A265283 -id:A265283 - cp's OEIS frontend

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.

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}]

A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 21, 22, 24, 25, 26, 27, 33, 34, 35, 36, 38, 39, 46, 48, 49, 51, 54, 55, 57, 58, 62, 65, 69, 72, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 108, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 144, 145, 146

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The enumeration of these partitions by sum is given by A265283.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}

Cf. A056239, A061395, A093641, A112798, A252464, A257541, A263297, A265283, A325224, A325225, A325227, A325230, A325232.

Select[Range[300],Min[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==2&]

A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

Original entry on oeis.org

6, 10, 12, 14, 22, 24, 26, 34, 38, 46, 48, 58, 62, 74, 82, 86, 94, 96, 106, 118, 122, 134, 142, 146, 158, 166, 178, 192, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 384, 386, 394, 398, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Also numbers n such that the sum of prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n is 1. 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, and their sum is A056239.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   48: {1,1,1,1,2}
   58: {1,10}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
   96: {1,1,1,1,1,2}
  106: {1,16}
  118: {1,17}

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Positions of 1's in A325223.

Cf. A001222, A056239, A060687, A061395, A093641, A112798, A174090, A257541, A265283, A325224, A325225, A325227, A325232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[primeMS[#]]-Max[Length[primeMS[#]],Max[primeMS[#]]]==1&]

from sympy import isprime
A325231_list = [n for n in range(6,10**6) if ((not n % 2) and isprime(n//2)) or (bin(n)[2:4] == '11' and bin(n).count('1') == 2)] # Chai Wah Wu, Apr 16 2019

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}]

A267460 Number of OFF (white) cells in the n-th iteration of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

0, 2, 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, Jan 15 2016

nonn

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

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267423.

Cf. A052928, A265283, A213222.

rule=133; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Length[catri[[k]]]-nbc[[k]],{k,1,rows}] (* Number of White cells in stage n *)

Conjectures from Colin Barker, Jan 16 2016: (Start)
a(n) = (2*n-(-1)^n+5)/2 for n>1.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>4.
G.f.: 2*x*(1+x-x^3) / ((1-x)^2*(1+x)).
(End)

cp's OEIS Frontend

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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A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

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A325231 Numbers of the form 2 * p or 3 * 2^k, p prime, k > 1.

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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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Mathematica

A267460 Number of OFF (white) cells in the n-th iteration of the "Rule 133" elementary cellular automaton starting with a single ON (black) cell.

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