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A309253 a(n) is the smallest number m with exactly n such divisors d that sigma(d) divides m.

%I A309253 #20 Jun 22 2025 21:31:51
%S A309253 1,6,30,12,60,84,1140,120,168,2340,1848,360,2184,1080,4368,840,10440,
%T A309253 1680,7920,2520,6552,3360,7560,5040,13104,27720,73440,36960,21840,
%U A309253 15120,72072,10080,95760,26208,63840,20160,146160,144144,87360,174720,1071360,166320
%N A309253 a(n) is the smallest number m with exactly n such divisors d that sigma(d) divides m.
%e A309253 For n = 3; a(3) = 30 because 30 is the smallest number with exactly 3 divisors d that sigma(d) is also its divisor, namely 1, 2 and 5 as sigma(1) = 1, sigma(2) = 3 and sigma(5) = 6, and all these (1, 3 and 6) are divisors of 30.
%t A309253 f[n_] := Count[Divisors[n], _?(Divisible[n, DivisorSigma[1, #]] &)]; m = 42; s = Table[0, {m}]; c = 0; n = 1; While[c < m, v = f[n]; If[v <=m && s[[v]] == 0, c++; s[[v]] = n]; n++]; s (* _Amiram Eldar_, Aug 08 2019 *)
%o A309253 (Magma) A309253:=func<n|exists(r){m:m in[1..200000] | #[d: d in Divisors(m) | IsIntegral(m / SumOfDivisors(d))] eq n }select r else 0>; [A309253(n): n in[1..40]];
%o A309253 (PARI) a(n) = my(m=1); while(sumdiv(m, d, !(m%sigma(d))) != n, m++); m; \\ _Jinyuan Wang_, Aug 08 2019
%Y A309253 Cf. A173441, A326696, A326697, A326698.
%K A309253 nonn
%O A309253 1,2
%A A309253 _Jaroslav Krizek_, Aug 08 2019