A062943 -id:A062943 - cp's OEIS frontend

A000792 a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1.

1, 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 729, 972, 1458, 2187, 2916, 4374, 6561, 8748, 13122, 19683, 26244, 39366, 59049, 78732, 118098, 177147, 236196, 354294, 531441, 708588, 1062882, 1594323, 2125764, 3188646, 4782969, 6377292

Offset: 0

N. J. A. Sloane

Numbers of the form 3^k, 2*3^k, 4*3^k with a(0) = 1 prepended.
If a set of positive numbers has sum n, this is the largest value of their product.
In other words, maximum of products of partitions of n: maximal value of Product k_i for any way of writing n = Sum k_i. To find the answer, take as many of the k_i's as possible to be 3 and then use one or two 2's (see formula lines below).
a(n) is also the maximal size of an Abelian subgroup of the symmetric group S_n. For example, when n = 6, one of the Abelian subgroups with maximal size is the subgroup generated by (123) and (456), which has order 9. [Bercov and Moser] - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
Also the maximum number of maximal cliques possible in a graph with n vertices (cf. Capobianco and Molluzzo). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 15 2001 [Corrected by Jim Nastos and Tanya Khovanova, Mar 11 2009]
Every triple of alternate terms {3*k, 3*k+2, 3*k+4} in the sequence forms a geometric progression with first term 3^k and common ratio 2. - Lekraj Beedassy, Mar 28 2002
For n > 4, a(n) is the least multiple m of 3 not divisible by 8 for which omega(m) <= 2 and sopfr(m) = n. - Lekraj Beedassy, Apr 24 2003
Maximal number of divisors that are possible among numbers m such that A080256(m) = n. - Lekraj Beedassy, Oct 13 2003
Or, numbers of the form 2^p*3^q with p <= 2, q >= 0 and 2p + 3q = n. Largest number obtained using only the operations +,* and () on the parts 1 and 2 of any partition of n into these two summands where the former exceeds the latter. - Lekraj Beedassy, Jan 07 2005
a(n) is the largest number of complexity n in the sense of A005520 (A005245). - David W. Wilson, Oct 03 2005
a(n) corresponds also to the ultimate occurrence of n in A001414 and thus stands for the highest number m such that sopfr(m) = n, for n >= 2. - Lekraj Beedassy, Apr 29 2002
a(n) for n >= 1 is a paradigm shift sequence with procedural length p = 0, in the sense of A193455. - Jonathan T. Rowell, Jul 26 2011
a(n) = largest term of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012
For n >= 2, a(n) is the largest number whose prime divisors (with multiplicity) add to n, whereas the smallest such number (resp. smallest composite number) is A056240(n) (resp. A288814(n)). - David James Sycamore, Nov 23 2017
For n >= 3, a(n+1) = a(n)*(1 + 1/s), where s is the smallest prime factor of a(n). - David James Sycamore, Apr 10 2018

			a{8} = 18 because we have 18 = (8-5)*a(5) = 3*6 and one can verify that this is the maximum.
a(5) = 6: the 7 partitions of 5 are (5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1) and the corresponding products are 5, 4, 6, 3, 4, 2 and 1; 6 is the largest.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 18*x^8 + ...

