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A324553
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a(n) = the smallest number m such that gcd(m, tau(m)) = n where tau(k) = the number of the divisors of k (A000005).
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2
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1, 2, 9, 8, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 384, 18939904, 180, 94633984, 240, 35721, 11264, 2218786816, 360, 10000, 53248, 26244, 1344, 225754218496, 720, 1031865892864, 1920, 7144929, 1114112, 1960000, 1260, 94076963651584, 4980736, 56070144, 1680, 1848279046291456, 4032, 8131987999031296, 33792, 3600, 96468992, 155444555888459776, 3360, 7529536, 30000
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OFFSET
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1,2
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COMMENTS
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a(n) = the smallest number m such that A009191(m) = n.
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
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LINKS
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FORMULA
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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
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EXAMPLE
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For n=3; a(3) = 9 because gcd(9, tau(9)) = gcd (9, 3) = 3 and 9 is the smallest.
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MATHEMATICA
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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 *)
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PROG
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(Magma) [Min([n: n in[1..10^6] | GCD(n, NumberOfDivisors(n)) eq k]): k in [1..16]]
(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), ", "))); };
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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