A325166 -id:A325166 - cp's OEIS frontend

A065770 Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

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

Offset: 1

Henry Bottomley, Nov 19 2001

nonn

It seems that a(n) <= A297113(n) for all n. Of the first 10000 positive natural numbers, 6454 are such that a(n) = A297113(n). - Antti Karttunen, Dec 31 2017

Also one plus the maximum number of unit steps East or South in the Young diagram of the integer partition with Heinz number n > 1, starting from the upper-left square and ending in a boundary square in the lower-right quadrant. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 06 2019

			a(50) = 4 since the cascade goes from 50 = 2^1 * 5^2 to 15 = 3^1 * 5^1 to 6 = 2^1 * 3^1 to 2 = 2^1 to 1.
From _Gus Wiseman_, Apr 06 2019: (Start)
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 longest path from (1,1) to (5,3) of length 6, so a(7865) = 7.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
FindStat, St000384: The maximal part of the shifted composition of an integer partition

Cf. A065769.
Differs from A297113 for the first time at n=20, where a(20) = 3, while A297113(20) = 4.
Cf. A001221, A001222, A052126, A056239, A064989, A112798, A174090, A257990, A325166, A325167, A325169.

Table[If[n==1,0,Max@@Total/@Position[PadRight[ConstantArray[1,#]&/@Sort[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],Greater]],1]-1],{n,100}] (* Gus Wiseman, Apr 06 2019 *)

(definec (A065770 n) (if (= 1 n) 0 (+ 1 (A065770 (A065769 n))))) ;; Antti Karttunen, Dec 31 2017

Inverse of primes, powers of 2 and primorials in sense that a(A000040(n))=n; a(A000079(n))=n; a(A002110(n))=n. If n>0: a(3^n)=n+1; a(2^n*3^k)=n+k; a(p(k)^n)=n+k-1; a(n!)=A022559(n).

a(1) = 0; and for n > 1, a(n) = 1 + A065769(n). - Antti Karttunen, Dec 31 2017

A325169 Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

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 minimum triangular partition contained inside the diagram.

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

FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Cf. A001221, A001222, A056239, A061395, A065770, A112798, A115994, A174090, A257990, A297113, A325166, A325167, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[otb[Reverse[primeMS[n]]],{n,100}]

A257990(n) <= a(n) <= 2 * A257990(n).

A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

From Gus Wiseman, Apr 06 2019: (Start)
Also the number of squares in the Young diagram of the integer partition with Heinz number n that are graph-distance 1 from the lower-right boundary, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (6,5,5,3) with Heinz number 7865 has diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with inner rim
            o
          o
        o o
  o o o
of size 7, so a(7867) = 7.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12721
Index entries for sequences computed from indices in prime factorization

One more than A297167 (after the initial term).
Cf. A001511, A008683, A033265, A064989, A156552, A252463, A252464, A297112, A297155.
Cf. also A297157, A297161, A297162.
Cf. A052126, A065770, A112798, A115994, A174090, A325166, A325167, A325169.

Table[If[n==1,0,PrimePi[FactorInteger[n][[-1,1]]]+PrimeOmega[n]-PrimeNu[n]],{n,100}] (* Gus Wiseman, Apr 06 2019 *)

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)};
A297113(n) = if(n<=2,n-1,if(n%2,1+A297113(A064989(n)), !(n%4)+A297113(n/2)));

\\ More complex way, after Moebius transform:
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
A297113(n) = if(1==n,0,1+valuation(A297112(n),2));

;; With memoization-macro definec.
(definec (A297113 n) (cond ((<= n 2) (- n 1)) ((= 2 (modulo n 4)) (A297113 (/ n 2))) (else (+ 1 (A297113 (A252463 n))))))

a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)) .
For n > 1, a(n) = A001511(A297112(n)), where A297112(n) = Sum_{d|n} moebius(n/d)*A156552(d).
a(n) = A252464(n) - A297155(n).
For n > 1, a(n) = 1+A033265(A156552(n)) = 1+A297167(n) = A046660(n) + A061395(n). - Last two sums added by Antti Karttunen, Sep 02 2018
Other identities. For all n >= 1:
a(A000040(n)) = n. [Each n occurs for the first time at the n-th prime.]

