A327165 -id:A327165 - cp's OEIS frontend

A327153 Number of divisors d of n such that sigma(d)*d is equal to n.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

a(n) tells how many times in total n occurs in A064987.

			336 has 20 divisors: [1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336]. Only two of them, d=12 and d=14, satisfy sigma(d) = (336/d), thus a(336) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A064987, A327165 (positions of nonzero terms).

Cf. also A324539.

A327153(n) = sumdiv(n, d, (n==d*sigma(d)));

a(n) = Sum_{d|n} [A000203(d)*d == n], where [ ] is the Iverson bracket.

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

A327599 Odd numbers k that have a divisor d such that sigma(d)*d is equal to k.

Original entry on oeis.org

1, 117, 775, 2793, 9801, 16093, 30927, 88723, 90675, 137541, 292537, 326781, 488125, 732511, 796797, 954273, 1882881, 1926183, 2164575, 2896363, 3500157, 3618459, 4985713, 6725201, 7595775, 8042167, 10380591, 12326221, 12472075, 14076543, 16092297, 20456373, 23968425, 25774633

Offset: 1

David A. Corneth, Sep 18 2019

nonn

We need d and sigma(d) odd which happens precisely when d is an odd square.

			As 9 * sigma(9) = 9 * (1 + 3 + 9) = 9 * 13 = 117 is odd, 117 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Odd terms of A327165.

Cf. A064987.

Select[2Range[0, 9999] + 1, MemberQ[(DivisorSigma[1, #] * # &)/@Divisors[#], #] &] (* Alonso del Arte, Sep 18 2019 *)

upto(n) = {my(res = List()); forstep(i = 1, sqrtnint(n, 4), 2, c = i^2*sigma(i^2); if(c <= n, listput(res, c))); listsort(res, 1); res}

A338520 Integers that can be expressed as a product d*sigma(d), where sigma is the sum of divisors function, in a single way.

Original entry on oeis.org

1, 6, 12, 28, 30, 56, 72, 117, 120, 132, 180, 182, 306, 360, 380, 496, 552, 672, 702, 775, 792, 840, 870, 992, 1080, 1092, 1406, 1440, 1568, 1584, 1680, 1722, 1836, 1892, 2016, 2160, 2184, 2256, 2280, 2793, 2862, 3276, 3312, 3510, 3540, 3600, 3672, 3696, 3782, 3960

Offset: 1

Michel Marcus, Nov 01 2020

nonn

Integers m such that A327153(m) = 1.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000203, A064987, A327153, A337873.
Subsequence of A327165.
Subsequences: A000396 (perfect numbers), A036690 (p*(p+1)).
Cf. A338519 (similar for number of divisors).

f(n) = sumdiv(n, d, d*sigma(d)==n); \\ A327153
isok(n) = f(n)==1;

A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

Original entry on oeis.org

1, 6, 12, 28, 30, 56, 117, 120, 132, 140, 182, 306, 380, 496, 552, 672, 775, 870, 992, 1080, 1287, 1406, 1428, 1680, 1722, 1892, 2016, 2184, 2256, 2480, 2793, 2862, 3276, 3540, 3640, 3782, 3960, 4060, 4556, 4560, 4650, 5112, 5382, 5402, 5460, 6120, 6320, 6552

Offset: 1

Marius A. Burtea, Dec 06 2020

nonn

All terms are nonprimes.
The sequence includes all numbers of the form p*(p + 1) with p prime. Indeed: sigma(p*(p + 1)) = sigma(p)*sigma(p + 1) = (p + 1)*sigma(p + 1). So A036690 is a subsequence. Thus, the sequence is infinite.
Let k >= 1. If p and q = 1 + p + ... + p^(2*k) are prime numbers, then m = p^(2*k)*q is a term. Indeed, sigma(m) = sigma(p^(2*k)*q) = sigma(p^(2*k))*sigma(q) = q*sigma(q).
p is in: A053182 (k = 1), A065509 (k = 2), A163268 (k = 3), and A240693 (k = 5).
For k = 4 there are no prime p because 1 + p + p^2 + p^3 + p^4 + p^5 + p^6 + p^7 + p^8 = (p^6 + p^3 + 1)*(p^2 + p + 1).
If m = 2^(p - 1)*(2^p - 1), p >= 1, (see A006516), then sigma(m) = sigma(2^(p - 1)*(2^p - 1)) = sigma(2^(p - 1))*sigma(2^p - 1) = (2^p - 1)*sigma(2^p - 1), so m is a term.
Thus, A006516(n) and A000396(n), for n >= 1, are terms.

			sigma(6) = 12 = 3*4 = 3*sigma(3), so 6 is a term.
sigma(12) = 28 = 4*7 = 4*sigma(4), so 12 is a term.
sigma(30) = 72 = 6*12 = 6*sigma(6), so 30 is a term.
sigma(56) = 120 = 8*15 = 8*sigma(8), so 56 is a term.
sigma(117) = 182 = 13*14 = 13*sigma(13), so 117 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000203, A000396, A006516, A036690, A053182, A064987, A065509, A163268, A327165, A327599.

