cp's OEIS Frontend

A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

%I A340606 #10 Jan 27 2021 22:37:35
%S A340606 1,2,4,6,8,9,16,20,24,32,36,50,54,56,64,81,84,96,125,126,128,144,160,
%T A340606 176,189,196,216,240,256,294,324,360,384,400,416,441,486,512,540,576,
%U A340606 600,624,686,729,810,864,896,900,936,968,1000,1024,1029,1040,1088,1215
%N A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).
%C A340606 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 A340606 The sequence of terms together with their prime indices begins:
%e A340606    1: {}
%e A340606    2: {1}
%e A340606    4: {1,1}
%e A340606    6: {1,2}
%e A340606    8: {1,1,1}
%e A340606    9: {2,2}
%e A340606   16: {1,1,1,1}
%e A340606   20: {1,1,3}
%e A340606   24: {1,1,1,2}
%e A340606   32: {1,1,1,1,1}
%e A340606   36: {1,1,2,2}
%e A340606   50: {1,3,3}
%e A340606   54: {1,2,2,2}
%e A340606   56: {1,1,1,4}
%e A340606   64: {1,1,1,1,1,1}
%e A340606   81: {2,2,2,2}
%e A340606   84: {1,1,2,4}
%e A340606   96: {1,1,1,1,1,2}
%t A340606 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A340606 Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]
%Y A340606 Note: Heinz numbers are given in parentheses below.
%Y A340606 The reciprocal version is A143773 (A316428).
%Y A340606 These partitions are counted by A340693.
%Y A340606 A120383 lists numbers divisible by all of their prime indices.
%Y A340606 A324850 lists numbers divisible by the product of their prime indices.
%Y A340606 A003963 multiplies together the prime indices of n.
%Y A340606 A018818 counts partitions of n into divisors of n (A326841).
%Y A340606 A047993 counts balanced partitions (A106529).
%Y A340606 A067538 counts partitions of n whose length divides n (A316413).
%Y A340606 A056239 adds up the prime indices of n.
%Y A340606 A061395 selects the maximum prime index.
%Y A340606 A067538 counts partitions of n whose maximum divides n (A326836).
%Y A340606 A072233 counts partitions by sum and length.
%Y A340606 A112798 lists the prime indices of each positive integer.
%Y A340606 A168659 = partitions whose length is divisible by their maximum (A340609).
%Y A340606 A168659 = partitions whose maximum is divisible by their length (A340610).
%Y A340606 A289509 lists numbers with relatively prime prime indices.
%Y A340606 A326842 = partitions of n whose length and parts all divide n (A326847).
%Y A340606 A326843 = partitions of n whose length and maximum both divide n (A326837).
%Y A340606 A340852 have a factorization with factors dividing length.
%Y A340606 Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.
%K A340606 nonn
%O A340606 1,2
%A A340606 _Gus Wiseman_, Jan 24 2021