A083874 -id:A083874 - cp's OEIS frontend

A105330 Numbers n such that 2^(n+1)+2n+1 is prime.

0, 1, 2, 3, 4, 7, 10, 13, 14, 26, 40, 49, 50, 110, 142, 170, 315, 349, 502, 842, 1251, 1630, 2054, 2906, 3482, 5110, 5227, 5620, 8224, 8788, 8912, 13027, 16243, 17222, 28557, 46532, 54974, 92866, 93093, 120855, 155416

Offset: 1

Farideh Firoozbakht, Apr 28 2005

If n is in the sequence & m=2^n*(2^(n+1)+2n+1) then sigma(m)+tau(m) =2m because sigma(m)=(2^(n+1)-1)*(2^(n+1)+2n+2), tau(m)=2*(n+1) so sigma(m)+tau(m)=(2^(n+1)-1)*(2^(n+1)+2n+2)+2*(n+1)=2m. Hence 2^A105330*(2^(A105330+1)+2*A105330+1) is a subsequence of A083874. A105331 is this subsequence. Next term is greater than 30500.
No other n < 10^5. -T. D. Noe, Jun 23 2008
No other n < 300000. - T. D. Noe, Apr 03 2009

			110 is in the sequence because 2^111+2*110+1=2596148429267413814265248164610269 is prime.

Cf. A083874, A105331.

Do[If[PrimeQ[2^(m + 1) + 2m + 1], Print[m]], {m, 0, 30500}]

is(n)=isprime(2^(n+1)+2*n+1) \\ Charles R Greathouse IV, Feb 20 2017

4 more terms from T. D. Noe, Jun 23 2008

Added two more terms -- T. D. Noe, Apr 03 2009

A238226 Numbers k such that if x = sigma(k) + tau(k) - k then k = sigma(x) + tau(x) - x.

Original entry on oeis.org

1, 3, 14, 52, 130, 144, 184, 274, 300, 586, 656, 8648, 10434, 11470, 12008, 15774, 17034, 18802, 19270, 21032, 22088, 22184, 23288, 34688, 35394, 36872, 38744, 39790, 65324, 65392, 67628, 68476, 153868, 163676, 188468, 198628, 254526, 263890, 379026, 463390

Offset: 1

Paolo P. Lava, Feb 20 2014

nonn

A083874 is a subset of this sequence: it lists the fixed points of the transform n -> sigma(n)+tau(n)-n.

			Fixed points: 1, 3, 14, 52, 130, 184, 656, 8648, 12008, 34688, ...
sigma(144) = 403, tau(144) = 15 and 403 + 15 - 144 = 274.
sigma(274) = 414, tau(274) = 4 and 414 + 4 - 274 = 144.

Amiram Eldar, Table of n, a(n) for n = 1..900 (terms 1..50 from Paolo P. Lava)

Cf. A000005, A000203, A083874, A238225, A238227-A238230.

with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)+tau(n)-n;
if sigma(a)+tau(a)-a=n then print(n);
fi; od; end: P(10^6);

f[n_] := DivisorSigma[0, n] + DivisorSigma[1, n] - n; s={}; Do[m = f[n]; If[f[m] == n, AppendTo[s, n]], {n, 1, 500000}]; s (* Amiram Eldar, Jul 12 2019 *)

A066229 f-perfect numbers, where f(m) = m + 1.

Original entry on oeis.org

4, 10, 44, 2336, 8896, 34432, 449295, 549775212544, 2251801457852416, 9007202677293056, 9223372167851278336, 20055918935605248255

