A360328 - OEIS

%I #14 Feb 04 2023 14:14:37

%S 7425,8415,22275,25245,37125,42075,46035,66825,75735,76725,81675,

%T 92565,101475,111375,126225,138105,143055,182655,185625,200475,210375,

%U 227205,230175,245025,260865,277695,304425,334125,345015,355725,378675,383625,408375,414315,429165

%N Numbers k such that A360327(k) > 2*k.

%C Analogous to abundant numbers (A005101) with divisors that are restricted to numbers that have only prime-indexed prime factors.

%C The abundancy index of numbers in A076610 (i.e., numbers whose prime factors are only prime-indexed primes) is bounded by P = Product_{p in A006450} p/(p-1) which seems to be less than 4 (see A076610). Therefore, there are no terms k of A076610 with sigma(k) >= 4*k, or equivalently, no even terms in this sequence, and all the terms of this sequence are in A076610. Also, assuming that P < 15/4 = 3.75, there are no terms in this sequence that are coprime to 15.

%C Since P > 3 there are terms that are not divisible by 3. The least of them must be larger than Product_{k=1..21826870} A006450(k) = 3 * 5 * 11 * ... * 8958801613 > 10^206662375, because Product_{k=2..m} A006450(k)/(A006450(k)-1) > 2 only for m >= 21826870.

%C The least term that is not divisible by 5 is 789909738655399955305165431.

%C The least term that is not divisible by 11 is a(30) = 355725.

%C The least squarefree term is 14093057715.

%H Amiram Eldar, <a href="/A360328/b360328.txt">Table of n, a(n) for n = 1..1000</a>

%t f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], s[#] > 2*# &]

%o (PARI) is(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1)) > 2*n;}

%Y Intersection of A005101 (or A005231) and A076610.

%Y Cf. A006450, A076610, A360327.

%K nonn

%O 1,1

%A _Amiram Eldar_, Feb 03 2023