A371165 - OEIS

%I #13 Mar 23 2024 22:13:01

%S 3,5,11,17,26,31,35,38,39,41,49,57,58,59,65,67,69,77,83,86,87,94,109,

%T 119,127,129,133,146,148,157,158,179,191,202,206,211,217,235,237,241,

%U 244,253,274,277,278,283,284,287,291,298,303,319,326,331,333,334,353

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

%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.

%F A000005(a(n)) = A370820(a(n)).

%e The terms together with their prime indices begin:

%e      3: {2}        67: {19}        158: {1,22}

%e      5: {3}        69: {2,9}       179: {41}

%e     11: {5}        77: {4,5}       191: {43}

%e     17: {7}        83: {23}        202: {1,26}

%e     26: {1,6}      86: {1,14}      206: {1,27}

%e     31: {11}       87: {2,10}      211: {47}

%e     35: {3,4}      94: {1,15}      217: {4,11}

%e     38: {1,8}     109: {29}        235: {3,15}

%e     39: {2,6}     119: {4,7}       237: {2,22}

%e     41: {13}      127: {31}        241: {53}

%e     49: {4,4}     129: {2,14}      244: {1,1,18}

%e     57: {2,8}     133: {4,8}       253: {5,9}

%e     58: {1,10}    146: {1,21}      274: {1,33}

%e     59: {17}      148: {1,1,12}    277: {59}

%e     65: {3,6}     157: {37}        278: {1,34}

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

%Y For prime factors instead of divisors on both sides we get A319899.

%Y For prime factors on LHS we get A370802, for distinct prime factors A371177.

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

%Y For (greater than) instead of (equal) we get A371166.

%Y For (less than) instead of (equal) we get A371167.

%Y Partitions of this type are counted by A371172.

%Y Other inequalities: A370348 (A371171), A371168 (A371173), A371169, A371170.

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A239312 counts divisor-choosable partitions, ranks A368110.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y A370320 counts non-divisor-choosable partitions, ranks A355740.

%Y A370814 counts divisor-choosable factorizations, complement A370813.

%Y Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 14 2024