A096502 -id:A096502 - cp's OEIS frontend

A181692 The smallest positive m such that 2^m-2^n-1 is prime, or 0 if such an m does not exist.

2, 3, 3, 4, 6, 6, 8, 8, 14, 12, 14, 13, 20, 14, 18, 24, 22, 18, 20, 20, 38, 24, 42, 28, 32, 32, 50, 59, 34, 32, 44, 32, 38, 38, 36, 40, 48, 42, 40, 45, 48, 45, 56, 45, 54, 48, 76, 52, 68, 66, 100, 89, 80, 74, 80, 57, 66, 78, 98, 83, 162, 62, 166, 77, 66, 77, 72, 76, 74, 153, 80, 89, 86, 77, 94, 83, 78, 88, 110, 115, 84

Offset: 0

Vladimir Shevelev, Nov 05 2010

Amiram Eldar, Table of n, a(n) for n = 0..3000

Cf. A096502.

A181692 := proc(n) for m from n to 100000 do if isprime(2^m-2^n-1) then return m; end if; end do: return 0 ; end proc:

m[n_]:=Module[{m=n+1},While[!PrimeQ[2^m-2^n-1],m++];m]
Table[m[i],{i,0,80}] (* Harvey P. Dale, Dec 18 2010 *)

for(n=0,80, for(m=n+1,oo, k=2^m-2^n-1; if(isprime(k),print1(m,", "); break))) \\ Hugo Pfoertner, Jan 12 2020

a(12) corrected and sequence extended by R. J. Mathar, Nov 17 2010

A294386 a(n) is the smallest number whose deficiency or abundance is equal to 2*n, or a(n) = 0 if such a number does not exist.

Original entry on oeis.org

6, 3, 5, 7, 22, 11, 13, 27, 17, 19, 46, 23, 112, 58, 29, 31, 250, 57, 37, 55, 41, 43, 94, 47, 60, 106, 53, 87, 84, 59, 61, 85, 108, 67, 142, 71, 73, 712, 158, 79, 156, 83, 405, 115, 89, 141, 406, 119, 97, 202, 101, 103, 214, 107, 109, 145, 113, 177, 418, 143, 120, 243, 192, 127, 262, 131, 261, 274, 137, 139, 574, 185

Offset: 0

Michel Marcus and Omar E. Pol, Oct 29 2017

nonn

If A096502(n) <> 0, i.e., there is a prime p of the form 2^k - 2*n - 1, then 0 < a(n) <= 2^(k-1)*p since 2^(k-1)*p has deficiency 2*n. - Robert Israel, Oct 29 2017

Michel Marcus, Table of n, a(n) for n = 0..8220 (terms <= 10^10) (terms 0..1644 from Robert Israel)

Bisection of A294347.
First differs from A217769 at a(12).
Cf. A000203, A000396, A005100, A005101, A033879, A033880, A096502, A294393, A294406.

N:= 100: # to get a(0)..a(N)
count:= 0:
for n from 1 while count < N+1 do
  d:= abs(2*n - numtheory:-sigma(n));
  if d::even and d <= 2*N and not assigned(A[d/2]) then
    count:= count+1; A[d/2]:= n;
  fi
od:
seq(A[i],i=0..N); # Robert Israel, Oct 29 2017

a033879(n) = 2*n-sigma(n)
a(n) = my(k=1); while(1, if(abs(a033879(k))==2*n, return(k)); k++) \\ Felix Fröhlich, Oct 29 2017

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 47, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 8, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 1

Eric Chen, Dec 14 2014

It is known that a(254602) = -1, because |509203-2^k| is always divisible by 3, 5, 7, 13, 17, or 241. a(1147) is the first unknown term.
a((A101036(n)+1)/2) = -1, so there are infinitely many n such that a(n) = -1.
a((A133122(n)+1)/2) = A096502((A133122(n)-1)/2).

			a(12) = 2 because 2*12-1 = 23 and that 23-2^1 = 21 is not prime but 23-2^2 = 19 is.
a(69) = 6 because 2*69-1 = 137, |137-2^k| is composite for k = 1, 2, 3, 4, 5 and prime for k = 6.
Even the smallest k can be also very large. For example, a(169) = 791.
a(1147) > 65536.

Jinyuan Wang, Table of n, a(n) for n = 1..1146

Cf. A006285, A096502, A133122, A188903.

Cf. A046067, A046069, A067760.

Table[k = 1; While[!PrimeQ[Abs[(2*n-1) - 2^k]], k++]; k, {n, 1, 1000}]

A252168(n)={ my(k=1); n=2*n-1; while(!ispseudoprime(abs(n-2^k)), k++); k }

a(19) corrected by Jinyuan Wang, Mar 25 2023

A096823 a(n) = p*(p+(2n-1))/2, where p = A096822(n) is the smallest primes of form 2^x-(2n-1).

Original entry on oeis.org

6, 20, 12, 151115727449904501489664, 56, 40, 24, 272, 1504, 208, 176, 1312, 112, 80, 48, 6208, 992, 928, 2059264, 5696, 736, 144115176533131264, 608, 544, 5056, 416, 352, 4672, 224, 160, 96, 24704, 24448, 3904, 3776, 487936, 112384, 3392, 22912

Offset: 1

Labos Elemer, Jul 13 2004

nonn

These numbers are clearly analogous to perfect numbers.

sigma(a(n)) mod a(n) = 2*n.

			a(1) = 6 is the first even perfect number;
a(7) = 24 corresponds to A096821(1) = 24;
a(4) = 151115727449904501489664 = 2^38*(2^39-7) = 274877906944*549755813881;

Cf. A096502, A096821, A096822.

Edited by Max Alekseyev, May 29 2025

A294682 Numbers n such that A294386(n) = 2^(k-1)(2^k - 2n - 1) for some k such that 2^k - 2*n - 1 is prime.

Original entry on oeis.org

0, 12, 62, 121, 126, 205, 241, 877, 1021, 1022, 1645, 2041, 2424, 2761, 2791, 2965, 3355, 3445, 3541, 4021, 4081, 4094, 4165, 4825, 5071, 5191, 5251, 5593, 6151, 6385, 6631, 7465, 7765, 7884, 8137, 8188

Offset: 1

Robert Israel, Nov 06 2017

			a(3) = 62 is in the sequence because A294386(62) = 192 = 2^6*3 where 2^7 - 2*62 - 1 = 3 is prime.

Cf. A096502, A294386.

# Assuming A294386[n] has been assigned for n from 0 to N
Res:= NULL:
for n from 0 to N do
  for k from ilog2(2*n+1)+1 do
    p:= 2^k - 2*n-1;
    if 2^(k-1)*p > A294386[n] then break fi;
    if isprime(p) then
      if A294386[n] = 2^(k-1)*p then Res:= Res, n fi;
      break
    fi
  od
od:
Res;

cp's OEIS Frontend

A181692 The smallest positive m such that 2^m-2^n-1 is prime, or 0 if such an m does not exist.

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A294386 a(n) is the smallest number whose deficiency or abundance is equal to 2*n, or a(n) = 0 if such a number does not exist.

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A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

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Mathematica

PARI

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A096823 a(n) = p*(p+(2n-1))/2, where p = A096822(n) is the smallest primes of form 2^x-(2n-1).

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A294682 Numbers n such that A294386(n) = 2^(k-1)*(2^k - 2*n - 1) for some k such that 2^k - 2*n - 1 is prime.

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Maple

A294682 Numbers n such that A294386(n) = 2^(k-1)(2^k - 2n - 1) for some k such that 2^k - 2*n - 1 is prime.