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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
1, 2, 9, 12, 18, 40, 100, 112, 125, 240, 250, 352, 360, 392, 405, 540, 600, 672, 675, 810, 832, 900, 1008, 1125, 1350, 1372, 1500, 1512, 1701, 1875, 1936, 2112, 2176, 2240, 2250, 2268, 2352, 2401, 3168, 3402, 3528, 3750, 3969, 4752, 4802, 4864, 4992, 5292 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

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).

LINKS

Table of n, a(n) for n=1..48.

FORMULA

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

EXAMPLE

The sequence of terms together with their prime indices begins:

    1: {}

    2: {1}

    9: {2,2}

   12: {1,1,2}

   18: {1,2,2}

   40: {1,1,1,3}

  100: {1,1,3,3}

  112: {1,1,1,1,4}

  125: {3,3,3}

  240: {1,1,1,1,2,3}

  250: {1,3,3,3}

  352: {1,1,1,1,1,5}

  360: {1,1,1,2,2,3}

  392: {1,1,1,4,4}

  405: {2,2,2,2,3}

  540: {1,1,2,2,2,3}

  600: {1,1,1,2,3,3}

  672: {1,1,1,1,1,2,4}

MAPLE

with(numtheory):

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

select(q, [$1..5600])[];  # Alois P. Heinz, Mar 07 2019

MATHEMATICA

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

CROSSREFS

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

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

Sequence in context: A031070 A225547 A325755 * A109297 A048768 A070226

Adjacent sequences:  A324567 A324568 A324569 * A324571 A324572 A324573

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 07 2019

STATUS

approved

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)