A324515 Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 40, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223

Offset

1,1

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.

Also Heinz numbers of the integer partitions enumerated by A324516. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

Example

The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  23: {9}
  29: {10}
  31: {11}
  37: {12}
  40: {1,1,1,3}
  41: {13}
  43: {14}
  45: {2,2,3}

Maple

filter:= proc(n) local F, Inds, t;
  if isprime(n) then return true fi;
  F:= ifactors(n)[2];
  Inds:= map(numtheory:-pi, F[.., 1]);
  max(Inds) - min(Inds) = add(t[2], t=F) - nops(F)
end proc:
select(filter, [$2..300]); # Robert Israel, Nov 19 2023

Mathematica

Select[Range[2, 100], With[{f=FactorInteger[#]}, PrimePi[f[[-1, 1]]]-PrimePi[f[[1, 1]]]==Total[Last/@f]-Length[f]]&]

Crossrefs

Cf. A001221, A001222, A006141, A046660, A047993, A055396, A056239, A061395, A106529, A112798, A243055.

Cf. A324516, A324517, A324519, A324521, A324522, A324560, A324562.

Sequence in context: A325337 A342119 A372742 * A212127 A307895 A028835

Adjacent sequences: A324512 A324513 A324514 * A324516 A324517 A324518

Keyword

nonn

Author

Gus Wiseman, Mar 06 2019

Status

approved