A181701 -id:A181701 - cp's OEIS frontend

A181702 Numbers of the form 2^r*p with p an odd prime, which are in A181595 but not in A181701.

Original entry on oeis.org

24, 40, 224, 15872, 1040384

Offset: 1

Vladimir Shevelev, Nov 06 2010

a(6) > 1.85*10^11, if it exists. - Amiram Eldar, Apr 29 2024

Cf. A181701, A181595, A181596, A000396.

32128 removed by R. J. Mathar, Dec 05 2010

a(5) from Amiram Eldar, Apr 29 2024

A181595 Abundant numbers n for which the abundance d = sigma(n) - 2*n is a proper divisor, that is, 0 < d < n and d | n.

Original entry on oeis.org

12, 18, 20, 24, 40, 56, 88, 104, 196, 224, 234, 368, 464, 650, 992, 1504, 1888, 1952, 3724, 5624, 9112, 11096, 13736, 15376, 15872, 16256, 17816, 24448, 28544, 30592, 32128, 77744, 98048, 122624, 128768, 130304, 174592, 396896, 507392, 521728, 522752, 537248

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

Named near-perfect numbers by sequence author.
Union of this sequence and A005820 is A153501.
Every even perfect number n = 2^(p-1)*(2^p-1), p and 2^p-1 prime, of A000396 generates three entries: 2*n, 2^p*n and (2^p-1)*n.
Every number M=2^(t-1)*P, where P is a prime of the form 2^t-2^k-1, is an entry for which (2^k)|M and sigma(M)-2^k=2*M (see A181701).
Conjecture 1: For every k>=1, there exist infinitely many entries m for which (2^k)|m and sigma(m)-2^k = 2*m.
Conjecture 2. All entries are even. [Proved to be false, see below. (Ed.)]
Conjecture 3. If the suitable (according to the definition) divisor d of an entry is not a power of 2, then it is not suitable divisor for any other entry.
Conjecture 4. If a suitable divisor for an even entry is odd, then it is a Mersenne prime (A000043).
If Conjectures 3 and 4 are true, then an entry with odd suitable divisor has the form 2^(p-1)*(2^p-1)^2, where p and 2^p-1 are primes. - Vladimir Shevelev, Nov 08 2010 to Dec 16 2010
The only odd term in this sequence < 2*10^12 is 173369889. - Donovan Johnson, Feb 15 2012
173369889 remains only odd term up to 1.4*10^19. - Peter J. C. Moses, Mar 05 2012
These numbers are obviously pseudoperfect (A005835) since they are equal to the sum of all the proper divisors except the one that is the same as the abundance. - Alonso del Arte, Jul 16 2012

			The abundance of 12 is A033880(12) = 4, which is a proper divisor of 12, so 12 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..200
Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
Yanbin Li and Qunying Liao, A class of new near-perfect numbers, J. Korean Math. Soc. 52 (2015), No. 4, pp. 751-763.
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046. arXiv preprint, arXiv:1011.6160 [math.NT], 2010-2012.
X.-Z. Ren, Y.-G. Chen, On near-perfect numbers with two distinct prime factors, Bulletin of the Australian Mathematical Society, No 3 (2013), available on CJO2013. doi:10.1017/S0004972713000178.
M. Tang, X. Z. Ren and M. Li, On near-perfect and deficient-perfect numbers, Colloq. Math. 133 (2013), 221-226.

Cf. A000396, A005101, A153501, A005820.

q:= n-> (t-> t>0 and tAlois P. Heinz, May 11 2023

Select[Range[550000], 0 < (d = DivisorSigma[1, #] - 2*#) < # && Divisible[#, d] &] (* Amiram Eldar, May 12 2023 *)

is_A181595(n)=my(d=sigma(n)-2*n); (d>0) && (dA181595(n)&&print1(n","))  \\ M. F. Hasler, Apr 14 2012; corrected by Michel Marcus, May 12 2023

Definition shortened, entries checked by R. J. Mathar, Nov 17 2010

A181741 Primes of the form 2^t-2^k-1, k>=1.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 29, 31, 47, 59, 61, 127, 191, 223, 239, 251, 383, 479, 503, 509, 991, 1019, 1021, 2039, 3583, 3967, 4079, 4091, 4093, 6143, 8191, 15359, 16127, 16319, 16381, 63487, 65407, 65519, 129023, 131063, 131071, 245759, 253951, 261631, 261887, 262079, 262111, 262127, 262139

Offset: 1

Vladimir Shevelev, Nov 08 2010

nonn

All Mersenne primes A000668(i) are in the sequence, parametrized by t=A000043(i)+1 and k=A000043(i).
If p is in the sequence, then the exponents t and k are unique.
For given k, the smallest value of t defines sequence A181692.
Every term p=2^t-2^k-1 in this sequence here generates an entry 2^(t-1)*p in A181595 (cf. A181701).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000, probable primes for n > 150
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046. - _N. J. A. Sloane_, Sep 04 2012
Vladimir Shevelev, Perfect and near-perfect numbers, arXiv:1011.6160 [math.NT], 2010-2012.

