^r+1)3^s.

Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217; arXiv:1804.07446 [math.NT], 2018.

Cf. A351467, A350723, A005245, A354024, A354025.

A354024 Terms m > 1 of A351467 such that A005245(m) == 1 (mod 3).

Original entry on oeis.org

4, 10, 12, 28, 30, 32, 36, 82, 84, 90, 96, 108, 244, 246, 252, 256, 270, 288, 324, 730, 732, 738, 756, 768, 810, 864, 972, 2188, 2190, 2196, 2214, 2268, 2304, 2430, 2592, 2916, 6562, 6564, 6570, 6588, 6642, 6804, 6912, 7290, 7776, 8748

Offset: 1

Harry Altman, May 14 2022

m appears in this list if and only if it can be written as 2^p*3^r for p in {2,5,8} or as (3^r+1)3^s for r > 1.

Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217; arXiv:1804.07446 [math.NT], 2018.

Cf. A351467, A350723, A005245, A354023, A354025.

a(46) corrected by David Radcliffe, Aug 04 2025

A354025 Terms m of A351467 such that A005245(m) == 2 (mod 3).

Original entry on oeis.org

2, 5, 6, 13, 14, 15, 16, 18, 37, 38, 39, 40, 42, 45, 48, 54, 109, 110, 111, 112, 114, 117, 120, 126, 128, 135, 144, 162, 325, 326, 327, 328, 330, 333, 336, 342, 351, 360, 378, 384, 405, 432, 486, 973, 974, 975, 976, 978, 981, 984, 990, 999

Offset: 1

Harry Altman, May 14 2022

m appears in this list if and only if it can be written as 2^p*3^r for p in {1,4,7,10} or as 4(3^r+1)3^s (r>0) or 2(2*3^r+1)3^s or (4*3^r+1)3^s.

Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217; arXiv:1804.07446 [math.NT], 2018.

Cf. A351467, A350723, A005245, A354023, A354024.

cp's OEIS Frontend

A350723 Integer defect of n: a(n) = A005245(n) - A349983(n).

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A189125 Primes p such that A005245(p) < A005245(p-1)+1.

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A244743 Smallest number n with ||n-1||-||n|| = k where ||n||=A005245(n) denotes the complexity of n.

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A117618 Least number with complexity height of n, under integer complexity A005245.

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A189123 Positive integers n such that A005245(n-i) + A005245(i) < A005245(n-1) + 1 for some i, 1 < i < n.

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A189124 Positive integers n such that A005245(n) < min{ A005245(n-1) + 1, A005245(x) + A005245(y) } where x,y range over positive integers with x*y = n.

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A195101 Solid numbers. Numbers m such that A005245(m) < A005245(k) + A005245(m-k) for 1<= k < m.

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A354023 Terms m of A351467 such that A005245(m) == 0 (mod 3).

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A354024 Terms m > 1 of A351467 such that A005245(m) == 1 (mod 3).

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A354025 Terms m of A351467 such that A005245(m) == 2 (mod 3).

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