A325229 -id:A325229 - cp's OEIS frontend

A325225 Lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 4, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, 3, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 1, 3, 2, 3, 2, 2, 2, 2, 1, 3, 3, 3, 1, 3, 1, 4, 3

Offset: 1

Gus Wiseman, Apr 12 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			88 has 4 prime indices {1,1,1,5}, the maximum of which is 5, so a(88) = min(4,5) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Positions of 1's are A174090. Positions of 2's are A325229.

Cf. A001222, A056239, A061395, A065770, A093641, A112798, A252464, A257541, A257990, A263297, A325169, A325224, A325227, A325232.

Table[Min[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]],{n,100}]

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325225(n) = min(bigomega(n), A061395(n)); \\ Antti Karttunen, Apr 14 2019

a(n) = min(A001222(n), A061395(n)).

More terms from Antti Karttunen, Apr 14 2019

A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 4, 1, 0, 0, 0, 2, 6, 3, 0, 0, 0, 0, 2, 6, 6, 1, 0, 0, 0, 0, 2, 8, 9, 3, 0, 0, 0, 0, 0, 2, 8, 13, 6, 1, 0, 0, 0, 0, 0, 2, 10, 16, 11, 3, 0, 0, 0, 0, 0, 0, 2, 10, 20, 17, 6, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 12 2019

			Triangle begins:
  1
  2  0
  2  1  0
  2  3  0  0
  2  4  1  0  0
  2  6  3  0  0  0
  2  6  6  1  0  0  0
  2  8  9  3  0  0  0  0
  2  8 13  6  1  0  0  0  0
  2 10 16 11  3  0  0  0  0  0
  2 10 20 17  6  1  0  0  0  0  0
  2 12 24 25 11  3  0  0  0  0  0  0
  2 12 28 33 19  6  1  0  0  0  0  0  0
  2 14 32 44 29 11  3  0  0  0  0  0  0  0
  2 14 38 53 43 19  6  1  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)          (54)        (333)      (4221)    (51111)
  (111111111)  (63)        (432)      (4311)
               (72)        (441)      (5211)
               (81)        (522)      (6111)
               (22221)     (531)      (42111)
               (222111)    (621)      (411111)
               (2211111)   (711)
               (21111111)  (3222)
                           (3321)
                           (32211)
                           (33111)
                           (321111)
                           (3111111)

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Column k = 2 is A265283. Column k = 3 is A325228.

Cf. A051924, A096771, A115720, A263297, A325188, A325189, A325192, A325193, A325194, A325224, A325225, A325229, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==k&]],{n,15},{k,n}]

A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68

Offset: 0

Robert Price, Dec 06 2015

From Gus Wiseman, Apr 13 2019: (Start)
Also the number of integer partitions of n + 3 such that lesser of the maximum part and the number of parts is 2. The Heinz numbers of these partitions are given by A325229. For example, the a(0) = 1 through a(7) = 10 partitions are:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (2111)  (222)    (2221)    (71)       (81)        (82)
                       (2211)   (22111)   (2222)     (22221)     (91)
                       (21111)  (211111)  (22211)    (222111)    (22222)
                                          (221111)   (2211111)   (222211)
                                          (2111111)  (21111111)  (2221111)
                                                                 (22111111)
                                                                 (211111111)
(End)

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row:
                        1                          =  1
                      1 1 1                        =  3
                    1 1 . 1 1                      =  4
                  1 1 1 . 1 1 1                    =  6
                1 1 . 1 . 1 . 1 1                  =  6
              1 1 1 . 1 . 1 . 1 1 1                =  8
            1 1 . 1 . 1 . 1 . 1 . 1 1              =  8
          1 1 1 . 1 . 1 . 1 . 1 . 1 1 1            = 10
        1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1          = 10
      1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1        = 12
    1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1      = 12
  1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1    = 14
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1  = 14
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Column k = 2 of A325227.

Cf. A004526, A051924, A118102, A263297, A325193, A325225, A325228, A325229, A325232.

rule = 94; rows = 30; Table[Total[Table[Take[CellularAutomaton[rule, {{1},0},rows-1,{All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
Total /@ CellularAutomaton[94, {{1}, 0}, 65] (* Michael De Vlieger, Dec 14 2015 *)

Conjectures from Colin Barker, Dec 07 2015 and Apr 16 2019: (Start)
a(n) = (5-(-1)^n+2*n)/2 = A213222(n+3) for n>1.
a(n) = n+2 for n>1 and even.
a(n) = n+3 for n>1 and odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>2.
G.f.: (1+2*x-x^4) / ((1-x)^2*(1+x)).
(End)

A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 9, 13, 16, 20, 24, 28, 32, 38, 42, 48, 54, 60, 66, 74, 80, 88, 96, 104, 112, 122, 130, 140, 150, 160, 170, 182, 192, 204, 216, 228, 240, 254, 266, 280, 294, 308, 322, 338, 352, 368, 384, 400, 416, 434, 450, 468, 486, 504, 522, 542, 560

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The a(5) = 1 through a(10) = 16 partitions:
  (311)  (321)   (322)    (332)     (333)      (433)
         (411)   (331)    (422)     (432)      (442)
         (3111)  (421)    (431)     (441)      (532)
                 (511)    (521)     (522)      (541)
                 (3211)   (611)     (531)      (622)
                 (31111)  (3221)    (621)      (631)
                          (3311)    (711)      (721)
                          (32111)   (3222)     (811)
                          (311111)  (3321)     (3322)
                                    (32211)    (3331)
                                    (33111)    (32221)
                                    (321111)   (33211)
                                    (3111111)  (322111)
                                               (331111)
                                               (3211111)
                                               (31111111)

Column k = 3 of A325227.

Cf. A051924, A096771, A115720, A265283, A325224, A325225, A325229, A325231, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==3&]],{n,30}]

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A325225 Lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

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A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

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A265283 Number of ON (black) cells in the n-th iteration of the "Rule 94" elementary cellular automaton starting with a single ON (black) cell.

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A325228 Number of integer partitions of n such that the lesser of the maximum part and the number of parts is 3.

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