login
A324758
Heinz numbers of integer partitions containing no prime indices of the parts.
32
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101
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 Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
20: {1,1,3}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
27: {2,2,2}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Intersection[primeMS[#], Union@@primeMS/@primeMS[#]]=={}&]
CROSSREFS
The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.
Sequence in context: A324759 A360550 A326621 * A305504 A316413 A360009
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 17 2019
STATUS
approved