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

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

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: A225547 A325755 A360453 * A109297 A048768 A070226

Adjacent sequences: A324567 A324568 A324569 * A324571 A324572 A324573

Keyword

nonn

Author

Gus Wiseman, Mar 07 2019

Status

approved