B. R. Barwell, Cutting String and Arranging Counters, J. Rec. Math., 4 (1971), 164-168.
B. R. Barwell, Journal of Recreational Mathematics, "Maximum Product": Solution to Prob. 2004;25(4) 1993, Baywood, NY.
M. Capobianco and J. C. Molluzzo, Examples and Counterexamples in Graph Theory, p. 207. North-Holland: 1978.
S. L. Greitzer, International Mathematical Olympiads 1959-1977, Prob. 1976/4 pp. 18;182-3 NML vol. 27 MAA 1978
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 396.
P. R. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. Amer., 1991, pp. 30-31 and 188.
L. C. Larson, Problem-Solving Through Problems. Problem 1.1.4 pp. 7. Springer-Verlag 1983.
D. J. Newman, A Problem Seminar. Problem 15 pp. 5;15. Springer-Verlag 1982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
R. Bercov and L. Moser, On Abelian permutation groups, Canad. Math. Bull. 8 (1965) 627-630.
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
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. See M_n.
J. Arias de Reyna and J. van de Lune, Algorithms for determining integer complexity, arXiv preprint arXiv:1404.2183 [math.NT], 2014.
Nigel Derby, 96.21 The MaxProduct partition, The Mathematical Gazette 96:535 (2012), pp. 148-151.
Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.
Hans Havermann, Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Andrew Kenney and Caroline Shapcott, Maximum Part-Products of Odd Palindromic Compositions, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.6.
E. F. Krause, Maximizing The Product of Summands, Mathematics Magazine, MAA Oct 1996, Vol. 69, no. 5 pp. 270-271.
MathPro, 20000 Problems Under the Sea, Problem 14856.Putnam 1979/A1.
J. W. Moon and L. Moser, On cliques in graphs, Israel J. Math. 3 (1965), 23-28.
Natasha Morrison and Alex Scott, Maximizing the number of induced cycles in a graph, Preprint, 2016. See f_2(n).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
F. Pluvinage, Developing problem solving experiences in practical action projects, The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos. 1 & 2, pp. 219-244.
D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.
J. T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, 18 (2015), #15.10.7.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, Integers (2020) Vol. 20A, Article #A13.
J. Scholes, 40th Putnam 1979 Problem A1.
J. Scholes, 18th IMO 1976 Problem 4.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.
V. Vatter, Maximal independent sets and separating covers, Amer. Math. Monthly, 118 (2011), 418-423.
Robert G. Wilson v, Letter to N. J. Sloane, circa 1991.
A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete Math., 15 (1976), 193-199.
Index to sequences related to the complexity of n
Index entries for linear recurrences with constant coefficients, signature (0,0,3).

See A007600 for a left inverse.
Cf. A000793, A009490, A034891, A062943, A007601, A062723, A069188, A087902.
Cf. array A064364, rightmost (nonvanishing) numbers in row n >= 2.
See A056240 and A288814 for the minimal numbers whose prime factors sums up to n.
A000792, A178715, A193286, A193455, A193456, and A193457 are closely related as paradigm shift sequences for (p = 0, ..., 5 respectively).
Cf. A202337 (subsequence).
Cf. A038754, A005245, A005520.

a000792 n = a000792_list !! n
a000792_list = 1 : f [1] where
   f xs = y : f (y:xs) where y = maximum $ zipWith (*) [1..] xs
-- Reinhard Zumkeller, Dec 17 2011

I:=[1,1,2,3,4]; [n le 5 select I[n] else 3*Self(n-3): n in [1..45]]; // Vincenzo Librandi, Apr 14 2015

A000792 := proc(n)
    m := floor(n/3) ;
    if n mod 3 = 0 then
        3^m ;
    elif n mod 3 = 1 then
        4*3^(m-1) ;
    else
        2*3^m ;
    end if;
    floor(%) ;
end proc: # R. J. Mathar, May 26 2013

a[1] = 1; a[n_] := 4* 3^(1/3 *(n - 1) - 1) /; (Mod[n, 3] == 1 && n > 1); a[n_] := 2*3^(1/3*(n - 2)) /; Mod[n, 3] == 2; a[n_] := 3^(n/3) /; Mod[n, 3] == 0; Table[a[n], {n, 0, 40}]
CoefficientList[Series[(1 + x + 2x^2 + x^4)/(1 - 3x^3), {x, 0, 50}], x] (* Harvey P. Dale, May 01 2011 *)
f[n_] := Max[ Times @@@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]]; f[1] = 1; Array[f, 43, 0] (* Robert G. Wilson v, Jul 31 2012 *)
a[ n_] := If[ n < 2, Boole[ n > -1], 2^Mod[-n, 3] 3^(Quotient[ n - 1, 3] + Mod[n - 1, 3] - 1)]; (* Michael Somos, Jan 23 2014 *)
Join[{1, 1}, LinearRecurrence[{0, 0, 3}, {2, 3, 4}, 50]] (* Jean-François Alcover, Jan 08 2019 *)
Join[{1,1},NestList[#+Divisors[#][[-2]]&,2,41]] (* James C. McMahon, Aug 09 2024 *)

