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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 (list; graph; refs; listen; history; text; internal format)
 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 LINKS 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 Cf. A000005, A009191. Cf. also A005179, A037019, A324554, A324555. Sequence in context: A092270 A249225 A191351 * A230283 A121067 A073904 Adjacent sequences:  A324550 A324551 A324552 * A324554 A324555 A324556 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

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)