Offset: 1

Joseph L. Pe, Dec 18 2001

f-perfect numbers are defined in A066218.
From Farideh Firoozbakht, Sep 18 2006: (Start)
n is in the sequence iff sigma(n) = 2*n - d(n) + 2, where d(n) is number of positive divisors of n.
If 2^(i+1) + 2*i - 1 is prime then n = 2^i*(2^(i+1) + 2*i - 1) is in the sequence because sigma(n) + d(n) - 2 = (2^(i+1) - 1)*(2^(i+1) + 2*i) + 2*(i+1) - 2 = 2^(2*i+2) + 2*i*2^(i+1) - 2^(i+1) = 2^(i+1)*(2^(i+1) + 2*i - 1) = 2*n, so sigma(n) = 2*n - d(n) + 2.
Hence if i is in {1, 2, 5, 6, 7, 19, 25, 26, 31, 38, 62, 80, 97, 110, 126, 133, 137, 409, 469, 685, 758, 1004, 1025, 1385, 2077, 2646, 2969, 3438, 7806, 8683, ...} then 2^i*(2^(i+1) + 2*i - 1) is in the sequence. 10, 44, 2336, 8896, 34432, 549775212544, 2251801457852416, 9007202677293056, 9223372167851278336, 151115727472444489859072, ... are such terms. (End)
Also, numbers n such that the number of nontrivial proper subgroups of a dihedral group D_{2n} is the same as its order. - Ivan Neretin, Jun 21 2016, after Dietrich Burde, see MSE link

			f(10) = 11 = 2 + 3 + 6 = f(1) + f(2) + f(5), hence 10 is a term of the sequence.

Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
Math StackExchange, Discussion on math.SE.

Cf. A066218, A066230, A083874.

Select[ Range[ 500000 ], DivisorSigma[ 1, # ] == 2# - DivisorSigma[ 0, # ] + 2 & ] (* Farideh Firoozbakht, Sep 18 2006 *)
f[x_] := x + 1; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

isok(m) = sigma(m) == 2*m-numdiv(m)+2; \\ Michel Marcus, Mar 13 2020

a(8) from Giovanni Resta, Aug 21 2013

a(9)-a(11) from Farideh Firoozbakht confirmed and a(12) added by Max Alekseyev, Jul 30 2025

A105331 Numbers of the form 2^n*(2^(n+1)+2n+1) where 2^(n+1)+2n+1 is prime.

Original entry on oeis.org

3, 14, 52, 184, 656, 34688, 2118656, 134438912, 537346048, 9007202811510784, 2417851639318318791262208, 633825300114170432793740312576, 2535301200456572518883997515776

Offset: 1

Farideh Firoozbakht, Apr 28 2005

This sequence is a subsequence of A083874 (see A083874 & A105330).

This is because these numbers satisfy tau(n) + sigma(n) = 2n when n = 2^k * p with p is prime; for instance tau(14) + sigma(14) = 4 + 24 = 28 = 2 x 14. [See References.] - Bernard Schott, Apr 07 2017

			9007202811510784 is in the sequence because 9007202811510784 = 2^26*(2^27 + 2*26 + 1) and 2^27 + 2*26 + 1 is prime.

J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Problème 723, page 93.

Cf. A007503, A083874, A105330.

Do[If[PrimeQ[2^(m + 1) + 2m + 1], Print[2^m(2^(m + 1) + 2m + 1)]], {m, 0, 110}]
2^# (2^(#+1)+2#+1)&/@Select[Range[0,100],PrimeQ[2^(#+1)+2#+1]&] (* Harvey P. Dale, Nov 13 2012 *)

a(n) = 2^A105330(n)*(2^(A105330(n)+1) + 2*A105330(n) + 1). - Bernard Schott, Apr 07 2017

A173168 Primes of the form 2^k + 2k - 1.

Original entry on oeis.org

3, 7, 13, 23, 41, 271, 2069, 16411, 32797, 134217781, 2199023255633, 1125899906842723, 2251799813685349, 2596148429267413814265248164610269, 11150372599265311570767859136324180752990493

Offset: 1

T. D. Noe, Feb 11 2010

nonn

Primes in A061761. The values of k-1 are in A105330. These primes, multiplied by 2^(k-1), produce A105331, which are a subset of the dihedral perfect numbers, A083874.

g[n_]:=2^n+2*n-1; Select[Array[g,100], PrimeQ]

A320457 Lesser members of dihedral amicable pairs: numbers (m, k) such that t(m) = t(k) = m + k, where t(k) = sigma(k) + d(k).

