

A305349


Numbers k such that sopfr(k) = tau(k)^3.


1



183, 295, 583, 799, 943, 7042, 10978, 13581, 18658, 20652, 22402, 22898, 29698, 40162, 43522, 48442, 54778, 59362, 62338, 68098, 74938, 82618, 87418, 89722, 97282, 99298, 102202, 108418, 110842, 113122, 116602, 118498, 122362, 123322, 123778, 128482, 128698
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OFFSET

1,1


COMMENTS

For numbers k that satisfy the condition, tau(k) will always be even because tau(k) is odd only if k is a square, but if k is a square then sopfr(k) is even (because every prime appears with an even exponent) and thus it cannot be equal to tau(k)^3 which is odd as tau(k).
A squarefree number k = p_1*...*p_j is in the sequence if p_1 + ... + p_j = 8^j. It is likely that 8^j is the sum of j distinct primes for all j >= 2.  Robert Israel, Dec 10 2018


LINKS



MAPLE

filter:= proc(n) local F;
F:= ifactors(n)[2];
add(t[1]*t[2], t=F) = mul(t[2]+1, t=F)^3
end proc:


MATHEMATICA

sopf[n_] := If[n==1, 0, Plus@@Times@@@FactorInteger@ n]; Select[Range[200000], sopf[#]==DivisorSigma[0, #]^3 &] (* Amiram Eldar, Nov 01 2018 *)


PROG

(PARI) sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



