cp's OEIS Frontend

A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

%I A371285 #9 Mar 22 2024 12:33:39
%S A371285 1,2,6,4,10,12,42,8,36,20,22,24,390,84,60,16,34,72,798,40,252,44,230,
%T A371285 48,100,780,216,168,1914,120,62,32,132,68,420,144,101010,1596,2340,80,
%U A371285 82,504,4386,88,360,460,5170,96,1764,200,204,1560,42294,432,220,336
%N A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.
%C A371285 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 A371285 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 A371285 If n = prime(x_1)*...*prime(x_k) then a(n) = A275700(x_1)*...*A275700(x_k).
%e A371285 The prime indices of 105 are {2,3,4}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
%e A371285 The terms together with their prime indices begin:
%e A371285           1: {}
%e A371285           2: {1}
%e A371285           6: {1,2}
%e A371285           4: {1,1}
%e A371285          10: {1,3}
%e A371285          12: {1,1,2}
%e A371285          42: {1,2,4}
%e A371285           8: {1,1,1}
%e A371285          36: {1,1,2,2}
%e A371285          20: {1,1,3}
%e A371285          22: {1,5}
%e A371285          24: {1,1,1,2}
%e A371285         390: {1,2,3,6}
%e A371285          84: {1,1,2,4}
%e A371285          60: {1,1,2,3}
%e A371285          16: {1,1,1,1}
%e A371285          34: {1,7}
%e A371285          72: {1,1,1,2,2}
%t A371285 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371285 Table[Times@@Prime/@Join@@Divisors/@prix[n],{n,100}]
%Y A371285 Product of A275700 applied to each prime index.
%Y A371285 The squarefree case is also A275700.
%Y A371285 The sorted version is A371286.
%Y A371285 A000005 counts divisors.
%Y A371285 A001221 counts distinct prime factors.
%Y A371285 A027746 lists prime factors, A112798 indices, length A001222.
%Y A371285 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A371285 A355741 counts choices of a prime factor of each prime index.
%Y A371285 Cf. A000720, A003963, A005179, A007416, A034729, A054973, A056239, A321898, A370820, A371165, A371181, A371284, A371288.
%K A371285 nonn,mult
%O A371285 1,2
%A A371285 _Gus Wiseman_, Mar 21 2024