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A337324 a(n) is the smallest number m such that gcd(m, tau(m), sigma(m), pod(m)) = n where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

%I A337324 #33 Sep 08 2022 08:46:25
%S A337324 1,6,18,24,5000,90,66339,56,288,3240,10036224,60,582160384,20412,
%T A337324 16200,3968,49030215219,612,4637065216,1520,142884,912384,
%U A337324 98881718827959,480,7543125,479232,3175200,5824,26559758051835904,76950,25796647321600,2688,491774976,1268973568
%N A337324 a(n) is the smallest number m such that gcd(m, tau(m), sigma(m), pod(m)) = n where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).
%C A337324 From _David A. Corneth_, Aug 24 2020: (Start)
%C A337324 a(35) <= 1289027059712000000.
%C A337324 a(36) <= 136064563937280.
%C A337324 a(37) = 207816012706349056.
%C A337324 a(38) <= 1835772101525504.
%C A337324 a(39) <= 418089296461824.
%C A337324 a(40) <= 11698803719536640.
%C A337324 gcd(m, tau(m), sigma(m), pod(m)) = gcd(m, tau(m), sigma(m)) which may ease the search.
%C A337324 (End)
%e A337324 For n = 6; a(6) = 90 because 90 is the smallest number with gcd(90, tau(90), sigma(90), pod(90)) = gcd(90, 12, 234, 531441000000) = 6.
%o A337324 (Magma) [Min([m: m in[1..10^5] | GCD([m, #Divisors(m), &+Divisors(m), &*Divisors(m)]) eq k]): k in [1..10]]
%o A337324 (PARI) a(n) = {for(i = 1, oo, f = factor(i); if(gcd([i, numdiv(f), sigma(f)]) == n, return(i)))} \\ _David A. Corneth_, Aug 24 2020
%Y A337324 Cf. A337323 (gcd(n, tau(n), sigma(n), pod(n))).
%Y A337324 Cf. A337325 (least m such that gcd(tau(m), sigma(m), pod(m)) = n).
%Y A337324 Cf. A000005, A000203, A007955.
%K A337324 nonn
%O A337324 1,2
%A A337324 _Jaroslav Krizek_, Aug 23 2020
%E A337324 a(11) from _Amiram Eldar_, Aug 24 2020
%E A337324 More terms from _Jaroslav Krizek_ and _David A. Corneth_, Aug 24 2020