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A339056 Smallest integer k such that d(k^2)/d(k) = 2n-1, where d(k) is the number of divisors of k.

%I A339056 #36 Oct 03 2021 08:15:44
%S A339056 1,144,3600,1511654400000000,1587600,13168189440000,
%T A339056 177844628505600000000,192099600,76839840000,4757720360193884160000,
%U A339056 439167347942400000000,5037669383908052497858560000000000,32464832400,811620810000,831099709440000
%N A339056 Smallest integer k such that d(k^2)/d(k) = 2n-1, where d(k) is the number of divisors of k.
%C A339056 This sequence is related to the 3rd problem of the 39th International Mathematical Olympiad in 1998 at Taipei (Taiwan) (see link IMO).
%C A339056 As the set of integer values of the ratio d(k^2)/d(k) is exactly the set of all positive odd integers (see Marcin E. Kuczma reference), there exists, for each odd number, a smallest number k for which d(k^2)/d(k) = 2n-1.
%C A339056 All terms are perfect squares and if a number k is such that d(k^2)/d(k) = m, then all numbers that have the same prime signature as k give the same ratio m (see examples below); nevertheless, numbers with other prime signatures can also give this same ratio m (see example a(4)).
%C A339056 More results found by _Amiram Eldar_:
%C A339056 a(16) > 3*10^46,
%C A339056 a(17) = 13194538987069440000,
%C A339056 a(18) = 74219281802265600000000,
%C A339056 a(19) = 31164973305898534502400000000000000,
%C A339056 a(20) = 440046121805632742400000000,
%C A339056 a(21) = 439167347942400000000,
%C A339056 a(22) > 3*10^46.
%D A339056 Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 134-135.
%H A339056 The IMO Compendium, <a href="https://imomath.com/othercomp/I/Imo1998.pdf">Problem 3</a>, 39th IMO 1998.
%H A339056 39th International Mathematical Olympiad, <a href="https://sms.math.nus.edu.sg/simo/IMO_Problems/98.pdf">Problems and solutions</a>, Taiwan, July 1998.
%H A339056 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%e A339056 All numbers k with prime signature = [4, 2] give a ratio d(k^2)/d(k) = (9*5)/(5*3) = 3, and the smallest one is a(2) = 2^4*3^2 = 144.
%e A339056 All numbers k with prime signature = [4, 2, 2] give a ratio d(k^2)/d(k) = (9*5*5)/(5*3*3) = 5, and the smallest one is a(3) = 2^4*3^2*5^2 = 3600.
%e A339056 All numbers k with prime signature = [16, 10, 8] or [24, 12, 6] or [38, 10, 6] give the same ratio d(k^2)/d(k) = (33*21*17)/(17*11*9) = (49*25*13)/(25*13*7) = (77*21*13)/(39*11*7) = 7, but the smallest one is a(4) = 1511654400000000 = 2^16*3^10*5^8 < 2^24*3^12*5^6 < 2^38*3^10*5^6.
%e A339056 The successive prime signatures of the first ten terms are [], [4, 2], [4, 2, 2], [16, 10, 8], [4, 4, 2, 2], [16, 8, 4, 2], [16, 10, 8, 6], [4, 4, 2, 2, 2], [8, 4, 4, 2, 2], [28, 14, 4, 2, 2].
%o A339056 (PARI) isok(k, n) = numdiv(k^2)/numdiv(k) == n;
%o A339056 a(n) = my(k=1, m=2*n-1); while (!isok(k^2, m), k++); k^2; \\ _Michel Marcus_, Nov 26 2020
%Y A339056 Cf. A000005, A025487, A048691, A339055.
%Y A339056 Subsequence of A217584.
%K A339056 nonn
%O A339056 1,2
%A A339056 _Bernard Schott_, Nov 25 2020
%E A339056 a(12) corrected by _Amiram Eldar_, Nov 26 2020