Original entry on oeis.org

144, 300, 10434, 15774, 17034, 21032, 22088, 35394, 36872, 65324, 67628, 153868, 188468, 254526, 379026, 483812, 492414, 905212, 1090528, 1198180, 1514212, 1634262, 1886046, 1898420, 2013414, 2184860, 2191588, 2316546, 2596448, 2816156, 3340024, 3854886

Offset: 1

Amiram Eldar, Dec 01 2018

nonn

Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.

			144 is in the sequence since (144, 274) is a pair of dihedral amicable numbers: sigma(144) + d(144) = 403 + 15 = 418, sigma(274) + d(274) = 414 + 4 = 418, and 144 + 274 = 418.

Amiram Eldar, Table of n, a(n) for n = 1..420
David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

Cf. A000005 (d), A000203 (sigma), A007503, A083874, A322254 (greater members).

t[n_] := DivisorSigma[0,n] + DivisorSigma[1,n] - n; s={}; Do[n = t[m]; If[n>m && t[n]==m, AppendTo[s,m]], {m, 1, 100000}];s

f(n) = numdiv(n) + sigma(n) - n;
isok(n) = my(nn = f(n)); (nn > n) && (n == f(nn)); \\ Michel Marcus, Dec 04 2018

A322254 Greater members of dihedral amicable pairs: numbers (m, k) such that t(m) = t(k) = m + k, where t(k) = sigma(k) + d(k).

Original entry on oeis.org

274, 586, 11470, 18802, 19270, 22184, 23288, 39790, 38744, 65392, 68476, 163676, 198628, 263890, 463390, 512116, 596258, 1070492, 1100384, 1342004, 1590452, 2139722, 2122946, 2262628, 2389562, 2562844, 2344436, 2831470, 2642656, 2949628, 3464008, 5476346

Offset: 1

Amiram Eldar, Dec 01 2018

nonn

Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.

The terms are ordered according to their lesser counterparts (A320457). - Amiram Eldar, Jul 03 2025

			274 is in the sequence since (144, 274) is a pair of dihedral amicable numbers: sigma(144) + d(144) = 403 + 15 = 418, sigma(274) + d(274) = 414 + 4 = 418, and 144 + 274 = 418.

Amiram Eldar, Table of n, a(n) for n = 1..420
David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

Cf. A000005 (d), A000203 (sigma), A007503, A083874, A320457 (lesser members).

t[n_] := DivisorSigma[0,n] + DivisorSigma[1,n]-n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s,n]], {m,1,100000}]; s

f(n) = numdiv(n) + sigma(n) - n;
isok(n) = my(nn = f(n)); (nn < n) && (n == f(nn)); \\ Michel Marcus, Dec 04 2018

A322256 Numbers k such that t(k) = t(k+1) where t(k) = tau(k) + sigma(k) = A007503(k) is the number of subgroups of the dihedral group of order 2k.

Original entry on oeis.org

14, 1334, 1634, 2685, 33998, 42818, 64665, 84134, 109214, 122073, 166934, 289454, 383594, 440013, 544334, 605985, 649154, 655005, 792855, 845126, 1642154, 2284814, 2305557, 2913105, 3571905, 3682622, 4701537, 5181045, 6431732, 6444873, 6771405, 10074477

Offset: 1

Amiram Eldar, Dec 01 2018

nonn

Jensen and Keane asked if this sequence is infinite. Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.

Supersequence of A054004. Terms that are not in it are 845126, 14392646, 10461888478, ...

Amiram Eldar, Table of n, a(n) for n = 1..400
David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

Cf. A007503, A054004, A083874.

[n: n in [1..2*10^6] | (NumberOfDivisors(n) + SumOfDivisors(n)) eq (NumberOfDivisors(n+1) + SumOfDivisors(n+1))]; // Vincenzo Librandi, Dec 08 2018

t[n_] := DivisorSigma[0, n] + DivisorSigma[1, n]; tQ[n_] := t[n] == t[n + 1]; Select[Range[1000000], tQ]

isok(n) = (numdiv(n)+sigma(n)) == (numdiv(n+1)+sigma(n+1)); \\ Michel Marcus, Dec 04 2018

A329104 Numbers m such that sigma(m) - tau(m) = 2m.

