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).

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

Author

Jaroslav Krizek, Mar 05 2019

Keywords

• nonn

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

Examples

			For n=3; a(3) = 9 because gcd(9, tau(9)) = gcd (9, 3) = 3 and 9 is the smallest.
		

Crossrefs

Cf. A000005, A009191.

Cf. also A005179, A037019, A324554, A324555.

Programs

Magma

[Min([n: n in[1..10^6] | GCD(n, NumberOfDivisors(n)) eq k]): k in [1..16]]

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 *)

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

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

Extensions

More terms from Antti Karttunen (terms a(17) and a(39) also computed by Jon E. Schoenfield), Mar 06 2019