A324515 - OEIS

%I #10 Nov 19 2023 23:44:00

%S 2,3,5,7,11,12,13,17,18,19,23,29,31,37,40,41,43,45,47,53,59,61,67,71,

%T 73,75,79,83,89,97,100,101,103,107,109,112,113,120,127,131,137,139,

%U 149,151,157,163,167,173,175,179,180,181,189,191,193,197,199,211,223

%N 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.

%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.

%C 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).

%H Robert Israel, <a href="/A324515/b324515.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%e    2: {1}

%e    3: {2}

%e    5: {3}

%e    7: {4}

%e   11: {5}

%e   12: {1,1,2}

%e   13: {6}

%e   17: {7}

%e   18: {1,2,2}

%e   19: {8}

%e   23: {9}

%e   29: {10}

%e   31: {11}

%e   37: {12}

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

%e   41: {13}

%e   43: {14}

%e   45: {2,2,3}

%p filter:= proc(n) local F, Inds, t;

%p   if isprime(n) then return true fi;

%p   F:= ifactors(n)[2];

%p   Inds:= map(numtheory:-pi, F[..,1]);

%p   max(Inds) - min(Inds) = add(t[2],t=F) - nops(F)

%p end proc:

%p select(filter, [$2..300]); # _Robert Israel_, Nov 19 2023

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

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 06 2019