Cf. A181595, A181692, A181701, A000043.

Cf. A010051, primes in A081118, see also A208083.

a181741 n = a181741_list !! (n-1)
a181741_list = filter ((== 1) . a010051) a081118_list
-- Reinhard Zumkeller, Feb 23 2012

isA000079 := proc(n) if n = 1 then true; elif type(n,'odd') then false; else if nops( numtheory[factorset](n) ) = 1 then  true;  else
false; end if; end if; end proc:
isA181741 := proc(p) if isprime(p) then k := A007814(p+1) ; (p+1)/2^k+1 ; return ( isA000079(%) and k >=1 ) ; else
false;  end if; end proc:
for i from 1 to 1000 do p := ithprime(i) ; if isA181741(p) then printf("%d,",p) ; end if; end do: # R. J. Mathar, Nov 18 2010

Select[Table[2^t-2^k-1, {t, 1, 20}, {k, 1, t-1}] // Flatten // Union, PrimeQ] (* Jean-François Alcover, Nov 16 2017 *)

lista(nn) = {for (n=3, nn, forstep(k=n-1, 1, -1, if (isprime(p=2^n-2^k-1), print1(p, ", "));););} \\ Michel Marcus, Dec 17 2018

from itertools import count, islice
from sympy import isprime
def A181741_gen(): # generator of terms
    m = 2
    for t in count(1):
        r=1<>=1
        m<<=1
A181741_list = list(islice(A181741_gen(),30)) # Chai Wah Wu, Jul 08 2022

Conjecture: equals the intersection of A000040 and A081118 or the intersection of A000040 and A089633. [R. J. Mathar, Nov 18 2010]

Corrected (251 and 1019 inserted) and extended by R. J. Mathar, Nov 18 2010

A181703 Numbers of the form 2^(t-1)*(2^t-3), where 2^t-3 is prime.

Original entry on oeis.org

20, 104, 464, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504, 196159429230833773869868419445529014560349481041922097152, 3450873173395281893717377931138512601610429881249330192849350210617344

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

This is a subsequence of A181595. [Proof: sigma(m) = (2^t-1)*(2^t-2) leads to an abundance of m which is 2.]
Numbers m such that the sum of the even divisors of m equals the square of the odd divisors of m.
Proof: let s0 the sum of the even divisors and s1 the sum of the odd divisors.
s1 = 2^t-2 because 2^t-3 is prime.
s0 = 2 + 4 + 8 + ... + 2^(t-1) + (2^t - 3)(2 + 4 + 8 + ... + 2^(t-1)) = (2^t - 2)^2 => s0 = s1^2. - Michel Lagneau, Apr 17 2013

Eric Chen, Table of n, a(n) for n = 1..28
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.

Cf. A181595, A181701, A000396, A050414, A050415.

with(numtheory):for n from 1 to 600000 do:x:=divisors(n):n0:=nops(x):s0:=0:s1:=0:for k from 1 to n0 do:if irem(x[k],2)=0 then s0:=s0+ x[k]:else s1:=s1+ x[k]:fi:od:if s0=s1^2 then print(n):else fi:od: # Michel Lagneau, Apr 17 2013

for(k=1, 200, if(ispseudoprime(2^k-3), print1(2^(k-1)*(2^k-3), ", "))) \\ Eric Chen, Jun 13 2018

a(n) = 2^(A050414(n)-1) * (2^A050414(n) - 3). - Max Alekseyev, Jul 31 2025

Edited and extended by D. S. McNeil, Nov 18 2010

Definition simplified by R. J. Mathar, Nov 18 2010

A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

Original entry on oeis.org

12, 88, 1888, 32128, 521728, 8378368, 34359083008, 549753192448, 2251799645913088, 9223372026117357568, 2361183241263023915008, 2596148429267413634121263069790208, 2722258935367507707522529418717050175488

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

All these numbers are in A181595 because their abundance is 4, a proper divisor of m.

Cf. A059608, A156560, A181595, A181701, A000396, A181703, A045769

Rest[2^(#-1) (2^#-5)&/@(Round[N[Log[#+5]/Log[2]]]&/@Select[Table[2^t-5,{t,120}],PrimeQ])] (* Harvey P. Dale, Dec 16 2010 *)

571728 replaced with 521728 by R. J. Mathar, Dec 05 2010

A181705 Numbers of the form 2^(t-1)*(2^t-9), where 2^t-9 is prime.