A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

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

			The partition (3,3) has Heinz number 25 and diagram
  o o o
  o o o
containing maximal triangular partition
  o o
  o
and contained in minimal triangular partition
  o o o o
  o o o
  o o
  o
so a(25) = 4 - 2 = 2.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325183, A325188, A325189, A325191, A325196, A325197, A325199, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[otbmax[primeptn[n]]-otb[primeptn[n]],{n,100}]

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)

A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 10, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 14, 21, 22, 23, 14, 21, 26, 21, 22, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 22, 41, 42, 43, 26, 42, 46, 47, 22, 55, 42, 51, 34, 53, 35, 55, 26, 57, 58, 59, 42, 61, 62, 66, 13, 65, 66, 67

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k 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
with origin-to-boundary graph-distances
  4 4 4 3 2 1
  3 3 3 2 1
  2 2 2 1 1
  1 1 1
giving the origin-to-boundary partition (7,5,4,3) with Heinz number 6545, so a(7865) = 6545.

Eric Weisstein's World of Mathematics, Graph Distance.

The only terms appearing only once are the primorials A002110.
The union consists of all squarefree numbers A005117.
Cf. A000245, A056239, A065770, A112798, A174090, A297113.
Cf. A325166, A325167, A325169, A325184, A325188, A325189, A325195.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
Table[Times@@Prime/@If[n==1,{},-Differences[Map[Total,Drop[FixedPointList[corpos,ptnmat[primeptn[n]]],-1],2]]],{n,30}]

A325196 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

Original entry on oeis.org

3, 4, 9, 10, 12, 15, 18, 20, 42, 45, 50, 60, 63, 70, 75, 84, 90, 100, 105, 126, 140, 150, 294, 315, 330, 350, 420, 441, 462, 490, 495, 525, 550, 588, 630, 660, 693, 700, 735, 770, 825, 882, 924, 980, 990, 1050, 1100, 1155, 1386, 1470, 1540, 1650, 2730, 3234

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The enumeration of these partitions by sum is given by A325191.

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}

Gus Wiseman, Young diagrams for their first 60 terms.
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
FindStat, St000384: The maximal part of the shifted composition of an integer partition.

Cf. A060687, A065770, A256617, A325166, A325169, A325179, A325181, A325183, A325185, A325188, A325189, A325191, A325195, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[1000],otbmax[primeptn[#]]-otb[primeptn[#]]==1&]

A325165 Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4, 0, 2, 0, 0, 0, 5, 0, 3, 2, 0, 0, 0, 6, 0, 4, 4, 0, 0, 0, 0, 7, 0, 5, 6, 3, 0, 0, 0, 0, 8, 0, 7, 8, 6, 0, 0, 0, 0, 0, 9, 0, 9, 10, 9, 4, 0, 0, 0, 0, 0, 10, 0, 13, 12, 12, 8, 0, 0, 0, 0, 0, 0, 11

Offset: 0

Gus Wiseman, Apr 05 2019

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. For example, the partition (6,5,5,3) has diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has diagonal distances from the lower-right boundary equal to
  3 3 3 2 1 1
  3 2 2 2 1
  2 2 1 1 1
  1 1 1
so the inner lining sequence is (9,6,4) with last part 4, so (6,5,5,3) is counted under T(19,4).

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  0  3
  0  1  0  0  4
  0  2  0  0  0  5
  0  3  2  0  0  0  6
  0  4  4  0  0  0  0  7
  0  5  6  3  0  0  0  0  8
  0  7  8  6  0  0  0  0  0  9
  0  9 10  9  4  0  0  0  0  0 10
  0 13 12 12  8  0  0  0  0  0  0 11
  0 17 16 15 12  5  0  0  0  0  0  0 12
  0 24 20 18 16 10  0  0  0  0  0  0  0 13
  0 31 28 21 20 15  6  0  0  0  0  0  0  0 14
  0 42 36 27 24 20 12  0  0  0  0  0  0  0  0 15
  0 54 50 33 28 25 18  7  0  0  0  0  0  0  0  0 16
  0 71 64 45 32 30 24 14  0  0  0  0  0  0  0  0  0 17
  0 90 86 57 40 35 30 21  8  0  0  0  0  0  0  0  0  0 18
Row n = 9 counts the following partitions (empty columns not shown):
  (72)       (63)      (54)     (9)
  (333)      (522)     (432)    (81)
  (621)      (531)     (441)    (711)
  (5211)     (4221)    (3222)   (6111)
  (42111)    (4311)    (3321)   (51111)
  (321111)   (32211)   (22221)  (411111)
  (2211111)  (33111)            (3111111)
             (222111)           (21111111)
                                (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000041. Column k = 1 is A188674.

