A325135 -id:A325135 - cp's OEIS frontend

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 4, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 5, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 5, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

a(n) tells how many iterations of A252463 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A005940 and A163511.

Similarly for A253553 in trees A253563 and A253565. - Antti Karttunen, Apr 14 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
and largest hook
  o o o o o o
  o
  o
both of which have size 8, so a(715) = 8.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000040, A000079, A001221, A001222, A005940, A029837, A061395, A156552 (A005941), A163511, A243071, A252461, A252463, A252735, A252736, A252759, A253553, A253563, A253565, A297113, A297155, A297167, A324870, A324872.
Right edge of irregular triangle A265146.
Cf. also A246348.
Cf. A052126, A056239, A093641, A112798, A257990, A325133, A325134, A325135.

Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]]-1,{n,100}] (* Gus Wiseman, Apr 02 2019 *)

A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = (bigomega(n) + A061395(n) - 1); \\ Antti Karttunen, Apr 14 2019

from sympy import primepi, primeomega, primefactors
def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023

a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
a(n) = A029837(1+A243071(n)). [a(n) = binary width of terms of A243071.]
a(n) = A029837(A005941(n)) = A029837(1+A156552(n)). [Also binary width of terms of A156552.]
Other identities. For all n >= 1:
a(A000040(n)) = n.
a(A001248(n)) = n+1.
a(A030078(n)) = n+2.
And in general, a(prime(n)^k) = n+k-1.
a(A000079(n)) = n. [I.e., a(2^n) = n.]
For all n >= 2:
a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).
a(n) = A325134(n) - 1. - Gus Wiseman, Apr 02 2019
From Antti Karttunen, Apr 14 2019: (Start)
a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
a(n) = A001221(n) + A297167(n) = A297113(n) + A297155(n).
(End).

A325134 a(1) = 1; a(n) = number of prime factors of n counted with multiplicity plus the largest prime index of n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 7, 6, 5, 5, 8, 5, 9, 6, 6, 7, 10, 6, 5, 8, 5, 7, 11, 6, 12, 6, 7, 9, 6, 6, 13, 10, 8, 7, 14, 7, 15, 8, 6, 11, 16, 7, 6, 6, 9, 9, 17, 6, 7, 8, 10, 12, 18, 7, 19, 13, 7, 7, 8, 8, 20, 10, 11, 7, 21, 7, 22, 14, 6, 11, 7, 9, 23

Offset: 1

Gus Wiseman, Apr 02 2019

nonn

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

Also one plus the size of the largest hook contained in the Young diagram of the integer partition with Heinz number n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Cf. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325135.

with(numtheory):
a:= n-> `if`(n=1, 1, bigomega(n)+pi(max(factorset(n)[]))):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 03 2019

Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]],{n,100}]

a(n) = A001222(n) + A061395(n).

a(n) = A252464(n) + 1.

A188674 Stack polyominoes with square core.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 4, 5, 7, 9, 13, 17, 24, 31, 42, 54, 71, 90, 117, 147, 188, 236, 298, 371, 466, 576, 716, 882, 1088, 1331, 1633, 1987, 2422, 2935, 3557, 4290, 5177, 6216, 7465, 8932, 10682, 12731, 15169, 18016, 21387, 25321, 29955, 35353, 41696, 49063, 57689, 67698, 79375, 92896, 108633, 126817, 147922, 172272

Offset: 0

Emanuele Munarini, Apr 08 2011

nonn

a(n) is the number of stack polyominoes of area n with square core.
The core of stack is the set of all maximal columns.
The core is a square when the number of columns is equal to their height.
Equivalently, a(n) is the number of unimodal compositions of n, where the number of the parts of maximum value equal the maximum value itself. For instance, for n = 10, we have the following stacks:
(1,3,3,3), (3,3,3,1), (1,1,1,1,1,1,2,2), (1,1,1,1,1,2,2,1), (1,1,1,1,2,2,1,1), (1,1,1,2,2,1,1,1), (1,1,2,2,1,1,1,1), (1,2,2,1,1,1,1,1), (2,2,1,1,1,1,1,1).
From Gus Wiseman, Apr 06 2019 and May 21 2022: (Start)
Also the number of integer partitions of n with final part in their inner lining partition equal to 1, where the k-th part of the inner lining partition of a partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. For example, the a(4) = 1 through a(10) = 9 partitions are:
  (22)  (32)   (42)    (52)     (62)      (72)       (82)
        (221)  (321)   (421)    (521)     (333)      (433)
               (2211)  (3211)   (4211)    (621)      (721)
                       (22111)  (32111)   (5211)     (3331)
                                (221111)  (42111)    (6211)
                                          (321111)   (52111)
                                          (2211111)  (421111)
                                                     (3211111)
                                                     (22111111)
Also partitions that have a fixed point and a conjugate fixed point, ranked by A353317. The strict case is A352829. For example, the a(0) = 0 through a(9) = 7 partitions are:
  ()  .  .  (21)  (31)   (41)    (51)     (61)      (71)
                  (211)  (311)   (411)    (511)     (332)
                         (2111)  (3111)   (4111)    (611)
                                 (21111)  (31111)   (5111)
                                          (211111)  (41111)
                                                    (311111)
                                                    (2111111)
Also partitions of n + 1 without a fixed point or conjugate fixed point.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.