Original entry on oeis.org

56, 7192, 7232, 7912, 10792, 17272, 30592, 114256, 2154584, 3428368, 89245784

Offset: 1

Jaroslav Krizek, Nov 04 2019

Abundant numbers m with abundance A(m) = tau(m).
Corresponding values of A(m) = tau(m): 8, 16, 14, 16, 16, 16, 16, 20, 32, 30, 32, ...
a(12) > 10^13, if it exists. - Giovanni Resta, Nov 07 2019

			Number 56 is in the sequence because sigma(56) - tau(56) = 2*56; 120 - 8 = 112.

Cf. A000005, A000203, A033880, A065608.

Cf. A083874 (deficient numbers m with deficiency D(m) = tau(m)).

[m: m in [1..10^5] | SumOfDivisors(m) - NumberOfDivisors(m) eq 2*m];

Select[Range[4*10^6], DivisorSigma[1, #] - DivisorSigma[0, #] == 2 # &] (* Amiram Eldar, Nov 04 2019 *)

isok(m) = my(f=factor(m)); sigma(f) - numdiv(f) == 2*m; \\ Michel Marcus, Nov 05 2019

a(11) from Amiram Eldar, Nov 04 2019

A328951 Numbers m such that sigma(m) + tau(m) = 3m.

Original entry on oeis.org

60, 5472, 2500704, 24361213461200

Offset: 1

Jaroslav Krizek, Nov 10 2019

Abundant numbers m with abundance A(m) = m - tau(m) = A049820(m), where A049820(n) is the number of non-divisors of n.
Subsequence of A056076.
Corresponding values of A(m) = m - tau(m): 48, 5436, 2500632, ...
4 is the only number m with deficiency D(m) = m - tau(m).
808989640739424 is also a term. - Giovanni Resta, Nov 14 2019

			60 is a term because sigma(60) + tau(60) = 3*60; 168 + 12 = 180 = 3*60.

Cf. A000005, A000203, A033880, A049820, A056076.
Cf. A083874 (numbers m such that sigma(m) + tau(m) = 2m).
Cf. A011251 (numbers m such that sigma(m) + phi(m) = 3m).
Cf. A329104 (numbers m with abundance A(m) = tau(m)).

[m: m in [1..10^7] | SumOfDivisors(m) - 2*m eq m - NumberOfDivisors(m)];

Select[Range[3*10^6], DivisorSigma[0, #] + DivisorSigma[1, #] == 3# &] (* Amiram Eldar, Nov 10 2019 *)

isok(m) = my(f=factor(m)); sigma(f) + numdiv(m) == 3*m; \\ Michel Marcus, Nov 13 2019

a(4) from Martin Ehrenstein, Jul 25 2023

cp's OEIS Frontend

A105330 Numbers n such that 2^(n+1)+2n+1 is prime.

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A238226 Numbers k such that if x = sigma(k) + tau(k) - k then k = sigma(x) + tau(x) - x.

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A066229 f-perfect numbers, where f(m) = m + 1.

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A105331 Numbers of the form 2^n*(2^(n+1)+2n+1) where 2^(n+1)+2n+1 is prime.

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A173168 Primes of the form 2^k + 2k - 1.

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A320457 Lesser members of dihedral amicable pairs: numbers (m, k) such that t(m) = t(k) = m + k, where t(k) = sigma(k) + d(k).

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A322254 Greater members of dihedral amicable pairs: numbers (m, k) such that t(m) = t(k) = m + k, where t(k) = sigma(k) + d(k).

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A322256 Numbers k such that t(k) = t(k+1) where t(k) = tau(k) + sigma(k) = A007503(k) is the number of subgroups of the dihedral group of order 2k.

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