A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.

1, 2, 9, 12, 40, 112, 125, 352, 360, 675, 832, 1008, 2176, 2401, 3168, 3969, 4864, 7488, 11776, 14000, 19584, 29403, 29696, 43776, 44000, 63488, 75600, 104000, 105984, 123201, 151552, 161051, 214375, 237600, 267264, 272000, 335872, 496125, 561600, 571392, 608000

Offset

1,2

Comments

These are a kind of self-describing numbers (cf. A001462, A304679). The increasing case is A109298.

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 ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base.

Also Heinz numbers of the integer partitions counted by A324572. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Each finite set of positive integers determines a unique term with those prime indices. For example, corresponding to {1,2,4,5} is 1397088 = prime(1)^5 * prime(2)^4 * prime(4)^2 * prime(5)^1.

Links

Table of n, a(n) for n=1..41.

Example

The sequence of terms together with their prime indices begins as follows. For example, we have 40: {1,1,1,3} because 40 = prime(1) * prime(1) * prime(1) * prime(3).
      1: {}
      2: {1}
      9: {2,2}
     12: {1,1,2}
     40: {1,1,1,3}
    112: {1,1,1,1,4}
    125: {3,3,3}
    352: {1,1,1,1,1,5}
    360: {1,1,1,2,2,3}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
   1008: {1,1,1,1,2,2,4}
   2176: {1,1,1,1,1,1,1,7}
   2401: {4,4,4,4}
   3168: {1,1,1,1,1,2,2,5}
   3969: {2,2,2,2,4,4}
   4864: {1,1,1,1,1,1,1,1,8}
   7488: {1,1,1,1,1,1,2,2,6}
  11776: {1,1,1,1,1,1,1,1,1,9}
  14000: {1,1,1,1,3,3,3,4}
  19584: {1,1,1,1,1,1,1,2,2,7}

Mathematica

Select[Range[1000], Reverse[PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]==Last/@If[#==1, {}, FactorInteger[#]]&]

Crossrefs

Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078.

Cf. A324524, A324525, A324572.

Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360, A304679.

Sequence in context: A253608 A126977 A102237 * A076505 A218073 A129345

Adjacent sequences: A324568 A324569 A324570 * A324572 A324573 A324574

Keyword

nonn

Author

Gus Wiseman, Mar 08 2019

Status

approved