A325225 -id:A325225 - cp's OEIS frontend

A331297 Lexicographically earliest infinite sequence such that a(i) = a(j) => A263297(i) = A263297(j) and A325225(i) = A325225(j) for all i, j.

1, 2, 3, 3, 4, 5, 6, 4, 5, 7, 8, 7, 9, 10, 7, 6, 11, 7, 12, 13, 10, 14, 15, 10, 7, 16, 7, 17, 18, 13, 19, 8, 14, 20, 10, 10, 21, 22, 16, 17, 23, 17, 24, 25, 13, 26, 27, 14, 10, 13, 20, 28, 29, 10, 14, 30, 22, 31, 32, 17, 33, 34, 17, 9, 16, 25, 35, 36, 26, 17, 37, 14, 38, 39, 13, 40, 14, 28, 41, 25, 10, 42, 43, 30, 20, 44, 31, 45, 46, 17, 16, 47, 34, 48, 22, 16, 49, 17, 25, 17

Offset: 1

Antti Karttunen, Jan 18 2020

nonn

Restricted growth sequence transform of the unordered pair [A001222(n), A061395(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A061395, A263297, A325225, A331170, A331298, A331299.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
Aux331297(n) = { my(a=bigomega(n),b=A061395(n)); [min(a,b),max(a,b)]; };
Aux331297(n) = Set([bigomega(n),A061395(n)]); \\ Alternatively.
v331297 = rgs_transform(vector(up_to, n, Aux331297(n)));
A331297(n) = v331297[n];

For all i, j:
  A331170(i) = A331170(j) => a(i) = a(j),
  A331298(i) = A331298(j) => a(i) = a(j),
  A331299(i) = A331299(j) => a(i) = a(j),
  a(i) = a(j) => A326846(i) = A326846(j).

A263297 The greater of bigomega(n) and maximal prime index in the prime factorization of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 2, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 3, 4, 5, 9, 4, 3, 6, 3, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 4, 13, 4, 14, 5, 3, 9, 15, 5, 4, 3, 7, 6, 16, 4, 5, 4, 8, 10, 17, 4, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 5, 21, 12, 3, 8, 5, 6, 22, 5

Offset: 1

Alexei Kourbatov, Oct 13 2015

nonn

Also: minimal m such that n is the product of at most m primes not exceeding prime(m). (Here the primes do not need to be distinct; cf. A263323.)
By convention, a(1)=0, as 1 is the empty product.
Those n with a(n) <= k fill a k-simplex whose 1-sides span from 0 to k.
For a similar construction with distinct primes (hypercube), see A263323.
Each nonnegative integer occurs finitely often; in particular:
- Terms a(n) <= k occur A000984(k) = (2*k)!/(k!)^2 times.
- The term a(n) = 0 occurs exactly once.
- The term a(n) = k > 0 occurs exactly A051924(k) = (3*k-2)*C(k-1) times, where C(k)=A000108(k) are Catalan numbers.

			a(6)=2 because 6 is the product of 2 primes (2*3), each not exceeding prime(2)=3.
a(8)=3 because 8 is the product of 3 primes (2*2*2), each not exceeding prime(3)=5.
a(11)=5 because 11 is prime(5).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000108, A000984, A001222, A051924, A061395, A263323, A325225, A331296 (ordinal transform), A331297.

seq(`if`(n=1,0,max(pi(max(factorset(n))),bigomega(n))),n=1..80); # Peter Luschny, Oct 15 2015

f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Plus @@ Last /@ FactorInteger@n]; Array[f, 80]

a(n)=if(n<2, return(0)); my(f=factor(n)); max(vecsum(f[,2]), primepi(f[#f~,1])) \\ Charles R Greathouse IV, Oct 13 2015

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

a(n) <= pi(n), with equality when n is 1 or prime.

A257541 The rank of the partition with Heinz number n.

Original entry on oeis.org

0, 1, -1, 2, 0, 3, -2, 0, 1, 4, -1, 5, 2, 1, -3, 6, -1, 7, 0, 2, 3, 8, -2, 1, 4, -1, 1, 9, 0, 10, -4, 3, 5, 2, -2, 11, 6, 4, -1, 12, 1, 13, 2, 0, 7, 14, -3, 2, 0, 5, 3, 15, -2, 3, 0, 6, 8, 16, -1, 17, 9, 1, -5, 4, 2, 18, 4, 7, 1, 19, -3, 20, 10, 0, 5

Offset: 2

Emeric Deutsch, May 09 2015

sign

The rank of a partition p is the largest part of p minus the number of parts of p.
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,1] the Heinz number is 2*2*2 = 8. Its rank is 1 - 3 = -2 = a(8). - Emeric Deutsch, Jun 09 2015
This is the Dyson rank (St000145), which is different from the Frobenius rank (St000183); see the FindStat links. - Gus Wiseman, Apr 13 2019

			a(24) = -2. Indeed, the partition corresponding to the Heinz number 24 = 2*2*2*3 is [1,1,1,2]; consequently, a(24)= 2 - 4 = -2.

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.

Alois P. Heinz, Table of n, a(n) for n = 2..10000
FindStat, St000145: The Dyson rank of a partition
FindStat, St000183: The side length of the Durfee square of an integer partition

Positions of 0's are A106529. Positions of 1's are A325233. Positions of -1's are A325234.
Cf. A000720, A001222, A061395, A215366.
Cf. A046660, A047993, A056239, A093641, A101198, A112798, A174090, A243055, A257990, A263297, A325134, A325178, A325225, A325235.

with(numtheory): a := proc(n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: seq(a(n), n = 2 .. 120);

Table[PrimePi@ FactorInteger[n][[-1, 1]] - PrimeOmega@ n, {n, 2, 76}] (* Michael De Vlieger, May 09 2015 *)

a(n) = q(largest prime factor of n) - bigomega(n), where q(p) is defined by q-th prime = p while bigomega(n) is the number of prime factors of n, including multiplicities.

