A338382 -id:A338382 - cp's OEIS frontend

A338383 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*tau(k) = A338382(n).

18, 27, 24, 32, 56, 64, 90, 135, 126, 189, 120, 160, 198, 297, 168, 192, 224, 234, 351, 306, 459, 342, 513, 264, 352, 280, 320, 414, 621, 312, 416, 400, 500, 522, 783, 408, 544, 558, 837, 456, 608, 666, 999, 450, 675, 360, 432, 552, 736, 738, 1107, 774, 1161, 616, 704

Offset: 1

Bernard Schott, Oct 26 2020

The map k -> k*tau(k) = m is not injective (A038040) and this sequence lists, in increasing order of m, the preimages of the integers m that have more than one preimage.

			The table begins:
   18,  27;
   24,  32;
   56,  64;
   90, 135;
  126, 189;
  120, 160;
  198, 297;
  168, 192, 224;
  ...
1st row is (18, 27) because 18 * tau(18) = 27 * tau(27) = 108 = A338382(1).
2nd row is (24, 32) because 24 * tau(24) = 32 * tau(32) = 192 = A338382(2).
8th row is (168, 192, 224), because 168 * tau(168) = 192 * tau(192) = 224 * tau(224) = 2688 = A338382(8); it is the first row with 3 preimages.

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.

Cf. A000005, A038040, A327166, A338381, A338382, A338384, A338385.

Cf. A337874 (similar for k*sigma(k)).

upto(n) = {m = Map(); res = List(); n = n\2; w = []; for(i = 1, n, c = i*numdiv(i); if(mapisdefined(m, c), listput(res, c); l = mapget(m, c); listput(l, i); mapput(m, c, l) , mapput(m, c, List(i)); ) ); listsort(res, 1); v = select(x -> x <= 2*(n+1), res); for(i = 1, #v, w = concat(w, Vec(mapget(m, v[i]))) ); w; } \\ Michel Marcus, Oct 27 2020

A338384 Integers that can be written m = ktau(k) = qtau(q) where (k, q) is a primitive solution of this equation and tau(k) is the number of divisors of k.

Original entry on oeis.org

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

Bernard Schott, Nov 03 2020

nonn

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.

			-> 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).

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.

Subsequence of A338382.
Cf. A000005, A038040, A338381, A338383, A338385, A338519.
Cf. A337875 (similar for k*sigma(k))

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

More terms from David A. Corneth, Nov 04 2020

A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), kA338384(n).

Original entry on oeis.org

18, 27, 24, 32, 56, 64, 192, 224, 400, 500, 360, 432, 540, 648, 972, 2187, 1875, 3125, 1458, 1701, 1296, 1350, 2160, 2400, 5120, 5632, 2880, 3024, 3840, 4608, 4032, 4704, 3780, 5184, 10240, 16384, 8448, 9216, 20000, 25000, 15680, 16464, 15876, 25515, 20412, 23814

Offset: 1

Bernard Schott, Nov 09 2020

As the multiplicativity of tau(k) ensures an infinity of solutions to the general equation k*tau(k) = q*tau(q) (see A338382), Richard K. Guy asked if there is an infinity of primitive solutions. A solution (k, q) with m = k*tau(k) = q*tau(q) is primitive in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, m' = m/(d*tau(d)), d>1 with m' = k' * tau(k') = q' * tau(q').

Warning, Richard K. Guy asked if "there is an infinity of primitive solutions (for k*tau(k) = q*tau(q)), in the sense that (k', q') is not a solution for any k' = k/d, q' = q/d, d>1". It appears that this definition is not enough well defined, because some solutions as (4032, 4704), (20000, 25000), (20412, 23814),... that are primitive are not obtained in this case (see detailed example (20000, 25000) below). The mathematical explanation is that tau satisfies the relation tau(r*s) = tau(r) * tau(s) * (t/tau(t)) where t = gcd(r,s).

			The table begins:
    18,  27;
    24,  32;
    56,  64;
   192, 224;
   400, 500;
   360, 432;
   ...
1st row is (18, 27) because 18 * tau(18) = 27 * tau(27) = 108 = A338384(1).
4th row is (192, 224) because 192 * tau(192) = 224 * tau(24) = 2688 = A338384(4); Note that 168 * tau(168) = 192 * tau(192) = 224 * tau(24) = 2688 = A338382(8) but (168, 192) and (168, 224) are not primitive solutions (see detailed example in A338384).
5th row is (400, 500) because 400 * tau(400) = 500 * tau(500) = 6000.
20th row is (20000, 25000) although (20000/50, 25000/50) = (400, 500) and that (400, 500) is the 5th row. Explanation: A338384(20) = 600000 = 20000*tau(20000) = 25000*tau(25000) and this pair is primitive, because for d = 50, we get 600000/(50*tau(50)) = 2000 <>  (20000/50)*tau(20000/50) = (25000/50)*tau(25000/50) = 6000. To be exhaustive, the two other pairs linked with 600000: (15000, 20000) and (15000, 25000) are not primitive.

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.

Cf. A000005, A038040, A338381, A338382, A338383, A338384.

Cf. A337876 (similar for k*sigma(k)).

A338519 Integers that can be expressed as a product d*tau(d), where tau is the number of divisors function, in a single way.

Original entry on oeis.org

1, 4, 6, 10, 12, 14, 22, 24, 26, 27, 32, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 75, 80, 82, 84, 86, 88, 94, 104, 106, 118, 120, 122, 132, 134, 136, 140, 142, 146, 147, 152, 156, 158, 166, 168, 178, 184, 194, 202, 204, 206, 214, 218, 220, 226, 228, 232, 240, 248, 254

Offset: 1

Michel Marcus, Nov 01 2020

nonn

Integers m such that A327166(m) = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A038040, A327166, A338382.
Subsequences: A100484 (2*p), A079705 (3*p^2) that gives odd terms.
Cf. A338520 (similar for sum of divisors).

f(n) = sumdiv(n, d, d*numdiv(d) == n); \\ A327166
isok(n) = f(n)==1;

cp's OEIS Frontend

A338383 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*tau(k) = A338382(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

A338384 Integers that can be written m = ktau(k) = qtau(q) where (k, q) is a primitive solution of this equation and tau(k) is the number of divisors of k.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions

A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), kA338384(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A338519 Integers that can be expressed as a product d*tau(d), where tau is the number of divisors function, in a single way.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A338383 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*tau(k) = A338382(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions

A338385 Table read by rows, in which the n-th row lists the primitive solutions (k, q), kA338384(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A338519 Integers that can be expressed as a product d*tau(d), where tau is the number of divisors function, in a single way.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A338384 Integers that can be written m = ktau(k) = qtau(q) where (k, q) is a primitive solution of this equation and tau(k) is the number of divisors of k.