Cf. A001523 (stacks).
Cf. A006918, A252464, A325135, A325163, A325165.
Positive crank: A001522, ranked by A352874.
Zero crank: A064410, ranked by A342192.
Nonnegative crank: A064428, ranked by A352873.
Fixed point but no conjugate fixed point: A118199, ranked by A353316.
A000041 counts partitions, strict A000009.
A002467 counts permutations with a fixed point, complement A000166.
A115720/A115994 count partitions by Durfee square, rank statistic A257990.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A000700, A088902, A114088, A300788, A330644, A352828, A352829, A352832.

a[n_]:=CoefficientList[Series[1+Sum[x^((k+1)^2)/Product[(1-x^i)^2,{i,1,k}],{k,0,n}],{x,0,n}],x]
(* second program *)
pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
Table[Length[Select[IntegerPartitions[n],pml[#]=={1}&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

G.f.: 1 + sum(k>=0, x^((k+1)^2)/((1-x)^2*(1-x^2)^2*...*(1-x^k)^2)).

A325163 Heinz number of the inner lining partition of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 5, 10, 7, 11, 7, 13, 11, 14, 7, 17, 14, 19, 11, 22, 13, 23, 11, 21, 17, 21, 13, 29, 22, 31, 11, 26, 19, 33, 22, 37, 23, 34, 13, 41, 26, 43, 17, 33, 29, 47, 13, 55, 33, 38, 19, 53, 33, 39, 17, 46, 31, 59, 26, 61, 37, 39, 13, 51, 34, 67, 23

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has diagonal distances
  3 3 3 2 1 1
  3 2 2 2 1
  2 2 1 1 1
  1 1 1
so the inner lining partition is (9,6,4), which has Heinz number 2093, so a(7865) = 2093.

Cf. A052126, A056239, A064989, A065770, A093641, A112798, A188674, A252464, A257990, A297113, A325133, A325135, A325164, A325167, A325169.

Table[Times@@Prime/@(-Differences[Total/@Take[FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,Reverse[Flatten[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{1,-2}]]),{n,100}]

A325166 Size of the internal portion of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 3, 0, 1, 3, 1, 0, 1, 3, 3, 2, 1, 0, 3, 3, 1, 2, 1, 0, 3, 0, 1, 3, 0, 3, 3, 0, 1, 2, 4, 0, 2, 0, 1, 4, 1, 4, 3, 0, 1, 3, 1, 0, 3, 3, 1, 2, 1, 0, 4, 4, 1, 2, 1, 3, 1, 0, 4, 3, 3, 0, 3, 0, 1, 5

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.

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

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with internal portion
  o o o o o
  o o o o
  o o o
of size 12, so a(7865) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Positions of zeros are A174090.

Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.

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

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325166(n) = (A056239(n) - A061395(n) - bigomega(n) + omega(n)); \\ Antti Karttunen, Apr 14 2019

a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).

a(n) = A056239(n) - A297113(n).

More terms from Antti Karttunen, Apr 14 2019

A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121

Offset: 0

Gus Wiseman, Apr 05 2019

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside it.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (22)   (32)    (33)     (43)      (44)       (54)        (55)
        (31)   (41)    (42)     (52)      (53)       (63)        (64)
        (211)  (221)   (51)     (61)      (62)       (72)        (73)
               (311)   (222)    (511)     (71)       (81)        (82)
               (2111)  (411)    (2221)    (611)      (711)       (91)
                       (2211)   (4111)    (2222)     (6111)      (811)
                       (3111)   (22111)   (5111)     (22221)     (7111)
                       (21111)  (31111)   (22211)    (51111)     (22222)
                                (211111)  (41111)    (222111)    (61111)
                                          (221111)   (411111)    (222211)
                                          (311111)   (2211111)   (511111)
                                          (2111111)  (3111111)   (2221111)
                                                     (21111111)  (4111111)
                                                                 (22111111)
                                                                 (31111111)
                                                                 (211111111)

Colin Barker, Table of n, a(n) for n = 0..1000
N. Guru Sharan, Rook decomposition of the Partition function, arXiv:2507.20260 [math.CO], 2025. See p. 5.
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]

concat([0,0,0], Vec(x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)) + O(x^80))) \\ Colin Barker, Apr 08 2019

From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.
a(n) = 2*n - 4 for n>4 and even.
a(n) = 2*n - 5 for n>4 and odd.
(End)

A325133 Heinz number of the integer partition obtained by removing the inner lining, or, equivalently, the largest hook, of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 5, 3, 2, 1, 1, 4, 3, 1, 2, 1, 1, 2, 1, 1, 4, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 1, 6, 1, 5, 2, 1, 1, 8, 1, 1, 2, 3, 1, 2, 1, 1, 4, 5, 1, 2, 1, 3, 1, 1, 5, 4, 3, 1, 2, 1, 1, 6

Offset: 1

Gus Wiseman, Apr 02 2019

nonn

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

			The partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
or largest hook
  o o o o o o
  o
  o
both of which have complement
  o o o o
  o o
which is the partition (4,2) with Heinz number 21, so a(715) = 21.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Positions of ones are A093641 (Heinz numbers of hooks). The number of iterations required to reach 1 starting with n is A257990(n).

Cf. A000720, A001222, A046660, A052126, A056239, A061395, A064989, A112798, A243055, A252464, A325134, A325135.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,1,Times@@Prime/@DeleteCases[Most[primeMS[n]]-1,0]],{n,100}]

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A325133(n) = A052126(A064989(n)); \\ Antti Karttunen, Apr 14 2019

a(n) = A064989(A052126(n)) = A052126(A064989(n)).

More terms from Antti Karttunen, Apr 14 2019

cp's OEIS Frontend

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

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A325134 a(1) = 1; a(n) = number of prime factors of n counted with multiplicity plus the largest prime index of n.

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A188674 Stack polyominoes with square core.

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A325163 Heinz number of the inner lining partition of the integer partition with Heinz number n.

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A325166 Size of the internal portion of the integer partition with Heinz number n.

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A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.

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A325133 Heinz number of the integer partition obtained by removing the inner lining, or, equivalently, the largest hook, of the integer partition with Heinz number n.

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