A337874 -id:A337874 - cp's OEIS frontend

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

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

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).

Original entry on oeis.org

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

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

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).

Original entry on oeis.org

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

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

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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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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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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).

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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).