A321898 -id:A321898 - cp's OEIS frontend

A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

1, 2, 6, 4, 10, 12, 42, 8, 36, 20, 22, 24, 390, 84, 60, 16, 34, 72, 798, 40, 252, 44, 230, 48, 100, 780, 216, 168, 1914, 120, 62, 32, 132, 68, 420, 144, 101010, 1596, 2340, 80, 82, 504, 4386, 88, 360, 460, 5170, 96, 1764, 200, 204, 1560, 42294, 432, 220, 336

Offset: 1

Gus Wiseman, Mar 21 2024

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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.

			The prime indices of 105 are {2,3,4}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
The terms together with their prime indices begin:
          1: {}
          2: {1}
          6: {1,2}
          4: {1,1}
         10: {1,3}
         12: {1,1,2}
         42: {1,2,4}
          8: {1,1,1}
         36: {1,1,2,2}
         20: {1,1,3}
         22: {1,5}
         24: {1,1,1,2}
        390: {1,2,3,6}
         84: {1,1,2,4}
         60: {1,1,2,3}
         16: {1,1,1,1}
         34: {1,7}
         72: {1,1,1,2,2}

Product of A275700 applied to each prime index.
The squarefree case is also A275700.
The sorted version is A371286.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A321898, A370820, A371165, A371181, A371284, A371288.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Join@@Divisors/@prix[n],{n,100}]

If n = prime(x_1)*...*prime(x_k) then a(n) = A275700(x_1)*...*A275700(x_k).

A321897 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 6, 3, 8, 6, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 24, 30, 20, 15, 20, 10, 1, 0, 0, 0, 1, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 0, 1, 720, 840, 504, 420, 630

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

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

			Triangle begins:
    1
    1
    1    1
    0    1
    2    3    1
    0    1    1
    6    3    8    6    1
    0    0    1
    0    1    0    2    1
    0    0    2    3    1
   24   30   20   15   20   10    1
    0    0    0    1    1
  120   90  144   40   15   90  120   45   40   15    1
    0    6    0    3    8    6    1
    0    0    2    3    2    4    1
    0    0    0    0    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    0    0    1    0    2    1
For example, row 14 gives: 12h(41) = 6p(41) + 3p(221) + 8p(311) + 6p(2111) + p(11111).

Wikipedia, Symmetric polynomial

Row sums are A321898.

Cf. A005651, A008480, A056239, A124794, A124795, A319193, A321742-A321765, A321896.

cp's OEIS Frontend

A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

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