A337875 -id:A337875 - cp's OEIS frontend

A337876 Table read by rows, in which the n-th row lists all the primitive solutions k, in increasing order, such that k*sigma(k) = A337875(n).

12, 14, 48, 62, 112, 124, 160, 189, 192, 254, 315, 351, 448, 508, 1984, 2032, 2560, 2728, 5580, 5616, 6156, 6534, 12288, 16382, 22464, 22860, 28672, 32764, 28800, 34000, 42000, 51200, 46500, 51200, 51200, 54250, 72800, 95697, 76230, 80028, 126976, 131056, 119700, 189875

Offset: 1

Bernard Schott, Oct 20 2020

As the multiplicativity of sigma(k) ensures an infinity of solutions to the general equation m = k*sigma(k) (see A337873), Leo Moser asked if k*sigma(k) = q*sigma(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 References). This sequence lists in increasing order of m the primitive solutions (k, q).

A subset of primitive solutions: if 2^p-1 and 2^r-1 are distinct Mersenne primes (A000668), then k = (2^p-1) * 2^(r-1) and q = (2^r-1) * 2^(p-1) satisfy k*sigma(k) = q*sigma(q) = m = (2^p-1) * (2^r-1) * 2^(p+r-1) [see first 2 examples]. Hence, there exists an infinity of primitive solutions if the sequence A000043 of Mersenne exponents is infinite.

			The table begins:
   12,   14;
   48,   62;
  112,  124;
  160,  189;
  192,  254;
  315,  351;
  ...
1st row is (12, 14) because 12 * sigma(12) = 14 * sigma(14) = 336 = A337875(1) with p = 2 and r = 3.
2nd row is (48, 62) because 48 * sigma(48) = 62 * sigma(62) = 5952 = A337875(2) with p = 2 and r = 5.
16th row is (42000, 51200), (46500, 51200), (51200, 54250) because 42000 * sigma(42000) = 51200 * sigma(51200), 46500 * sigma(46500) = 51200 * sigma(51200) and  51200 * sigma(51200) = sigma54250 * sigma(54250) = 649999584000 = A337875(16). These 3 primitive solutions corresponding to the smallest m = 649999584000 have been found by _Michel Marcus_. The three other possible solutions (42000, 46500), (42000, 54250), (46500, 54250) are not primitive.
18th row is (76230, 80028) because 76230 * sigma(76230) = 80028 * sigma(80028) = 18979440480 = A337875(18). Note that 76230 * sigma(76230) = 80028 * sigma(80028) = 84942 * sigma(84942) = 18979440480 = A337873(3266) but (76230, 84942) and (80028, 84942) are not primitive solutions (see detailed example in A337875). These case have been found by _Jinyuan Wang_.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.

Cf. A000203, A000668, A064987.

Cf. A337873, A337874, A337875.

process(x, y, resp) = {my(vresp = Vec(resp)); for (i=1, #vresp, if (x/vresp[i][1] == y/vresp[i][2], return(resp));); listput(resp, [x, y]); resp;}
findprim(res, mx) = {my(mp = Map()); my(resp = List()); for (i=1, #res, my(vx = mapget(mx, res[i])); for (j=1, #vx-1, for (k=j+1, #vx, resp = process(vx[j], vx[k], resp);););); resp;}
upto(n) = {my(m = Map(), mx = Map(), res = List(), n = sqrtint(n), resp); for(i = 1, n, my(c = i*sigma(i)); if(mapisdefined(m, c), listput(res, c); mapput(m, c, mapget(m, c) + 1); mapput(mx, c, concat(mapget(mx, c), i)), mapput(m, c, 1); mapput(mx, c, [i]);)); listsort(res, 1); res = Vec(select(x -> x <= (n+1)^2, res)); Vec(findprim(res, mx));}
upto(10^11) \\ Michel Marcus, Oct 20 2020

More terms from Michel Marcus, Oct 20 2020

A337873 Numbers m such that the equation m = k*sigma(k) has more than one solution.

Original entry on oeis.org

336, 5952, 10080, 27776, 44352, 60480, 61152, 97536, 102816, 127680, 178560, 185472, 196560, 260400, 292320, 333312, 455168, 472416, 578592, 635712, 758016, 785664, 833280, 961632, 1083264, 1179360, 1189440, 1270752, 1330560, 1530816, 1717632, 1815072, 1821312, 1834560

Offset: 1

Bernard Schott, Sep 27 2020

The map k -> k*sigma(k) = m is not injective (A064987), this sequence lists in increasing order the integers m that have several preimages.
These terms m satisfy A327153(m) > 1.
If 2^p-1 and 2^r-1 are distinct Mersenne primes (A000668), then k = (2^p-1)* 2^(r-1) and q = (2^r-1) * 2^(p-1) satisfy k*sigma(k) = q*sigma(q) = m = (2^p-1) * (2^r-1) * 2^(p+r-1) [see examples a(1) and a(2)].
The multiplicativity of sigma(k) ensures an infinity of solutions and thus of terms m [see example a(3)].

