A337876 -id:A337876 - cp's OEIS frontend

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).

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

A337875 Integers that can be written m = ksigma(k) = qsigma(q) where (k, q) is a primitive solution of this equation and sigma(m) is the sum of divisors of (m).

Original entry on oeis.org

336, 5952, 27776, 60480, 97536, 196560, 455168, 8062976, 15713280, 97493760, 104282640, 402604032, 1597639680, 1878818816, 2959632000, 6499584000, 15923980800, 18979440480, 33281933312, 54027792000, 102953410560, 103078428672, 103448378880

Offset: 1

Bernard Schott, Oct 09 2020

nonn

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 and 3rd example).
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 Examples a(1) and a(2)]. Hence, there exists an infinity of primitive solutions if the sequence A000043 of Mersenne exponents is infinite.
There exist terms m in A337873 that have three solutions like A337873(16) = 333312 = 336 * sigma(336) = 372 * sigma(372) = 434 * sigma(434) whose solutions (336,372), (336,434) and (372,434) are not primitive, but Jinyuan Wang has found some terms m in A337873 with 3 preimages as A337873(3266) = 18979440480 from which one pair is primitive and the two others not primitive [see example a(18)].

			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.
10080 is not a term: 60 * sigma(60) = 70 * sigma(70) = 10080 but as 60/5 = 12 and 70/5 = 14, hence, this solution that is generated by the first example is not primitive.
For a(4): 160 * sigma(160) = 189 * sigma(189) = 60480 is the smallest example with gcd(k,q) = 1 with k = 2^5*5 = 160 and q = 3^3*7 = 189.
For a(6): 315 * sigma(315) = 351 * sigma(351) = 196560 is the smallest example with k and q both odd.
For a(18): 76230 * sigma(76230) = 80028 * sigma(80028) = 84942 * sigma(84942) = A337873(3266) = 18979440480.
  -> 1) for k=76230 and q=84942; with d=11^2, k/11^2=630 and q/11^2=702.
  630 * sigma(630) = 702 * sigma(702) = 1179360, hence (76230, 84942) is not a primitive solution;
  -> 2) for k=80028 and q=84942; with d=13, k/13=6156 and q/13=6534.
  6156 * sigma(6156) = 6534 * sigma(6534) = 104282640, hence (80028, 84942) is not a primitive solution; but
  -> 3) for k=76230 and q=80028, there is no common divisor d such that k/d and q/d can satisfy (k/d)*sigma(k/d) = (q/d)*sigma(q/d), so (76239, 80028) is a primitive solution linked to m = 18979440480 that is the term a(18).

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

Mac Tutor History of Mathematics, Leo Moser.

Cf. A000203, A000668, A064987.
Subset of A337873.
Cf. A337874, A337876 (primitive solutions).

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)); resp = findprim(res, mx); vresp = Vec(resp); vecsort(vector(#vresp, k, vresp[k][1]*sigma(vresp[k][1])),,8);}
upto(10^12) \\ Michel Marcus, Oct 17 2020

More terms from Jinyuan Wang, Oct 10 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)).

A371419 Lesser member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

Original entry on oeis.org

12, 48, 112, 160, 192, 448, 1984, 12288, 28672, 126976, 196608, 458752, 520192, 786432, 1835008, 2031616, 8126464, 8323072, 33292288, 536805376, 2147221504, 3221225472, 7516192768, 33285996544, 34359476224, 136365211648

Offset: 1

Amiram Eldar, Mar 23 2024

Analogous to amicable numbers (A002025 and A002046) with the largest aliquot divisor of the sum of divisors (A371418) instead of the sum of aliquot divisors (A001065).
Carmichael (1921) proposed this function (A371418) for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> A371418(x). The chains of cycle 2 are analogous to amicable numbers.
Carmichael noted that if q < p are two different Mersenne exponents (A000043), then 2^(p-1)*(2^q-1) and 2^(q-1)*(2^p-1) are an amicable pair. With the 51 Mersenne exponents that are currently known it is possible to calculate 51 * 50 / 2 = 1275 amicable pairs. (160, 189) is a pair that is not of this "Mersenne form". Are there any other pairs like it? There are no other such pairs with lesser member below a(26).
a(27) <= 8795019280384.
The greater counterparts are in A371420.

			12 is a term since A371418(12) = 14 > 12, and A371418(14) = 12.

Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A000043, A001065, A002046, A337874, A337876, A371418, A371420.

Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

r[n_] := n/FactorInteger[n][[1, 1]]; s[n_] := r[DivisorSigma[1, n]]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 1, 10^6}]; seq

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m); for(n = 1, nmax, m = f(n); if(m > n && f(m) == n, print1(n, ", ")));}

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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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A337875 Integers that can be written m = k*sigma(k) = q*sigma(q) where (k, q) is a primitive solution of this equation and sigma(m) is the sum of divisors of (m).

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

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A371419 Lesser member of Carmichael's variant of amicable pair: numbers k < m such that s(k) = m and s(m) = k, where s(k) = A371418(k).

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A337875 Integers that can be written m = ksigma(k) = qsigma(q) where (k, q) is a primitive solution of this equation and sigma(m) is the sum of divisors of (m).

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.