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A324553
a(n) = the smallest number m such that gcd(m, tau(m)) = n where tau(k) = the number of the divisors of k (A000005).
2
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
OFFSET
1,2
COMMENTS
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
FORMULA
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
EXAMPLE
For n=3; a(3) = 9 because gcd(9, tau(9)) = gcd (9, 3) = 3 and 9 is the smallest.
MATHEMATICA
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 *)
PROG
(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), ", "))); };
A324553search_and_print(2^29, 127); \\ Antti Karttunen, Mar 06 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Mar 05 2019
EXTENSIONS
More terms from Antti Karttunen (terms a(17) and a(39) also computed by Jon E. Schoenfield), Mar 06 2019
STATUS
approved