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%I #26 Sep 08 2022 08:46:24
%S 1,2,9,8,400,12,3136,24,36,80,123904,60,692224,448,2025,384,18939904,
%T 180,94633984,240,35721,11264,2218786816,360,10000,53248,26244,1344,
%U 225754218496,720,1031865892864,1920,7144929,1114112,1960000,1260,94076963651584,4980736,56070144,1680,1848279046291456,4032,8131987999031296,33792,3600,96468992,155444555888459776,3360,7529536,30000
%N a(n) = the smallest number m such that gcd(m, tau(m)) = n where tau(k) = the number of the divisors of k (A000005).
%C a(n) = the smallest number m such that A009191(m) = n.
%C The sequence is well-defined. Proof: Let p_1^e_1 * p_2^e2 *...* pk^ek = n then gcd(n * p_(k+1)^(p1-1) * p_(k+2)^(p2-1) * ... *p_(2k)^(pk-1), tau(n * p_(k+1)^(p1-1) * p_(k+2)^(p2-1) * ... *p_(2k)^(pk-1)) = n where p_i is prime and j < m <=> p_j < p_m. Q.E.D. - _David A. Corneth_, Mar 07 2019
%F For primes p >= 5, a(p) = p^2 * 2^(p-1). For odd primes p, a(2*p) = p * 2^(p-1). - _Antti Karttunen_, Mar 06 2019
%e For n=3; a(3) = 9 because gcd(9, tau(9)) = gcd (9, 3) = 3 and 9 is the smallest.
%t Array[If[And[# > 3, PrimeQ@ #], #^2*2^(# - 1), Block[{m = 1}, While[GCD[m, DivisorSigma[0, m]] != #, m++]; m]] &, 32] (* _Michael De Vlieger_, Mar 24 2019 *)
%o (Magma) [Min([n: n in[1..10^6] | GCD(n, NumberOfDivisors(n)) eq k]): k in [1..16]]
%o (PARI) A324553search_and_print(searchlimit,primes_up_to) = { my(m = Map(), k); forprime(p=5,primes_up_to,mapput(m,p,(p^2 * 2^(p-1))); mapput(m,2*p,p * 2^(p-1))); for(n=1,searchlimit,k=gcd(n,numdiv(n)); if(!mapisdefined(m,k), mapput(m,k,n), if(mapget(m,k)>n, print("Presuppositions failed: the first occurrence of ", k, " is already at ",n," not at ", mapget(m,k), " !"); return(1/0)))); for(k=1, oo, if(!mapisdefined(m,k), break, print1(mapget(m,k), ", "))); };
%o A324553search_and_print(2^29,127); \\ _Antti Karttunen_, Mar 06 2019
%Y Cf. A000005, A009191.
%Y Cf. also A005179, A037019, A324554, A324555.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Mar 05 2019
%E More terms from _Antti Karttunen_ (terms a(17) and a(39) also computed by _Jon E. Schoenfield_), Mar 06 2019
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