			For a(1): 12 * sigma(12) = 14 * sigma(14) = 336 with p=2 and r=3.
For a(2): 48 * sigma(48) = 62 * sigma(62) = 5952 with p=2 and r=5.
For a(3): 60 * sigma(60) = 70 * sigma(70) = 10080 with 60/12 = 70/14 = 5.
a(16) = 333312 is the smallest term with 3 preimages because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.

David A. Corneth, Table of n, a(n) for n = 1..10000
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Cf. A000203, A000668, A064987.
Cf. A327153. Subsequence of A327165.
Cf. A212490, A337874 (preimages), A337875 (primitive terms).

m = 2*10^6; v = Table[0, {m}]; Do[i = n*DivisorSigma[1, n]; If[i <= m, v[[i]]++], {n, 1, Floor@Sqrt[m]}]; Position[v, ?(# > 1 &)] // Flatten (* _Amiram Eldar, Sep 28 2020 *)

upto(n) = {m = Map(); res = List(); n = sqrtint(n); for(i = 1, n, c = i*sigma(i); if(mapisdefined(m, c), listput(res, c); mapput(m, c, mapget(m, c) + 1) , mapput(m, c, 1); ) ); listsort(res, 1); select(x -> x <= (n+1)^2, res) } \\ David A. Corneth, Sep 27 2020

isok(m) = {my(nb=0); fordiv(m, d, if (d*sigma(d) == m, nb++; if (nb>1, return(1)));); return (0);} \\ Michel Marcus, Sep 29 2020

More terms from David A. Corneth, Sep 27 2020

A337874 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*sigma(k) = A337873(n).

Original entry on oeis.org

12, 14, 48, 62, 60, 70, 112, 124, 132, 154, 160, 189, 156, 182, 192, 254, 204, 238, 228, 266, 240, 310, 276, 322, 315, 351, 300, 350, 348, 406, 336, 372, 434, 448, 508, 444, 518, 492, 574, 516, 602, 564, 658, 528, 682, 560, 620, 636, 742

Offset: 1

Bernard Schott, Oct 06 2020

The map k -> k*sigma(k) = m is not injective (A064987) and this sequence lists, in increasing order of m, the preimages of the integers m that have more than one preimage.

If 2^p-1 and 2^r-1 are distinct Mersenne primes (A000668), then k = (2^p-1) * 2^(r-1) and q = (2^r-1) * 2^(p-1) satisfy k*sigma(k) = q*sigma(q) = m = (2^p-1) * (2^r-1) * 2^(p+r-1) [see first 2 examples].

			The table begins:
   12,  14;
   48,  62;
   60,  70;
  112, 124;
  132, 154;
  160, 189;
  ...
1st row is (12, 14) because 12 * sigma(12) = 14 * sigma(14) = 336 = A337873(1) with p = 2 and r = 3.
2nd row is (48, 62) because 48 * sigma(48) = 62 * sigma(62) = 5952 = A337873(2) with p = 2 and r = 5.
16th row is (336, 372, 434) because 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) = 333312 = A337873(16).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B11, p. 101-102.

Cf. A000203, A064987, A327153.

Cf. A337873, A337875, A337876.

m = 10^6; v = Table[{}, {m}]; Do[i = n*DivisorSigma[1, n]; If[i <= m, AppendTo[v[[i]], n]], {n, 1, Floor@Sqrt[m]}]; Select[v, Length[#] > 1 &] // Flatten (* Amiram Eldar, Oct 06 2020 *)

upto(n) = {m = Map(); res = List(); n = sqrtint(n); w = []; for(i = 1, n, c = i*sigma(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 <= (n+1)^2, res); for(i = 1, #v, w = concat(w, Vec(mapget(m, v[i]))) ); w } \\ David A. Corneth, Oct 07 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

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A337876 Table read by rows, in which the n-th row lists all the primitive solutions k, in increasing order, such that k*sigma(k) = A337875(n).

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A337873 Numbers m such that the equation m = k*sigma(k) has more than one solution.

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A337874 Table read by rows, in which the n-th row lists all the preimages k, in increasing order, such that k*sigma(k) = A337873(n).

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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.

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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.