cp's OEIS Frontend

A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

%I A371177 #7 Mar 19 2024 08:38:03
%S A371177 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 A371177 64,66,68,72,80,82,84,88,90,96,100,102,108,110,118,120,124,126,128,
%U A371177 132,134,136,144,150,160,162,164,166,168,170,176,180,186,192,198,200
%N A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.
%C A371177 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 A371177 Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).
%F A371177 A001221(a(n)) = A370820(a(n)).
%e A371177 The terms together with their prime indices begin:
%e A371177     1: {}
%e A371177     2: {1}
%e A371177     4: {1,1}
%e A371177     6: {1,2}
%e A371177     8: {1,1,1}
%e A371177    10: {1,3}
%e A371177    12: {1,1,2}
%e A371177    16: {1,1,1,1}
%e A371177    18: {1,2,2}
%e A371177    20: {1,1,3}
%e A371177    22: {1,5}
%e A371177    24: {1,1,1,2}
%e A371177    30: {1,2,3}
%e A371177    32: {1,1,1,1,1}
%e A371177    34: {1,7}
%e A371177    36: {1,1,2,2}
%e A371177    40: {1,1,1,3}
%e A371177    42: {1,2,4}
%e A371177    44: {1,1,5}
%e A371177    48: {1,1,1,1,2}
%t A371177 Select[Range[100],PrimeNu[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
%Y A371177 The LHS is A001221, distinct case of A001222.
%Y A371177 The RHS is A370820, for prime factors A303975.
%Y A371177 For bigomega on the LHS we have A370802, counted by A371130.
%Y A371177 For divisors on the LHS we have A371165, counted by A371172.
%Y A371177 Partitions of this type are counted by A371178, strict A371128.
%Y A371177 The complement is A371179, counted by A371132.
%Y A371177 A000005 counts divisors.
%Y A371177 A000041 counts integer partitions, strict A000009.
%Y A371177 A008284 counts partitions by length.
%Y A371177 A305148 counts partitions without divisors, strict A303362, ranks A316476.
%Y A371177 Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.
%K A371177 nonn
%O A371177 1,2
%A A371177 _Gus Wiseman_, Mar 18 2024