OFFSET
1,2
COMMENTS
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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 08 2019
STATUS
approved