A047988 -id:A047988 - cp's OEIS frontend

A048182 Variation on A047988, where division by d costs d points instead of 0 points.

0, 1, 2, 3, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 8, 8, 7, 8, 8, 7, 8, 9, 8, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 10, 9, 10, 10, 10, 11, 11, 10, 11, 11, 10, 10, 11, 11, 12, 11, 11, 11, 11, 10, 11, 11, 11, 11, 12, 11, 12, 11, 10, 11, 12, 11, 12

Offset: 2

David W. Wilson

nonn

A061390 Use same rules as A047988. Sequence gives smallest numbers which require n steps to reach 2.

Original entry on oeis.org

3, 5, 7, 14, 38, 172, 823, 6185, 87223, 1940494

Offset: 1

Jason Earls, Jun 09 2001

Some values calculated by Joseph Devincentis.

a(11) > 5000000. - David Wasserman, Jun 21 2002

Erich Friedman, Math. Magic

Cf. A047984.

More terms from David Wasserman, Jun 21 2002

A213534 Numbers k such that A047988(k) = A047988(k+1) = 4.

Original entry on oeis.org

11872177333, 128907074146, 153592049893, 173277915346, 218180421013, 372099798262, 468167023417, 496310532853, 499081630666, 533381128933, 569616671893, 571632935206, 579535129813, 829161163813, 866580745333, 945276981106, 945635700586, 960970734706, 975903758266

Offset: 1

Martin Renner, Jun 13 2012

nonn

Kantke calls these numbers "Vierwertzwillinge", i.e., value-4-twins.

Thomas Kantke, Das Spiel Minimum, in: Spektrum der Wissenschaft Spezial Physik - Mathematik - Technik 2/2012, pp. 57-66.

Thomas Kantke, The game minimum and the decomposition of natural numbers, Spectrum of Science 4/1993, page 11.

Cf. A047987, A047988.

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

A048823 a(n) = value of n defined as follows: start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by subtracting 1. The division steps are free, but subtracting 1 costs 1 point. The "value" of n is the smallest cost to reach 2.

Original entry on oeis.org

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

Offset: 2

Christian G. Bower, May 15 1999

nonn

Sean A. Irvine, Java program (github)

Cf. A047988. Same formula except you cannot add 1 (only subtract).

Missing a(2)=0 inserted by Sean A. Irvine, Jul 09 2021

A048825 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding 1. The division steps are free, but adding 1 costs 1 point. a(n) is the smallest cost to reach 2.

Original entry on oeis.org

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

Offset: 2

Christian G. Bower, May 15 1999

nonn

Cf. A047988. Same formula except you cannot subtract 1 (only add).

Edited by Max Alekseyev, Jun 13 2011

cp's OEIS Frontend

A048182 Variation on A047988, where division by d costs d points instead of 0 points.

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A061390 Use same rules as A047988. Sequence gives smallest numbers which require n steps to reach 2.

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A213534 Numbers k such that A047988(k) = A047988(k+1) = 4.

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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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A048823 a(n) = value of n defined as follows: start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by subtracting 1. The division steps are free, but subtracting 1 costs 1 point. The "value" of n is the smallest cost to reach 2.

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Extensions

A048825 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding 1. The division steps are free, but adding 1 costs 1 point. a(n) is the smallest cost to reach 2.

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Crossrefs

Extensions

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.