cp's OEIS Frontend

A371167 Positive integers with more divisors (A000005) than distinct divisors of prime indices (A370820).

%I A371167 #11 Mar 23 2024 22:12:46
%S A371167 1,2,4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,27,28,30,32,33,34,36,40,
%T A371167 42,44,45,46,48,50,51,52,54,55,56,60,62,63,64,66,68,70,72,75,76,78,80,
%U A371167 81,82,84,85,88,90,92,93,96,98,99,100,102,104,105,108,110
%N A371167 Positive integers with more divisors (A000005) than distinct divisors of prime indices (A370820).
%C A371167 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.
%F A371167 A000005(a(n)) > A370820(a(n)).
%e A371167 The prime indices of 814 are {1,5,12}, and there are 8 divisors (1,2,11,22,37,74,407,814) and 7 distinct divisors of prime indices (1,2,3,4,5,6,12), so 814 is in the sequence.
%e A371167 The prime indices of 1859 are {5,6,6}, and there are 6 divisors (1,11,13,143,169,1859) and 5 distinct divisors of prime indices (1,2,3,5,6), so 1859 is in the sequence.
%e A371167 The terms together with their prime indices begin:
%e A371167      1: {}
%e A371167      2: {1}
%e A371167      4: {1,1}
%e A371167      6: {1,2}
%e A371167      8: {1,1,1}
%e A371167      9: {2,2}
%e A371167     10: {1,3}
%e A371167     12: {1,1,2}
%e A371167     14: {1,4}
%e A371167     15: {2,3}
%e A371167     16: {1,1,1,1}
%e A371167     18: {1,2,2}
%e A371167     20: {1,1,3}
%e A371167     21: {2,4}
%e A371167     22: {1,5}
%e A371167     24: {1,1,1,2}
%e A371167     25: {3,3}
%e A371167     27: {2,2,2}
%e A371167     28: {1,1,4}
%e A371167     30: {1,2,3}
%t A371167 Select[Range[100],Length[Divisors[#]]>Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
%Y A371167 For prime factors on the LHS we have A370348, counted by A371171.
%Y A371167 The RHS is A370820, for prime factors instead of divisors A303975.
%Y A371167 For (equal to) instead of (greater than) we get A371165, counted by A371172.
%Y A371167 For (less than) instead of (greater than) we get A371166.
%Y A371167 Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
%Y A371167 Other inequalities: A371168 (A371173), A371169, A371170.
%Y A371167 A000005 counts divisors.
%Y A371167 A001221 counts distinct prime factors.
%Y A371167 A027746 lists prime factors, A112798 indices, length A001222.
%Y A371167 A239312 counts divisor-choosable partitions, ranks A368110.
%Y A371167 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A371167 A370320 counts non-divisor-choosable partitions, ranks A355740.
%Y A371167 A370814 counts divisor-choosable factorizations, complement A370813.
%Y A371167 Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.
%K A371167 nonn
%O A371167 1,2
%A A371167 _Gus Wiseman_, Mar 14 2024