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A339886
Numbers whose prime indices cover an interval of positive integers starting with 2.
5
1, 3, 9, 15, 27, 45, 75, 81, 105, 135, 225, 243, 315, 375, 405, 525, 675, 729, 735, 945, 1125, 1155, 1215, 1575, 1875, 2025, 2187, 2205, 2625, 2835, 3375, 3465, 3645, 3675, 4725, 5145, 5625, 5775, 6075, 6561, 6615, 7875, 8085, 8505, 9375, 10125, 10395, 10935
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.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2}
9: {2,2}
15: {2,3}
27: {2,2,2}
45: {2,2,3}
75: {2,3,3}
81: {2,2,2,2}
105: {2,3,4}
135: {2,2,2,3}
225: {2,2,3,3}
243: {2,2,2,2,2}
315: {2,2,3,4}
375: {2,3,3,3}
405: {2,2,2,2,3}
525: {2,3,3,4}
675: {2,2,2,3,3}
729: {2,2,2,2,2,2}
735: {2,3,4,4}
945: {2,2,2,3,4}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[100], normQ[primeMS[#]-1]&]
CROSSREFS
The version starting at 1 is A055932.
The partitions with these Heinz numbers are counted by A264396.
Positions of 1's in A339662.
A000009 counts partitions covering an initial interval.
A000070 counts partitions with a selected part.
A016945 lists numbers with smallest prime index 2.
A034296 counts gap-free (or flat) partitions.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A107428 counts gap-free compositions (initial: A107429).
A286469 and A286470 give greatest difference for Heinz numbers.
A325240 lists numbers with smallest prime multiplicity 2.
A342050/A342051 have prime indices with odd/even least gap.
Sequence in context: A256388 A082897 A363523 * A147516 A372509 A233819
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 20 2021
STATUS
approved