A181692 -id:A181692 - cp's OEIS frontend

A181701 Near-perfect numbers (A181595) of the form 2^(t-1)*(2^t-2^k-1), where 2^t-2^k-1 is prime, k>=1, t>k.

12, 20, 56, 88, 104, 368, 464, 992, 1504, 1888, 1952, 16256, 24448, 28544, 30592, 32128, 98048, 122624, 128768, 130304, 507392, 521728, 522752, 2087936, 7337984, 8124416, 8353792, 8378368, 8382464, 25161728, 67100672, 125820928, 132112384, 133685248, 134193152

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

There exist near-perfect numbers of the form 2^r*p, where p is prime, which are not in the sequence (e.g., 24,40,224). For given k, the smallest value of t gives sequence A181692.

Donovan Johnson, Table of n, a(n) for n = 1..1000
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:1011.6160
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. A181595, A181596, A000396, A181692.

s = Sort@ Flatten@ Table[p = (2^t - 2^k - 1); If[PrimeQ@ p, 2^(t - 1) p, Nothing], {t, 2, 14}, {k, t - 1}]; Select[Select[s, DivisorSigma[1, #] > 2 # &], MemberQ[Divisors@ #, DivisorSigma[1, #] - 2 #] &] (* Michael De Vlieger, Sep 23 2015, after Alonso del Arte at A181595 *)

mx=2^269*(2^270-2^122-1); v=vector(1000); n=0; for(k=1, 269, for(t=k+1, 270, p=2^t-2^k-1; m=2^(t-1)*p; if(m>mx, next(2)); if(isprime(p), n++; v[n]=m))); v=vecsort(v); for(n=1, 1000, write("b181701.txt", n " " v[n])) /* Donovan Johnson, May 24 2013 */

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

A331205 a(n) = least prime of the form 2^m - 2^n + 1.

Original entry on oeis.org

2, 3, 5, 549755813881, 17, 97, 193, 140737488355201, 257, 7681, 15361, 134215681, 12289, 8380417, 114689

Offset: 0

Hugo Pfoertner, Jan 12 2020

nonn

a(15) = 2^447 - 2^15 + 1 is too large to be represented in the data.

			See A331204.

Hugo Pfoertner, Table of n, a(n) for n = 0..62

Cf. A181692, A331204 (corresponding values of m).

for(n=0,14, for(m=n+1,oo, k=2^m-2^n+1; if(isprime(k), print1(k,", "); break)))

A331204 Least m > n such that 2^m - 2^n + 1 is prime.

Original entry on oeis.org

1, 2, 3, 39, 5, 7, 8, 47, 9, 13, 14, 27, 14, 23, 17, 447, 17, 23, 20, 31, 23, 27, 34, 39, 31, 31, 29, 31, 43, 41, 32, 191, 40, 43, 49, 59, 38, 41, 42, 255, 64, 43, 65, 331, 48, 59, 62, 111, 52, 79, 53, 91, 55, 75, 61, 199, 71, 65, 86, 99, 65, 127, 74

Offset: 0

Hugo Pfoertner, Jan 12 2020

If it exists, a(63) > 10000.

			a(0) = 1: 2^1 - 2^0 + 1 = 2 = A331205(0) is prime,
a(1) = 2: 2^2 - 2^1 + 1 = 3 = A331205(1) is prime,
a(2) = 3: 2^3 - 2^2 + 1 = 5 = A331205(2) is prime,
a(3) = 39: 2^39 - 2^3 + 1 = 549755813881 = A331205(3) is prime, whereas all smaller values of m give composite sums: 9, 25, 57, 121, 249, 505, ..., 274877906937.

Cf. A181692, A331205 (corresponding primes).

for(n=0,62, for(m=n+1,oo, k=2^m-2^n+1; if(isprime(k), print1(m,", "); break)))

A331217 a(n) is the least prime of the form 2^m - 2^n - 1.

Original entry on oeis.org

2, 5, 3, 7, 47, 31, 191, 127, 16127, 3583, 15359, 6143, 1044479, 8191, 245759, 16744447, 4128767, 131071, 786431, 524287, 274876858367, 14680063, 4398042316799, 260046847, 4278190079, 4261412863, 1125899839733759, 576460752169205759, 16911433727

Offset: 0

Hugo Pfoertner, Jan 12 2020

nonn

			a(1) = 2: 2^2 - 2^0 - 1 = 2, thus exponent 2 = A181692(0);
a(2) = 5: 2^3 - 2^1 - 1 = 5, 2^2 - 2^1 - 1 = 1 is not a prime, A181692(1) = 3;
a(4) = 47: 2^6 - 2^4 - 1 = 31, whereas the first candidate 2^5 - 2^4 - 1 = 15 is composite.

Robert Israel, Table of n, a(n) for n = 0..680

Cf. A181692 (corresponding values of m), A331204, A331205.

a:=[]; for n in [0..30] do m:=n+1; while not IsPrime(2^m-2^n-1) do m:=m+1; end while; Append(~a,2^m-2^n-1); end for; a; // Marius A. Burtea, Jan 13 2020

f:= proc(n) local m, p;
   p:= -1;
   for m from n do
     p:= p + 2^m;
     if isprime(p) then return p fi
    od
end proc:
map(f, [$0..30]); # Robert Israel, Jan 13 2020

a[n_] := For[m = n+1, True, m++, If[PrimeQ[p = 2^m-2^n-1], Return[p]]];
a /@ Range[0, 28] (* Jean-François Alcover, Oct 25 2020 *)

for(n=0,28, for(m=n+1,oo, k=2^m-2^n-1; if(isprime(k), print1(k,", "); break)))

cp's OEIS Frontend

A181701 Near-perfect numbers (A181595) of the form 2^(t-1)*(2^t-2^k-1), where 2^t-2^k-1 is prime, k>=1, t>k.

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

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A331205 a(n) = least prime of the form 2^m - 2^n + 1.

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A331204 Least m > n such that 2^m - 2^n + 1 is prime.

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A331217 a(n) is the least prime of the form 2^m - 2^n - 1.

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