A325233 Heinz numbers of integer partitions with Dyson rank 1.

Original entry on oeis.org

3, 10, 15, 25, 28, 42, 63, 70, 88, 98, 105, 132, 147, 175, 198, 208, 220, 245, 297, 308, 312, 330, 343, 462, 468, 484, 495, 520, 544, 550, 693, 702, 726, 728, 770, 780, 816, 825, 1053, 1078, 1089, 1092, 1144, 1155, 1170, 1210, 1216, 1224, 1300, 1352, 1360

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index is one greater than their number of prime indices counted with multiplicity.

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:
     3: {2}
    10: {1,3}
    15: {2,3}
    25: {3,3}
    28: {1,1,4}
    42: {1,2,4}
    63: {2,2,4}
    70: {1,3,4}
    88: {1,1,1,5}
    98: {1,4,4}
   105: {2,3,4}
   132: {1,1,2,5}
   147: {2,4,4}
   175: {3,3,4}
   198: {1,2,2,5}
   208: {1,1,1,1,6}
   220: {1,1,3,5}
   245: {3,4,4}
   297: {2,2,2,5}
   308: {1,1,4,5}

FindStat, St000145: The Dyson rank of a partition

Positions of 1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325234, A325235.

Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==1&]

A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 3, 3, 6, 3, 7, 2, 4, 4, 8, 3, 4, 5, 4, 3, 9, 3, 10, 4, 5, 6, 5, 4, 11, 7, 6, 3, 12, 4, 13, 4, 4, 8, 14, 4, 6, 4, 7, 5, 15, 5, 6, 3, 8, 9, 16, 4, 17, 10, 5, 5, 7, 5, 18, 6, 9, 5, 19, 5, 20, 11, 5, 7, 7, 6, 21, 4, 6, 12

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is smaller, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - min(4,5) = 4.

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

The number of times k appears in the sequence is A325232(k).

Cf. A001222, A052126, A056239, A061395, A064989, A065770, A112798, A174090, A257990, A263297, A325134, A325169, A325223, A325225, A325227.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Total[primeMS[n]]-Min[Length[primeMS[n]],Max[primeMS[n]]]],{n,100}]

a(n) = A056239(n) - min(A001222(n), A061395(n)) = A056239(n) - A325225(n).

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}]

A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.

Original entry on oeis.org

2, 3, 6, 10, 18, 27, 44, 64, 97, 138, 200, 276, 390, 528, 724, 968, 1301, 1712, 2266, 2946, 3842, 4947, 6372, 8122, 10362, 13094, 16544, 20754, 26010, 32392, 40308, 49876, 61648, 75845, 93178, 114006, 139308, 169586, 206158, 249814, 302267, 364664, 439330

Offset: 0

Gus Wiseman, Apr 13 2019

nonn

			The a(0) = 1 through a(4) = 18 partitions:
  ()   (2)   (3)    (4)     (5)
  (1)  (11)  (22)   (32)    (33)
       (21)  (31)   (41)    (42)
             (111)  (221)   (51)
             (211)  (321)   (222)
             (311)  (411)   (322)
                    (1111)  (331)
                    (2111)  (421)
                    (3111)  (511)
                    (4111)  (2211)
                            (3211)
                            (4211)
                            (5111)
                            (11111)
                            (21111)
                            (31111)
                            (41111)
                            (51111)

Giovanni Resta, Table of n, a(n) for n = 0..75
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Number of times n appears in A325224.

Cf. A051924, A257541, A263297, A325193, A325194, A325224, A325225, A325227.

nn=30;
mindif[ptn_]:=If[ptn=={},0,Total[ptn]-Min[Length[ptn],Max[ptn]]];
allip=Array[IntegerPartitions,2*nn+2,0,Join];
Table[Length[Select[allip,mindif[#]==n&]],{n,0,nn}]

For n > 0, a(n) = Sum_{k > 0} A325227(n + k, k).

More terms from Giovanni Resta, Apr 15 2019

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)

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&]

A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

Original entry on oeis.org

6, 10, 14, 15, 18, 21, 22, 26, 33, 34, 35, 38, 39, 46, 50, 51, 54, 55, 57, 58, 62, 65, 69, 74, 75, 77, 82, 85, 86, 87, 91, 93, 94, 95, 98, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 147, 155, 158, 159, 161, 162, 166, 177, 178

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   54: {1,2,2,2}
   55: {3,5}
   57: {2,8}
   58: {1,10}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's in A325226.

Cf. A056239, A093641, A112798, A174090, A257541, A307517, A325223, A325224, A325225, A325231.

filter:= proc(n) local F;
   F:= sort(ifactors(n)[2],(a,b)-> a[1]Robert Israel, Apr 14 2019

Select[Range[100],PrimeOmega[#/Power@@FactorInteger[#][[-1]]]==1&]

from sympy import factorint
A325230_list = [n for n, m in ((n, factorint(n)) for n in range(2,10**6)) if len(m) == 2 and m[min(m)] == 1] # Chai Wah Wu, Apr 16 2019

cp's OEIS Frontend

A331297 Lexicographically earliest infinite sequence such that a(i) = a(j) => A263297(i) = A263297(j) and A325225(i) = A325225(j) for all i, j.

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A263297 The greater of bigomega(n) and maximal prime index in the prime factorization of n.

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A257541 The rank of the partition with Heinz number n.

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A325233 Heinz numbers of integer partitions with Dyson rank 1.

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A325224 Sum of prime indices of n minus the 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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A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.

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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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