OFFSET
1,1
COMMENTS
As the multiplicativity of tau(k) ensures an infinity of solutions to the general equation m = k*tau(k) (see A338382), Richard K. Guy asked if, as for k*sigma(k) = q*sigma(q) (A337875, A337876), k*tau(k) = q*tau(q) has an infinity of primitive solutions, in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, d>1 (see reference Guy's book and 3rd example). The answer to this question seems not to be known today.
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, p. 102-103.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 168, page 127.
EXAMPLE
-> For a(1): 18 * tau(18) = 27 * tau(27) = 108.
-> For a(2): 24 * tau(24) = 32 * tau(32) = 192.
-> Why 1080 = A338382(4) is not a term? 90 * tau(90) = 135 * tau(135) = 1080 but as 90/5 = 18 and 135/5 = 27, this solution that is generated by the first example is not primitive.
-> For a(4) : 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = A338382(8) = 2688.
1) for k=168 and q=192; with d=3, k/3=56 and q/3=64, with 56 * tau(56) = 64 * tau(64) = 448 = a(3), hence (168, 192) is not a primitive solution;
2) for k=168 and q=224; with d=7, k/7=24 and q/7=32, with 24 * tau(24) = 32 * tau(32) = 192 = a(2), hence (24, 32) is not a primitive solution; but
3) for k=192 and q=224, there is no common divisor d such that 192/d and 224/d can satisfy (192/d)*tau(192/d) = (224/d)*tau(224/d), so (192, 224) is a primitive solution linked to m = 2688 that is the term a(4).
PROG
(PARI) is(n) = {my(l, d); l = List(); d = divisors(n); for(i = 1, #d, if(d[i]*numdiv(d[i]) == n, listput(l, d[i]); ) ); forvec(x = vector(2, i, [1, #l]), if(isprimitive(l[x[1]], l[x[2]], n), return(1) ) , 2 ); 0 }
isprimitive(m, n, t) = { my(g = gcd(m, n), d = divisors(g)); for(i = 2, #d, if(m/d[i]*numdiv(m/d[i]) == t/d[i]/numdiv(d[i]) && n/d[i]*numdiv(n/d[i]) == t/d[i]/numdiv(d[i]), return(0) ) ); 1 } \\ David A. Corneth, Nov 06 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Nov 03 2020
EXTENSIONS
More terms from David A. Corneth, Nov 04 2020
STATUS
approved