{a(n) = floor( 3^(n - 4 - (n - 4) \ 3 * 2) * 2^( -n%3))}; /* Michael Somos, Jul 23 2002 */

lista(nn) = {print1("1, 1, "); print1(a=2, ", "); for (n=1, nn, a += a/divisors(a)[2]; print1(a, ", "););} \\ Michel Marcus, Apr 14 2015

A000792(n)=if(n>1,3^((n-2)\3)*(2+(n-2)%3),1) \\ M. F. Hasler, Jan 19 2019

G.f.: (1 + x + 2*x^2 + x^4)/(1 - 3*x^3). - Simon Plouffe in his 1992 dissertation.
a(3n) = 3^n; a(3*n+1) = 4*3^(n-1) for n > 0; a(3*n+2) = 2*3^n.
a(n) = 3*a(n-3) if n > 4. - Henry Bottomley, Nov 29 2001
a(n) = n if n <= 2, otherwise a(n-1) + Max{gcd(a(i), a(j)) | 0 < i < j < n}. - Reinhard Zumkeller, Feb 08 2002
A007600(a(n)) = n; Andrew Chi-Chih Yao attributes this observation to D. E. Muller. - Vincent Vatter, Apr 24 2006
a(n) = 3^(n - 2 - 2*floor((n - 1)/3))*2^(2 - (n - 1) mod 3) for n > 1. - Hieronymus Fischer, Nov 11 2007
From Kiyoshi Akima (k_akima(AT)hotmail.com), Aug 31 2009: (Start)
a(n) = 3^floor(n/3)/(1 - (n mod 3)/4), n > 1.
a(n) = 3^(floor((n - 2)/3))*(2 + ((n - 2) mod 3)), n > 1. (End)
a(n) = (2^b)*3^(C - (b + d))*(4^d), n > 1, where C = floor((n + 1)/3), b = max(0, ((n + 1) mod 3) - 1), d = max(0, 1 - ((n + 1) mod 3)). - Jonathan T. Rowell, Jul 26 2011
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x / (1 + x / (1 + x^2 / (1 + x))))))). - Michael Somos, May 12 2012
3*a(n) = 2*a(n+1) if n > 1 and n is not divisible by 3. - Michael Somos, Jan 23 2014
a(n) = a(n-1) + largest proper divisor of a(n-1), n > 2. - Ivan Neretin, Apr 13 2015
a(n) = max{a(i)*a(n-i) : 0 < i < n} for n >= 4. - Jianing Song, Feb 15 2020
a(n+1) = a(n) + A038754(floor( (2*(n-1) + 1)/3 )), for n > 1. - Thomas Scheuerle, Oct 27 2022

More terms and better description from Therese Biedl (biedl(AT)uwaterloo.ca), Jan 19 2000

A062923 Numbers k that, when expressed in base 4 and then interpreted in base 8, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 12, 16, 30, 32, 48, 64, 120, 128, 166, 192, 256, 278, 480, 512, 664, 765, 768, 1024, 1112, 1390, 1682, 1803, 1920, 2048, 2426, 2656, 3060, 3072, 4096, 4365, 4448, 5446, 5560, 6728, 7212, 7441, 7680, 8192, 9704, 9945, 10624, 12240, 12288, 16384, 17460

Offset: 1

Erich Friedman, Jul 21 2001

			8 in base 4 is 20, which interpreted in base 8 is 16 = 2*8.

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062925, A062928, A062929, A062930, A062931, A062934, A062937, A062939, A062942, A062943, A062944.