Cf. A006918, A115720, A115994, A117485, A252464, A257990, A325163, A325166, A325168.

pml[ptn_]:=If[ptn=={},{},FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,ptn][[-3]]];
Table[Length[Select[IntegerPartitions[n],Total[pml[#]]==k&]],{n,0,10},{k,0,n}]

T(n) = {my(v=Vec(1+sum(k=1, sqrtint(n), x^(k^2)/((1-y*x^k)*prod(j=1, k-1, 1 - x^j + O(x^(n+1-k^2))))^2))); vector(#v, i, Vecrev(v[i], -i))}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 19 2023

G.f.: A(x,y) = 1 + Sum_{k>=1} x^(k^2)/((1 - y*x^k) * Product_{j=1..k-1} (1 - x^j))^2. - Andrew Howroyd, Jan 19 2023

A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

Original entry on oeis.org

0, 0, 2, 0, 3, 3, 0, 4, 6, 4, 0, 5, 10, 10, 5, 0, 6, 15, 20, 15, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 28, 56, 70, 56, 28, 8, 0, 9, 36, 84, 126, 126, 84, 36, 9, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 11, 55, 165, 330, 462

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325196.

Under the Bulgarian solitaire step, these partitions form cycles of length >= 2. Length >= 2 means not the length=1 self-loop which occurs from the triangular partition when n is a triangular number. See A074909 for self-loops included. - Kevin Ryde, Sep 27 2019

			The a(2) = 2 through a(12) = 10 partitions (empty columns not shown):
  (2)   (22)   (32)   (322)   (332)   (432)   (4322)   (4332)
  (11)  (31)   (221)  (331)   (422)   (3321)  (4331)   (4422)
        (211)  (311)  (421)   (431)   (4221)  (4421)   (4431)
                      (3211)  (3221)  (4311)  (5321)   (5322)
                              (3311)          (43211)  (5331)
                              (4211)                   (5421)
                                                       (43221)
                                                       (43311)
                                                       (44211)
                                                       (53211)

FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
Eric Weisstein's World of Mathematics, Graph Distance

Column k=1 of A325200.

Cf. A060687, A065770, A071724, A256617, A325166, A325169, A325178, A325179, A325181, A325187, A325188, A325189, A325195, A325196.

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

a(n) = my(t=ceil(sqrtint(8*n+1)/2), r=n-t*(t-1)/2); if(r==0,0, binomial(t,r)); \\ Kevin Ryde, Sep 27 2019

Positions of zeros are A000217 = n * (n + 1) / 2.

a(n) = A074909(n) - A010054(n). - Kevin Ryde, Sep 27 2019

A325197 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

Original entry on oeis.org

5, 8, 14, 21, 24, 25, 27, 28, 35, 36, 40, 54, 56, 66, 98, 99, 110, 120, 125, 132, 135, 147, 154, 165, 168, 175, 180, 189, 196, 198, 200, 220, 225, 231, 245, 250, 252, 264, 270, 275, 280, 297, 300, 308, 375, 378, 385, 390, 392, 396, 440, 450, 500, 546, 585, 594

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The enumeration of these partitions by sum is given by A325199.

			The sequence of terms together with their prime indices begins:
    5: {3}
    8: {1,1,1}
   14: {1,4}
   21: {2,4}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   35: {3,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   66: {1,2,5}
   98: {1,4,4}
   99: {2,2,5}
  110: {1,3,5}
  120: {1,1,1,2,3}
  125: {3,3,3}
  132: {1,1,2,5}

Cf. A195086, A065770, A325166, A325168, A325169, A325170, A325180, A325182, A325188, A325189, A325195, A325196, A325198, A325199, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[1000],otbmax[primeptn[#]]-otb[primeptn[#]]==2&]

cp's OEIS Frontend

A065770 Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

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A325169 Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

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A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)).

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PARI

PARI

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A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram 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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A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

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A325196 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

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A325165 Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.

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