Magma
```
s:=func; [n:n in [1..6600]|s(n)];
```

q[n_] := Module[{d = Divisors[n], s}, s = Plus @@ d; AnyTrue[d, #*DivisorSigma[1, #] == s &]]; Select[Range[7000], q] (* Amiram Eldar, Dec 06 2020 *)

isok(k) = my(sk=sigma(k)); fordiv(k, d, if (d*sigma(d) == sk, return(1))); \\ Michel Marcus, Dec 06 2020

A348031 Multiply-perfect numbers k that have a divisor d such that sigma(d)*d = k.

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 23569920, 33550336, 142990848, 1379454720, 8589869056, 66433720320, 137438691328, 153003540480, 403031236608, 14942123276641920, 275502900594021408, 2305843008139952128, 71065075104190073088, 154345556085770649600, 203820700083634254643200, 34384125938411324962897920

Offset: 1

Antti Karttunen, Sep 26 2021

nonn

Multiply-perfect numbers (A007691) that are present in A064987.
Note how the first three terms of A005820 are included here, while the next three are in A348032.
At least the even terms of A000396 are all present.

Index entries for sequences related to sigma(n)

Intersection of A007691 and A327165.

Cf. A000396, A005820, A064987, A348032 (complement in A007691), A348035 (subsequence).

A348034 Numbers k that have at least one unitary divisor d such that sigma(d)*d is equal to k.

Original entry on oeis.org

1, 6, 12, 28, 30, 56, 117, 120, 132, 182, 306, 380, 496, 552, 672, 775, 870, 992, 1080, 1406, 1680, 1722, 1892, 2016, 2184, 2256, 2793, 2862, 3276, 3540, 3782, 3960, 4556, 4560, 4650, 5112, 5402, 5460, 6320, 6552, 6972, 7392, 8010, 8128, 9180, 9300, 9506, 9801, 10302, 10712, 11556, 11904, 11990, 12882, 16093, 16256

Offset: 1

Antti Karttunen, Sep 26 2021

nonn

			120 = 2^3 * 3 * 5 has a unitary divisor 8 for which sigma(8) = 15 = 120/8, thus 120 is included in the sequence.
672 = 2^5 * 3 * 7 has a unitary divisor 21 for which sigma(21) = 32 = 120/21, thus 672 is included in the sequence.

Index entries for sequences related to sigma(n)

Subsequence of A327165. Even terms of A000396 are all present.
Positions of nonzero terms in A348033.
Cf. also A348035 (multiply-perfect numbers in this sequence).

q[n_] := DivisorSum[n, 1 &, CoprimeQ[#, n/#] && #*DivisorSigma[1, #] == n &] > 0; Select[Range[16256], q] (* Amiram Eldar, Sep 27 2021 *)

isA348034(n) = { fordiv(n, d, if(1==gcd(d,n/d)&&n==d*sigma(d),return(1))); (0); };

A348032 Multiply-perfect numbers k that do not have a divisor d such that sigma(d)*d = k.

Original entry on oeis.org

2178540, 45532800, 459818240, 1476304896, 14182439040, 31998395520, 43861478400, 51001180160, 518666803200, 704575228896, 13661860101120, 30823866178560, 181742883469056, 740344994887680, 796928461056000, 6088728021160320, 20158185857531904, 212517062615531520, 622286506811515392, 69357059049509038080, 87934476737668055040

Offset: 1

Antti Karttunen, Sep 26 2021

nonn

Numbers k in A007691 for which A327153(k) = 0, that are not in A327165.

Question: Is A323653 a subsequence of this sequence? See also conjecture in A348033.

Index entries for sequences related to sigma(n)

Cf. A348031 (complement in A007691).

Cf. A327153, A327165, A348033.

With[{s = DeleteCases[Map[ToExpression[#] &, Import["https://oeis.org/A007691/b007691.txt", "Data"]], ?(Length[#] == 0 &)][[All, -1]]}, Select[s[[1 ;; 45]], Function[k, NoneTrue[Divisors[k], # DivisorSigma[1, #] == k &]]]] (* _Michael De Vlieger, Sep 28 2021, using b-file at A007691 *)

cp's OEIS Frontend

A327153 Number of divisors d of n such that sigma(d)*d is equal to 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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A327599 Odd numbers k that have a divisor d such that sigma(d)*d is equal to k.

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A338520 Integers that can be expressed as a product d*sigma(d), where sigma is the sum of divisors function, in a single way.

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A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

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A348031 Multiply-perfect numbers k that have a divisor d such that sigma(d)*d = k.

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A348034 Numbers k that have at least one unitary divisor d such that sigma(d)*d is equal to k.

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A348032 Multiply-perfect numbers k that do not have a divisor d such that sigma(d)*d = k.

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