|
%I #12 Apr 22 2021 01:43:01
%S 1,3,9,15,27,45,75,81,105,135,225,243,315,375,405,525,675,729,735,945,
%T 1125,1155,1215,1575,1875,2025,2187,2205,2625,2835,3375,3465,3645,
%U 3675,4725,5145,5625,5775,6075,6561,6615,7875,8085,8505,9375,10125,10395,10935
%N Numbers whose prime indices cover an interval of positive integers starting with 2.
%C 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.
%e The sequence of terms together with their prime indices begins:
%e 3: {2}
%e 9: {2,2}
%e 15: {2,3}
%e 27: {2,2,2}
%e 45: {2,2,3}
%e 75: {2,3,3}
%e 81: {2,2,2,2}
%e 105: {2,3,4}
%e 135: {2,2,2,3}
%e 225: {2,2,3,3}
%e 243: {2,2,2,2,2}
%e 315: {2,2,3,4}
%e 375: {2,3,3,3}
%e 405: {2,2,2,2,3}
%e 525: {2,3,3,4}
%e 675: {2,2,2,3,3}
%e 729: {2,2,2,2,2,2}
%e 735: {2,3,4,4}
%e 945: {2,2,2,3,4}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t Select[Range[100],normQ[primeMS[#]-1]&]
%Y The version starting at 1 is A055932.
%Y The partitions with these Heinz numbers are counted by A264396.
%Y Positions of 1's in A339662.
%Y A000009 counts partitions covering an initial interval.
%Y A000070 counts partitions with a selected part.
%Y A016945 lists numbers with smallest prime index 2.
%Y A034296 counts gap-free (or flat) partitions.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A073491 lists numbers with gap-free prime indices.
%Y A107428 counts gap-free compositions (initial: A107429).
%Y A286469 and A286470 give greatest difference for Heinz numbers.
%Y A325240 lists numbers with smallest prime multiplicity 2.
%Y A342050/A342051 have prime indices with odd/even least gap.
%Y Cf. A001223, A001522, A006128, A007052, A124010, A257989, A257993, A264401, A317090, A317589, A339737.
%K nonn
%O 1,2
%A _Gus Wiseman_, Apr 20 2021
|
