cp's OEIS Frontend

A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

%I A371165 #13 Mar 23 2024 22:13:01
%S A371165 3,5,11,17,26,31,35,38,39,41,49,57,58,59,65,67,69,77,83,86,87,94,109,
%T A371165 119,127,129,133,146,148,157,158,179,191,202,206,211,217,235,237,241,
%U A371165 244,253,274,277,278,283,284,287,291,298,303,319,326,331,333,334,353
%N A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).
%C A371165 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 A371165 A000005(a(n)) = A370820(a(n)).
%e A371165 The terms together with their prime indices begin:
%e A371165      3: {2}        67: {19}        158: {1,22}
%e A371165      5: {3}        69: {2,9}       179: {41}
%e A371165     11: {5}        77: {4,5}       191: {43}
%e A371165     17: {7}        83: {23}        202: {1,26}
%e A371165     26: {1,6}      86: {1,14}      206: {1,27}
%e A371165     31: {11}       87: {2,10}      211: {47}
%e A371165     35: {3,4}      94: {1,15}      217: {4,11}
%e A371165     38: {1,8}     109: {29}        235: {3,15}
%e A371165     39: {2,6}     119: {4,7}       237: {2,22}
%e A371165     41: {13}      127: {31}        241: {53}
%e A371165     49: {4,4}     129: {2,14}      244: {1,1,18}
%e A371165     57: {2,8}     133: {4,8}       253: {5,9}
%e A371165     58: {1,10}    146: {1,21}      274: {1,33}
%e A371165     59: {17}      148: {1,1,12}    277: {59}
%e A371165     65: {3,6}     157: {37}        278: {1,34}
%t A371165 Select[Range[100],Length[Divisors[#]] == Length[Union@@Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
%Y A371165 For prime factors instead of divisors on both sides we get A319899.
%Y A371165 For prime factors on LHS we get A370802, for distinct prime factors A371177.
%Y A371165 The RHS is A370820, for prime factors instead of divisors A303975.
%Y A371165 For (greater than) instead of (equal) we get A371166.
%Y A371165 For (less than) instead of (equal) we get A371167.
%Y A371165 Partitions of this type are counted by A371172.
%Y A371165 Other inequalities: A370348 (A371171), A371168 (A371173), A371169, A371170.
%Y A371165 A000005 counts divisors.
%Y A371165 A001221 counts distinct prime factors.
%Y A371165 A027746 lists prime factors, A112798 indices, length A001222.
%Y A371165 A239312 counts divisor-choosable partitions, ranks A368110.
%Y A371165 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A371165 A370320 counts non-divisor-choosable partitions, ranks A355740.
%Y A371165 A370814 counts divisor-choosable factorizations, complement A370813.
%Y A371165 Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.
%K A371165 nonn
%O A371165 1,1
%A A371165 _Gus Wiseman_, Mar 14 2024