Original entry on oeis.org

56, 368, 128768, 2087936, 8589344768, 2199013818368, 36893488108764397568, 904625697166532776746648320380374279912262923807289020860114158381451706368

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

Subsequence of A181595.
(Proof: Let m=2^(t-1)*(2^t-9) be the entry. By the multiplicative property of the sigma-function, sigma(m)=(2^t-1)*(2^t-8).
The abundance sigma(m)-2*m is therefore 8, and since all t involved are >=4, 8 is a divisor of m because 8 divides 2^(t-1).)

Cf. A059610, A181595, A181701, A000396, A181703, A181704

2^(#-1) (2^#-9)&/@Select[Range[3,130],PrimeQ[2^#-9]&] (* Harvey P. Dale, Oct 24 2011 *)

Edited by R. J. Mathar, Sep 12 2011

A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).

Original entry on oeis.org

24, 224, 15872, 1040384, 274844352512, 1125891316908032, 72057456598974464, 4951760154835678090382802944, 6129982163463555430774932117031404988667342368173719552

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..14

Subsequence of A181595 and A181702.

Cf. A000043, A000396, A147538, A181596, A000396, A181701, A181702.

With[{p = MersennePrimeExponent[Range[10]]}, 2^(2*p - 1)*(2^p - 1)] (* Amiram Eldar, Apr 29 2024 *)

a(n) = A147538(A000043(n)). - Amiram Eldar, Apr 29 2024

a(6)-a(9) from Amiram Eldar, Apr 29 2024

A181706 Numbers of the form 2^(t-1)*(2^t-17), where 2^t-17 is prime.

Original entry on oeis.org

1504, 30592, 8353792, 2146926592, 34357510144, 549746900992, 8796057370624, 140737345748992, 9223372000347553792, 2361183240850707054592, 9671406556879650002305024, 154742504910523000781012992

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

All entries are near-perfect numbers (A181595). The proof follows as in A181705, but this time the abundance is 16.

Cf. A059611, A181595, A181701, A000396, A181703, A181704, A181705.

A181707 Numbers of the form m=2^(t-1)*(2^t-33), where 2^t-33 is prime.

Original entry on oeis.org

992, 28544, 122624, 507392, 34355412992, 8796023816192, 140737211531264, 144115179217485824, 9671406556844465630216192, 162259276829213066154002603835392, 11417981541647679048463794346093005918389141504

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

Generated by t= 6, 8, 9, 10, 18, 22, 24, 29, 42, 54, 77, 90, 102, 137,...

A subsequence of A181595 because the abundance of m is 32, and 32 divides 2^(t-1) and therefore divides m.

Cf. A181595, A181701, A000396, A181703, A181704, A181705, A181706, A175989.

Definition simplified and more terms added by R. J. Mathar, Nov 18 2010

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).

Original entry on oeis.org

18, 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, 4951760152529835082242850816, 6129982163463555428116476125461573244012649752219877376

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

The associated exponents k are in A000043: 2, 3, 5, 7, 13, 17, 19 ,31, 61, ...

One can prove that, if m = 2^(k-1)*(2^k-1) is a perfect number, then m*2^k and m*(2^k-1) are both in A181595. Thus every even term in A000396 is a difference of two terms in A181595.

			With k=3, m = 2^(k-1)*(2^k - 1) = 2^2*(8 - 1) = 28 is a perfect number (A000396), so m*(2^k - 1) = 28*7 = 196 is in the sequence. - _Michael B. Porter_, Jul 19 2016

Cf. A000043, A000396, A181595, A181596, A181701, A181710.

If odd perfect numbers do not exist, then a(n) = A181710(n) - A000396(n).

a(n) = A019279(n)*(A000668(n))^2 if there are no odd superperfect numbers. - César Aguilera, Jun 13 2017

Definition condensed by R. J. Mathar, Dec 05 2010

cp's OEIS Frontend

A181702 Numbers of the form 2^r*p with p an odd prime, which are in A181595 but not in A181701.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A181595 Abundant numbers n for which the abundance d = sigma(n) - 2*n is a proper divisor, that is, 0 < d < n and d | n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A181741 Primes of the form 2^t-2^k-1, k>=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A181703 Numbers of the form 2^(t-1)*(2^t-3), where 2^t-3 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A181705 Numbers of the form 2^(t-1)*(2^t-9), where 2^t-9 is prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A181710 Near-perfect numbers (A181595) of the form m*2^p, where m = 2^(p-1)*(2^p-1) is a perfect number (A000396).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A181706 Numbers of the form 2^(t-1)*(2^t-17), where 2^t-17 is prime.

Views

Author

Keywords

A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).