A324851 - OEIS

%I #13 Mar 20 2019 04:25:55

%S 2,4,6,12,15,16,20,30,35,36,42,48,56,88,99,112,120,126,130,135,143,

%T 144,160,162,180,192,210,216,220,221,228,231,242,250,256,270,275,280,

%U 288,297,300,308,322,330,338,360,396,400,408,429,435,440,455,468,480,493

%N Numbers > 1 divisible by the sum of their prime indices.

%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. The sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.

%C For any k>=2, let d be a divisor of k such that d > A056239(k).  Then 2^(d-A056239(k))*k is in the sequence.  Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - _Robert Israel_, Mar 19 2019

%H Robert Israel, <a href="/A324851/b324851.txt">Table of n, a(n) for n = 1..10000</a>

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

%e     2: {1}

%e     4: {1,1}

%e     6: {1,2}

%e    12: {1,1,2}

%e    15: {2,3}

%e    16: {1,1,1,1}

%e    20: {1,1,3}

%e    30: {1,2,3}

%e    35: {3,4}

%e    36: {1,1,2,2}

%e    42: {1,2,4}

%e    48: {1,1,1,1,2}

%e    56: {1,1,1,4}

%e    88: {1,1,1,5}

%e    99: {2,2,5}

%e   112: {1,1,1,1,4}

%e   120: {1,1,1,2,3}

%e   126: {1,2,2,4}

%e   130: {1,3,6}

%e   135: {2,2,2,3}

%p filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2],t=ifactors(n)[2]) = 0 end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Mar 19 2019

%t Select[Range[2,100],Divisible[#,Plus@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]]&]

%o (PARI) isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k,1])*f[k,2]));} \\ _Michel Marcus_, Mar 19 2019

%Y Cf. A003963, A036844, A056239, A112798, A120383, A290822, A306844.

%Y Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 18 2019