Join[{0},Select[Range[18000],Mod[FromDigits[IntegerDigits[#,4],8],#]==0&]] (* Harvey P. Dale, Feb 10 2024 *)

Offset changed to 1 and more terms from Georg Fischer, Mar 13 2023

A062925 Numbers k that, when expressed in base 4 and then interpreted in base 9, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 15, 25, 75, 100, 125, 355, 435, 500, 1775, 2415, 3675, 5825, 9660, 14700, 17074, 20786, 22382, 23300, 27300, 79716, 83144, 87087, 97860, 103930, 125460, 172105, 331275, 332576, 348348, 415720, 1325100, 1330304, 1531980

Offset: 1

Erich Friedman, Jul 21 2001

			5 in base 4 is 11, which interpreted in base 9 is 10 = 2*5.

Georg Fischer, Table of n, a(n) for n = 1..53 [First 45 terms from Harry J. Smith]

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062928, A062929, A062930, A062931, A062934, A062937, A062939, A062942, A062943, A062944.

Join[{0},Select[Range[1540000],Divisible[FromDigits[IntegerDigits[#,4],9],#]&]] (* Harvey P. Dale, Sep 03 2021 *)

select(n->n==0 || fromdigits(digits(n,4), 9) % n == 0, [0..100000]) \\ Andrew Howroyd, Jun 28 2018

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 by Georg Fischer, Mar 13 2023

A062928 Numbers k that, when expressed in base 5 and then interpreted in base 6, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 697, 704, 705, 764, 765, 1469, 1470, 1477, 1537, 2242, 2309, 2310, 2377, 3074, 3082, 590567, 591229, 595982, 2361731, 6900704, 7111031, 11808655, 34503520, 35555155, 65205900, 70204260, 70854060

Offset: 1

Erich Friedman, Jul 21 2001

			704 in base 5 is 10304, which interpreted in base 6 is 1408 = 2*704.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062929, A062930, A062931, A062934, A062937, A062939, A062942, A062943, A062944.

Join[{0},Select[Range[12000000],Divisible[FromDigits[ IntegerDigits[ #,5],6],#]&]] (* Harvey P. Dale, Jul 20 2014 *)

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 and a(28)-a(32) from Georg Fischer, Mar 13 2023

A062944 Numbers k that, when expressed in base 7 and then interpreted in base 10, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 51, 102, 105, 153, 156, 207, 210, 258, 261, 312, 315, 2589, 2590, 2601, 2753, 5354, 5355, 5506, 8108, 8259, 8260, 10861, 11013, 11025, 13614, 13615, 13626, 13766, 13778, 16379, 16380, 16531, 33045

Offset: 1

Erich Friedman, Jul 21 2001

Zero followed by A032549. [From R. J. Mathar, Oct 02 2008]

There are only 47 terms up to 10 million, the largest of which is 7581525. - Harvey P. Dale, Jul 09 2016

			51 in base 7 is 102, which interpreted in base 10 is 102=2*51

Georg Fischer, Table of n, a(n) for n = 1..64 [First 47 terms from Harvey P. Dale]

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062929, A062930, A062931, A062934, A062937, A062939, A062942, A062943.

Join[{0},Select[Range[35000],Divisible[FromDigits[IntegerDigits[ #,7]], #]&]] (* Harvey P. Dale, Jul 09 2016 *)

Offset changed to 1 by Georg Fischer, Mar 13 2023

A062929 Numbers k that, when expressed in base 5 and then interpreted in base 7, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 3640, 7863, 7894, 8186, 11830, 18200, 39315, 39470, 40930, 59150, 2521602, 3278326, 4167678, 13196470, 17870857, 17992485, 20838390, 36724952, 58516102, 73231902, 89354285, 89962425, 105564531, 200046408, 301764432, 446771425, 449812125, 633387186

Offset: 1

Erich Friedman, Jul 21 2001

			3640 in base 5 is 104030, which interpreted in base 7 is 18200=5*3640.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062930, A062931, A062934, A062937, A062939, A062942, A062943, A062944.

