cp's OEIS Frontend

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.

%I A324515 #10 Nov 19 2023 23:44:00
%S A324515 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 A324515 73,75,79,83,89,97,100,101,103,107,109,112,113,120,127,131,137,139,
%U A324515 149,151,157,163,167,173,175,179,180,181,189,191,193,197,199,211,223
%N 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.
%C A324515 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 A324515 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 A324515 Robert Israel, <a href="/A324515/b324515.txt">Table of n, a(n) for n = 1..10000</a>
%F A324515 A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).
%e A324515 The sequence of terms together with their prime indices begins:
%e A324515    2: {1}
%e A324515    3: {2}
%e A324515    5: {3}
%e A324515    7: {4}
%e A324515   11: {5}
%e A324515   12: {1,1,2}
%e A324515   13: {6}
%e A324515   17: {7}
%e A324515   18: {1,2,2}
%e A324515   19: {8}
%e A324515   23: {9}
%e A324515   29: {10}
%e A324515   31: {11}
%e A324515   37: {12}
%e A324515   40: {1,1,1,3}
%e A324515   41: {13}
%e A324515   43: {14}
%e A324515   45: {2,2,3}
%p A324515 filter:= proc(n) local F, Inds, t;
%p A324515   if isprime(n) then return true fi;
%p A324515   F:= ifactors(n)[2];
%p A324515   Inds:= map(numtheory:-pi, F[..,1]);
%p A324515   max(Inds) - min(Inds) = add(t[2],t=F) - nops(F)
%p A324515 end proc:
%p A324515 select(filter, [$2..300]); # _Robert Israel_, Nov 19 2023
%t A324515 Select[Range[2,100],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]-PrimePi[f[[1,1]]]==Total[Last/@f]-Length[f]]&]
%Y A324515 Cf. A001221, A001222, A006141, A046660, A047993, A055396, A056239, A061395, A106529, A112798, A243055.
%Y A324515 Cf. A324516, A324517, A324519, A324521, A324522, A324560, A324562.
%K A324515 nonn
%O A324515 1,1
%A A324515 _Gus Wiseman_, Mar 06 2019