cp's OEIS Frontend

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

%I A370802 #12 Mar 23 2024 22:13:23
%S A370802 1,2,6,9,10,22,25,28,30,34,42,45,62,63,66,75,82,92,98,99,102,104,110,
%T A370802 118,121,134,140,147,152,153,156,166,170,186,210,218,228,230,232,234,
%U A370802 246,254,260,275,276,279,289,308,310,314,315,330,342,343,344,348,350
%N A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).
%C A370802 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 A370802 All squarefree terms are even.
%F A370802 A001222(a(n)) = A370820(a(n)).
%e A370802 The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 is in the sequence.
%e A370802 The terms together with their prime indices begin:
%e A370802     1: {}
%e A370802     2: {1}
%e A370802     6: {1,2}
%e A370802     9: {2,2}
%e A370802    10: {1,3}
%e A370802    22: {1,5}
%e A370802    25: {3,3}
%e A370802    28: {1,1,4}
%e A370802    30: {1,2,3}
%e A370802    34: {1,7}
%e A370802    42: {1,2,4}
%e A370802    45: {2,2,3}
%e A370802    62: {1,11}
%e A370802    63: {2,2,4}
%e A370802    66: {1,2,5}
%e A370802    75: {2,3,3}
%e A370802    82: {1,13}
%e A370802    92: {1,1,9}
%e A370802    98: {1,4,4}
%e A370802    99: {2,2,5}
%e A370802   102: {1,2,7}
%e A370802   104: {1,1,1,6}
%t A370802 Select[Range[100],PrimeOmega[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
%Y A370802 For factors instead of divisors on the RHS we have A319899.
%Y A370802 A version for binary indices is A367917.
%Y A370802 For (greater than) instead of (equal) we have A370348, counted by A371171.
%Y A370802 The RHS is A370820, for prime factors instead of divisors A303975.
%Y A370802 Partitions of this type are counted by A371130, strict A371128.
%Y A370802 For divisors instead of factors on LHS we have A371165, counted by A371172.
%Y A370802 For only distinct prime factors on LHS we have A371177, counted by A371178.
%Y A370802 Other inequalities: A371166, A371167, A371169, A371170.
%Y A370802 A000005 counts divisors.
%Y A370802 A001221 counts distinct prime factors.
%Y A370802 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370802 A239312 counts divisor-choosable partitions, ranks A368110.
%Y A370802 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370802 A370320 counts non-divisor-choosable partitions, ranks A355740.
%Y A370802 Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.
%K A370802 nonn
%O A370802 1,2
%A A370802 _Gus Wiseman_, Mar 14 2024