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A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

%I A371283 #12 Aug 25 2024 19:40:23
%S A371283 2,6,10,22,34,42,62,82,118,134,166,218,230,254,314,358,382,390,422,
%T A371283 482,554,566,662,706,734,798,802,862,922,1018,1094,1126,1174,1198,
%U A371283 1234,1418,1478,1546,1594,1718,1754,1838,1914,1934,1982,2062,2126,2134,2174,2306
%N A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.
%C A371283 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A371283 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.
%H A371283 Joseph Likar, <a href="/A371283/b371283.txt">Table of n, a(n) for n = 1..10000</a>
%e A371283 The terms together with their prime indices begin:
%e A371283      2: {1}
%e A371283      6: {1,2}
%e A371283     10: {1,3}
%e A371283     22: {1,5}
%e A371283     34: {1,7}
%e A371283     42: {1,2,4}
%e A371283     62: {1,11}
%e A371283     82: {1,13}
%e A371283    118: {1,17}
%e A371283    134: {1,19}
%e A371283    166: {1,23}
%e A371283    218: {1,29}
%e A371283    230: {1,3,9}
%e A371283    254: {1,31}
%e A371283    314: {1,37}
%e A371283    358: {1,41}
%e A371283    382: {1,43}
%e A371283    390: {1,2,3,6}
%t A371283 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371283 Select[Range[2,100],SameQ[prix[#],Divisors[Last[prix[#]]]]&]
%Y A371283 Partitions of this type are counted by A054973.
%Y A371283 The unsorted version is A275700.
%Y A371283 These numbers have products A371286, unsorted version A371285.
%Y A371283 Squarefree case of A371288, counted by A371284.
%Y A371283 A000005 counts divisors.
%Y A371283 A001221 counts distinct prime factors.
%Y A371283 A027746 lists prime factors, A112798 indices, length A001222.
%Y A371283 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A371283 A355741 counts choices of a prime factor of each prime index.
%Y A371283 Cf. A000720, A005179, A007416, A034729, A370820, A371131, A371177.
%K A371283 nonn
%O A371283 1,1
%A A371283 _Gus Wiseman_, Mar 21 2024