%I #7 Mar 19 2024 08:38:03

%S 1,2,4,6,8,10,12,16,18,20,22,24,30,32,34,36,40,42,44,48,50,54,60,62,

%T 64,66,68,72,80,82,84,88,90,96,100,102,108,110,118,120,124,126,128,

%U 132,134,136,144,150,160,162,164,166,168,170,176,180,186,192,198,200

%N Positive integers whose prime indices include all distinct divisors of all prime indices.

%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.

%C Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

%F A001221(a(n)) = A370820(a(n)).

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 4: {1,1}

%e 6: {1,2}

%e 8: {1,1,1}

%e 10: {1,3}

%e 12: {1,1,2}

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

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 22: {1,5}

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

%e 30: {1,2,3}

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

%e 34: {1,7}

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

%e 40: {1,1,1,3}

%e 42: {1,2,4}

%e 44: {1,1,5}

%e 48: {1,1,1,1,2}

%t Select[Range[100],PrimeNu[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

%Y The LHS is A001221, distinct case of A001222.

%Y The RHS is A370820, for prime factors A303975.

%Y For bigomega on the LHS we have A370802, counted by A371130.

%Y For divisors on the LHS we have A371165, counted by A371172.

%Y Partitions of this type are counted by A371178, strict A371128.

%Y The complement is A371179, counted by A371132.

%Y A000005 counts divisors.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length.

%Y A305148 counts partitions without divisors, strict A303362, ranks A316476.

%Y Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 18 2024