A047985 -id:A047985 - cp's OEIS frontend

A047988 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. a(n) is the smallest cost to reach 2.

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

Offset: 2

Thomas Kantke (bytes.more(AT)ibm.net)

			From _David A. Corneth_, Jun 05 2020: (Start)
For n = 19 we get to 2 via
n := 19 - 1 = 2 (+1 point as we subtract 1 from n)
n := 18/3 = 6 (+0 point as we divide by a divisor)
n := 6/2 = 3 (+0 point, we divide by a divisor. we can't divide by 3 as 3 > sqrt(6)).
n := 3 - 1 = 2 (+1 point; steps end here; we have n = 2)
We earned 2 points in these steps. As this is the minimal number of points starting at n = 19, a(19) = 2. (End)

David A. Corneth, Table of n, a(n) for n = 2..10001
David A. Corneth, PARI program
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047984, A047985, A047986, A047987.

Cf. A048823, A048824, A048825, A048826, A048827.

\\ see Corneth link \\ David A. Corneth, Jun 05 2020

More terms from David W. Wilson

A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1p2p3p4p5...pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1p2...*pi >= p(i+1) for all i < k.

Original entry on oeis.org

2, 4, 8, 12, 16, 24, 32, 36, 40, 48, 56, 60, 64, 72, 80, 84, 96, 108, 112, 120, 128, 132, 144, 160, 168, 176, 180, 192, 200, 208, 216, 224, 240, 252, 256, 264, 280, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 420, 432, 440, 448

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of n (A047988) is the smallest cost to reach 2. Sequence gives numbers with value 0.
a(n) is also the length of the largest Dyck path of the symmetric representation of sigma of the n-th number whose symmetric representation of sigma has only one part. For an illustration see A317305. (Cf. A237593.) - Omar E. Pol, Aug 25 2018
This sequence can be defined equivalently as the increasing terms of the set containing 2 and all the integers such that if n is in the set, then all m * n are in the set for all m <= n. - Giuseppe Melfi, Oct 21 2019
The subsequence giving the largest term with k prime factors (k >= 1) starts 2, 4, 12, 132, 17292, 298995972, ... . - Peter Munn, Jun 04 2020

			Starting at 24 we divide by 3, 2, then 2, reaching 2.

T. D. Noe, Table of n, a(n) for n = 1..1000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Spektrum der Wissenschaft, No. 4, 1993, pp. 11-13.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.

Cf. A047984, A047985, A047986, A047987, A047988, A052287, A237593, A317305.

import Data.List.Ordered (union)
a047836 n = a047836_list !! (n-1)
a047836_list = f [2] where
   f (x:xs) = x : f (xs `union` map (x *) [2..x])
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

nMax = 100; A174973 = Select[Range[10*nMax], AllTrue[Rest[dd = Divisors[#]] / Most[dd], Function[r, r <= 2]]&]; a[n_] := 2*A174973[[n]]; Array[a, nMax] (* Jean-François Alcover, Nov 10 2016, after Reinhard Zumkeller *)

a(n) = 2 * A174973(n). - Reinhard Zumkeller, Sep 28 2011

The number of terms <= x is c*x/log(x) + O(x/(log(x))^2), where c = 0.612415..., and a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 1.63287... This follows from the formula just above. - Andreas Weingartner, Jun 30 2021

More terms from David W. Wilson

A047984 Integers n such that A047988(n)=1.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 37, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 57, 59, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 88, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

nonn

			7 has value 1 since we add 1 to get 8, then divide by 2 twice.

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047985, A047986, A047987, A047988.

More terms from David W. Wilson

A047986 Integers n such that A047988(n)=3.

Original entry on oeis.org

173, 347, 823, 907, 1237, 1697, 2137, 2333, 2423, 2473, 2767, 2777, 3253, 3413, 3547, 3559, 3623, 3767, 4243, 4273, 4457, 4547, 4723, 5091, 5297, 5861, 6185, 6197, 6593, 6823, 6827, 6871, 7093, 7247, 7537, 7717, 7817, 7823, 7879, 8243

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

nonn

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047985, A047986, A047987, A047988.

More terms from David W. Wilson

A213531 Number of k < 10^n such that A047988(k) = 2.

Original entry on oeis.org

0, 11, 201, 2592, 29916, 328988, 3550745, 37432690, 390065916, 4034529147, 41532029309, 425608837164

Offset: 1

Martin Renner, Jun 13 2012

nonn

Thomas Kantke, Mathematische Unterhaltungen, in: Spektrum der Wissenschaft 4/1993, p. 11-13.
Thomas Kantke, Das Spiel Minimum, in: Spektrum der Wissenschaft Spezial Physik - Mathematik - Technik 2/2012, pp. 57-66.

Cf. A047985, A047988, A212524, A213530, A213532, A213533.

cp's OEIS Frontend

A047988 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. a(n) is the smallest cost to reach 2.

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A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1*p2*p3*p4*p5*...*pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1*p2*...*pi >= p(i+1) for all i < k.

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A047984 Integers n such that A047988(n)=1.

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A047986 Integers n such that A047988(n)=3.

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A213531 Number of k < 10^n such that A047988(k) = 2.

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A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1p2p3p4p5...pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1p2...*pi >= p(i+1) for all i < k.