A340606 - OEIS

%I #10 Jan 27 2021 22:37:35

%S 1,2,4,6,8,9,16,20,24,32,36,50,54,56,64,81,84,96,125,126,128,144,160,

%T 176,189,196,216,240,256,294,324,360,384,400,416,441,486,512,540,576,

%U 600,624,686,729,810,864,896,900,936,968,1000,1024,1029,1040,1088,1215

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

%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    1: {}

%e    2: {1}

%e    4: {1,1}

%e    6: {1,2}

%e    8: {1,1,1}

%e    9: {2,2}

%e   16: {1,1,1,1}

%e   20: {1,1,3}

%e   24: {1,1,1,2}

%e   32: {1,1,1,1,1}

%e   36: {1,1,2,2}

%e   50: {1,3,3}

%e   54: {1,2,2,2}

%e   56: {1,1,1,4}

%e   64: {1,1,1,1,1,1}

%e   81: {2,2,2,2}

%e   84: {1,1,2,4}

%e   96: {1,1,1,1,1,2}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]

%Y Note: Heinz numbers are given in parentheses below.

%Y The reciprocal version is A143773 (A316428).

%Y These partitions are counted by A340693.

%Y A120383 lists numbers divisible by all of their prime indices.

%Y A324850 lists numbers divisible by the product of their prime indices.

%Y A003963 multiplies together the prime indices of n.

%Y A018818 counts partitions of n into divisors of n (A326841).

%Y A047993 counts balanced partitions (A106529).

%Y A067538 counts partitions of n whose length divides n (A316413).

%Y A056239 adds up the prime indices of n.

%Y A061395 selects the maximum prime index.

%Y A067538 counts partitions of n whose maximum divides n (A326836).

%Y A072233 counts partitions by sum and length.

%Y A112798 lists the prime indices of each positive integer.

%Y A168659 = partitions whose length is divisible by their maximum (A340609).

%Y A168659 = partitions whose maximum is divisible by their length (A340610).

%Y A289509 lists numbers with relatively prime prime indices.

%Y A326842 = partitions of n whose length and parts all divide n (A326847).

%Y A326843 = partitions of n whose length and maximum both divide n (A326837).

%Y A340852 have a factorization with factors dividing length.

%Y Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 24 2021