A243503 -id:A243503 - cp's OEIS frontend

A243504 Product of parts of integer partitions as ordered by the table A241918: a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

1, 1, 1, 2, 1, 4, 1, 3, 2, 8, 1, 9, 1, 16, 4, 4, 1, 6, 1, 27, 8, 32, 1, 16, 2, 64, 3, 81, 1, 18, 1, 5, 16, 128, 4, 12, 1, 256, 32, 64, 1, 54, 1, 243, 9, 512, 1, 25, 2, 12, 64, 729, 1, 8, 8, 256, 128, 1024, 1, 48, 1, 2048, 27, 6, 16, 162, 1, 2187, 256, 36, 1, 20, 1, 4096

Offset: 1

Antti Karttunen, Jun 05 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

The positions of ones after a(1)=1 is given by A000040 (primes).

Cf. A243503 (the sum of parts), A241918, A227184, A075158, A003963, A241909.

a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).
a(n) = A003963(A241909(n)).
a(n) = A227184(A075158(n-1)).
a(A000040(n)) = 1 for all n.
a(A000079(n)) = n for all n.

A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 20 2022

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.

			Triangle begins:
   1:   .
   2:   .
   3:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  2 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
  17:   .
  18:  2 1
  19:   .
  20:  3 3
For example, the prime indices of 900 are (1,1,2,2,3,3), so row 900 is 3 - (1,1,2,2,3) + 1 = (3,3,2,2,1).

Row lengths are A001222(n) - 1.
Indices of empty rows are A008578.
Even-indexed rows have sums A243503.
Row sums are A326844(n) + A001222(n) - 1.
An opposite version is A356958, Heinz numbers A246277.
Heinz numbers of the rows are A358195, even bisection A241916.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
Cf. A055396, A124010, A253565, A325351, A325352, A355534, A355536, A358137.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,{},1+Last[primeMS[n]]-Most[primeMS[n]]],{n,100}]

A331170 a(n) = min(n, A122111(n)), where A122111 conjugates the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 9, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 20, 21, 22, 23, 14, 25, 26, 25, 28, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 28, 41, 42, 43, 44, 45, 46, 47, 22, 49, 45, 51, 52, 53, 35, 55, 56, 57, 58, 59, 42, 61, 62, 63, 13, 65, 66, 67, 68, 69, 70, 71, 33, 73, 74, 75, 76, 77, 78, 79, 44, 49, 82, 83, 84, 85, 86, 87, 88, 89, 70, 91, 92, 93, 94, 95, 26, 97, 98, 99, 63

Offset: 1

Antti Karttunen, Jan 12 2020

nonn

For all i, j:
  a(i) = a(j) => A056239(i) = A056239(j),
  a(i) = a(j) => A243503(i) = A243503(j).

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

Cf. A056239, A064989, A122111, A243503.

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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A331170(n) = min(n, A122111(n));

A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 27 2022

nonn

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), which has the following diagram. The 3 X 4 rectangle is shown in dots.
  . . . o o o
  . . . o o
  . . . o o
  . . .
The size of the complement is 7, so a(7865) = 7.

The opposite version is A326844.
Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326846 = size of the smallest rectangle containing the prime indices of n.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.

Table[If[n==1,0,With[{q=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[q]-q[[1]]*Length[q]]],{n,100}]

a(n) = A056239(n) - A001222(n) * A055396(n).

a(n) = A056239(n) - A359360(n).

cp's OEIS Frontend

A243504 Product of parts of integer partitions as ordered by the table A241918: a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

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A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

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A331170 a(n) = min(n, A122111(n)), where A122111 conjugates the prime factorization of n.

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PARI

A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

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