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A324570 Numbers where the sum of distinct prime indices (A066328) is equal to the number of prime factors counted with multiplicity (A001222). 4

%I

%S 1,2,9,12,18,40,100,112,125,240,250,352,360,392,405,540,600,672,675,

%T 810,832,900,1008,1125,1350,1372,1500,1512,1701,1875,1936,2112,2176,

%U 2240,2250,2268,2352,2401,3168,3402,3528,3750,3969,4752,4802,4864,4992,5292

%N Numbers where the sum of distinct prime indices (A066328) is equal to the number of prime factors counted with multiplicity (A001222).

%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. For example, 540 = prime(1)^2 * prime(2)^3 * prime(3)^1 has sum of distinct prime indices 1 + 2 + 3 = 6, while the number of prime factors counted with multiplicity is 2 + 3 + 1 = 6, so 540 belongs to the sequence.

%C Also Heinz numbers of the integer partitions counted by A114638. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%F A066328(a(n)) = A001222(a(n)).

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 9: {2,2}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 40: {1,1,1,3}

%e 100: {1,1,3,3}

%e 112: {1,1,1,1,4}

%e 125: {3,3,3}

%e 240: {1,1,1,1,2,3}

%e 250: {1,3,3,3}

%e 352: {1,1,1,1,1,5}

%e 360: {1,1,1,2,2,3}

%e 392: {1,1,1,4,4}

%e 405: {2,2,2,2,3}

%e 540: {1,1,2,2,2,3}

%e 600: {1,1,1,2,3,3}

%e 672: {1,1,1,1,1,2,4}

%p with(numtheory):

%p q:= n-> is(add(pi(p), p=factorset(n))=bigomega(n)):

%p select(q, [$1..5600])[]; # _Alois P. Heinz_, Mar 07 2019

%t Select[Range[1000],Total[PrimePi/@First/@FactorInteger[#]]==PrimeOmega[#]&]

%Y Cf. A001221, A001222, A056239, A066328, A112798, A114638, A117144, A276078.

%Y Cf. A109298, A324524, A324525, A324570, A324571, A324572.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 07 2019

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Last modified August 1 06:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)