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A338384 Integers that can be written m = k*tau(k) = q*tau(q) where (k, q) is a primitive solution of this equation and tau(k) is the number of divisors of k. 3
108, 192, 448, 2688, 6000, 8640, 12960, 17496, 18750, 20412, 32400, 86400, 112640, 120960, 138240, 169344, 181440, 245760, 304128, 600000, 658560, 714420, 857304, 979776, 1350000, 1632960, 1778112, 2073600, 2361960, 3359232, 3500000, 4561920, 7112448 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..33.

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

Subsequence of A338382.

Cf. A000005, A038040, A338381, A338383, A338385, A338519.

Cf. A337875 (similar for k*sigma(k))

Sequence in context: A208088 A323548 A338382 * A344702 A044340 A044721

Adjacent sequences:  A338381 A338382 A338383 * A338385 A338386 A338387

KEYWORD

nonn

AUTHOR

Bernard Schott, Nov 03 2020

EXTENSIONS

More terms from David A. Corneth, Nov 04 2020

STATUS

approved

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Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)