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.
This sequence is finite. Proof: k = p_1*p_2*...*p_t is a term iff q_1*q_2*...*q_t = 2*(q_1 + q_2 + ... + q_t), where q_i = pi(p_i) and q_1 < q_2 < ... < q_t. If t = 2, then 1/2 = 1/q_1 + 1/q_2. Thus q_1 <= 3, we have k = prime(3)*prime(6) = 65. If t = 3, then 1/2 = 1/(q_1*q_2) + 1/(q_1*q_3) + 1/(q_2*q_3). Thus q_1*q_2 <= 5, we have k = prime(1)*prime(4)*prime(5) = 154 or k = prime(1)*prime(3)*prime(8) = 190. If t > 3, then 1/2 = Sum_{i=1..t} q_i/(q_1*q_2*...*q_t) < Sum_{i=1..t} i/t! < 1/2, a contradiction. - Jinyuan Wang, Jun 27 2020
EXAMPLE
The sequence of terms together with their prime indices starts:
65: {3,6}
154: {1,4,5}
190: {1,3,8}
MAPLE
q:= n-> (l-> andmap(i-> i[2]=1, l) and (h-> mul(i, i=h)=2*add(i,
i=h))(map(i-> numtheory[pi](i[1]), l)))(ifactors(n)[2]):
select(q, [$1..1000])[]; # Alois P. Heinz, Sep 12 2019
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], SquareFreeQ[#]&&SameQ[Times@@primeMS[#], 2*Plus@@primeMS[#]]&]
KEYWORD
nonn,bref,fini,full
AUTHOR
Gus Wiseman, Sep 12 2019
STATUS
approved