Select[Range[0,20*10^6],Divisible[FromDigits[IntegerDigits[#,5],7],#]&] (* Harvey P. Dale, Jul 18 2015 *)

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 and a(22)-a(33) from Georg Fischer, Mar 13 2023

A062930 Numbers k that, when expressed in base 5 and then interpreted in base 8, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 129, 130, 259, 260, 389, 390, 400, 519, 520, 529, 530, 720, 774, 780, 1560, 2340, 2394, 2400, 3120, 3354, 3380, 4680, 6734, 6760, 7017, 9360, 10397, 10400, 10657, 14037, 16770, 16900, 18720, 20280, 24654, 33670, 33800, 35085, 40560, 51985, 52000

Offset: 1

Erich Friedman, Jul 21 2001

			129 in base 5 is 1004, which interpreted in base 8 is 516 = 4*129.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062929, A062931, A062934, A062937, A062939, A062942, A062943, A062944.

Offset changed to 1 by Georg Fischer, Mar 13 2023

A062931 Numbers k that, when expressed in base 5 and then interpreted in base 9, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 28, 30, 58, 60, 88, 90, 118, 120, 168, 179, 180, 348, 359, 360, 840, 895, 900, 1740, 1795, 1800, 5370, 5400, 11726, 11984, 16200, 21142, 26850, 27000, 38340, 57574, 137128, 183960, 214207, 293628, 421560, 750288, 866700, 1043027, 1304280, 1468140

Offset: 1

Erich Friedman, Jul 21 2001

			28 in base 5 is 103, which interpreted in base 9 is 84 = 3*28.

Join[{0},Select[Range[11*10^5],Divisible[FromDigits[ IntegerDigits[ #,5],9],#]&]] (* Harvey P. Dale, Apr 27 2015 *)

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 by Georg Fischer, Mar 13 2023

A062934 Numbers k that, when expressed in base 6 and then interpreted in base 7, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 14027, 14028, 14103, 14112, 14867, 14876, 14951, 14952, 10099698, 10099846, 20210795, 30310661, 30311795, 50522741, 723825175, 142569349356, 482364801576, 486288289536, 972577541899, 1945641402768, 2474129673299

Offset: 1

Erich Friedman, Jul 21 2001

1.6 * 10^13 < a(28) <= 24335728984305. - Delbert L. Johnson, May 16 2024

			14027 in base 6 is 144535, which interpreted in base 7 is 28054 = 2*14027.

More terms from Naohiro Nomoto, Aug 06 2001
Offset changed to 1 and a(17)-a(20) from Georg Fischer, Mar 13 2023
a(21)-a(27) from Delbert L. Johnson, May 16 2024

A062937 Numbers k that, when expressed in base 6 and then interpreted in base 8, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1459, 1564, 3023, 3129, 3139, 3244, 4703, 4704, 4809, 6383, 6384, 8754, 9384, 18138, 18774, 18834, 19464, 28218, 28224, 28854, 38298, 38304, 79802, 326236, 2463293, 2628864, 14779758, 15773184, 22011172, 88678548, 94639104, 209918592

Offset: 1

Erich Friedman, Jul 21 2001

			1459 in base 6 is 10431, which interpreted in base 8 is 4377=3*1459.

Join[{0},Select[Range[21*10^7],Mod[FromDigits[IntegerDigits[#,6],8],#]==0&]] (* Harvey P. Dale, Apr 21 2024 *)

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 and a(33)-a(38) from Georg Fischer, Mar 13 2023

cp's OEIS Frontend

A000792 a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1.

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A062923 Numbers k that, when expressed in base 4 and then interpreted in base 8, give a multiple of k.

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A062925 Numbers k that, when expressed in base 4 and then interpreted in base 9, give a multiple of k.

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A062928 Numbers k that, when expressed in base 5 and then interpreted in base 6, give a multiple of k.

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A062944 Numbers k that, when expressed in base 7 and then interpreted in base 10, give a multiple of k.

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A062929 Numbers k that, when expressed in base 5 and then interpreted in base 7, give a multiple of k.

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A062930 Numbers k that, when expressed in base 5 and then interpreted in base 8, give a multiple of k.

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A062931 Numbers k that, when expressed in base 5 and then interpreted in base 9, give a multiple of k.

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