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.
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
Offset: 1
Keywords
Examples
-> 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).
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.
Crossrefs
Programs
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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
Extensions
More terms from David A. Corneth, Nov 04 2020
Comments