A090748 Numbers n such that 2^(n+1) - 1 is prime.

1, 2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656

Offset

1,2

Comments

Perfect numbers A000396(n) = 2^A133033(n) - 2^a(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 24 2008

Number of proper divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 28 2008

Base 2 logarithm of n-th even superperfect number A061652(n). Also base 2 logarithm of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Apr 11 2008

Number of 0's in binary expansion of n-th even perfect number (see A135650). - Omar E. Pol, May 04 2008

Links

Ivan Panchenko, Table of n, a(n) for n = 1..47

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Formula

2^a(n) = A051027(2^(n+1)). - Juri-Stepan Gerasimov, Aug 21 2016

Example

1 is in the sequence because 2^2 - 1 = 3 is prime.

Mathematica

Select[Range[0, 10^4], PrimeQ[2^(# + 1) - 1] &] (* Vincenzo Librandi, Jul 28 2016

Prog

(Magma) [n: n in [1..5*10^3] |IsPrime(2^(n+1)-1)]; // Vincenzo Librandi, Jul 28 2016
(PARI) is(n)=ispseudoprime(2^(n+1)-1) \\ Charles R Greathouse IV, Aug 21 2016

Crossrefs

a(n) = A000043(n) - 1. A000043 is the main entry for this sequence.

Cf. A000396, A133033, A000668, A019279, A061652, A135650, A051027.

Keyword

nonn

Author

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 03 2004

Extensions

Edited, corrected and extended by Robert G. Wilson v, Feb 09 2004

Updated (a(39)) by Omar E. Pol, Jan 23 2009

a(40)-a(44) from Ivan Panchenko, Apr 11 2018

Status

approved

A019280 Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.

1, 2, 4, 6, 12, 16, 18, 30, 60

Offset

1,2

Comments

Cohen and te Riele prove that any even (2,2)-perfect number (a "superperfect" number) must be of the form 2^(p-1) with 2^p-1 prime (Suryanarayana) and the converse also holds. Any odd superperfect number must be a perfect square (Kanold). Searches up to > 10^20 did not find any odd examples. - Ralf Stephan, Jan 16 2003

See also the Cohen-te Riele links under A019276.

Links

Table of n, a(n) for n=1..9.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Formula

Coincides with A000043(n) - 1 unless odd superperfect numbers exist.

Crossrefs

Cf. A019278, A019279.

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

a(8)-a(9) from Jud McCranie, Jun 01 2000

Status

approved

A050584 Numbers n such that 117*2^n-1 is prime.

1, 2, 4, 6, 12, 16, 18, 20, 22, 24, 37, 40, 48, 49, 57, 62, 154, 172, 184, 236, 265, 374, 409, 445, 478, 664, 718, 928, 1186, 1369, 1804, 2006, 2140, 2618, 4456, 4846, 12604, 17680, 18318, 18812, 20320, 20660, 21494, 24977, 27749, 30734, 31720, 41742, 51928, 54216, 60386, 61882, 64408, 72078, 111698, 129820, 135318, 151594, 225165, 373045, 511361, 560848, 697458, 834916

Offset

1,2

Links

Table of n, a(n) for n=1..64.

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300

Wilfrid Keller, List of primes k.2^n - 1 for k < 300

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Kosmaj, Riesel list k<300.

Prog

(PARI) is(n)=ispseudoprime(117*2^n-1) \\ Charles R Greathouse IV, Jun 13 2017

Keyword

hard,nonn

Author

N. J. A. Sloane, Dec 29 1999

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008

Status

approved

A260698 Practical numbers of the form p - 1 where p is a prime.

1, 2, 4, 6, 12, 16, 18, 28, 30, 36, 40, 42, 60, 66, 72, 78, 88, 96, 100, 108, 112, 126, 150, 156, 162, 180, 192, 196, 198, 210, 228, 240, 256, 270, 276, 280, 306, 312, 330, 336, 348, 352, 378, 396, 400, 408, 420, 432, 448, 456, 460, 462, 486, 520, 522, 540, 546

Offset

1,2

Comments

Intersection of A005153 and A006093. - Michel Marcus, Nov 16 2015

Links

Table of n, a(n) for n=1..57.

Example

a(5)=12 as 12 is a practical number and 12+1=13 is prime. It is the 5th such practical number.

Mathematica

PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[Table[Prime[n]-1, {n, 1, 200}], PracticalQ] (* using T. D. Noe's program A005153 *)

Prog

(PARI) is(n) = bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return);
forprime(p=2, 1000, if(is(p-1), print1(p-1", "))) \\ Altug Alkan, Nov 16 2015

Crossrefs

Cf. A005153, A006093, A225223.

Keyword

nonn

Author

Frank M Jackson, Nov 16 2015

Status

approved

A309096 Increasing positive integers with prime factorization exponents all appearing earlier in the sequence.

1, 2, 4, 6, 12, 16, 18, 30, 36, 48, 60, 64, 90, 144, 150, 162, 180, 192, 210, 240, 300, 324, 420, 450, 576, 630, 720, 810, 900, 960, 1050, 1200, 1260, 1296, 1458, 1470, 1620, 1680, 2100, 2310, 2880, 2916, 2940, 3150, 3600, 3750, 4050, 4096, 4410, 4620, 4800

Offset

1,2

Comments

Because non-existing primes in the a factorization are recorded with exponent 0 here, and because 0 is not in the sequence, all entries must have a full set of prime divisors from 2 up to their largest prime: this is a subsequence of A055932. - R. J. Mathar, May 05 2023

3 and 5 do not appear in the sequence, so entries of A176297 or A362831 are not in the sequence. - R. J. Mathar, May 05 2023

Links

Table of n, a(n) for n=1..51.

Formula

a(1) = 1; a(n) = least positive integer x > a(n-1) where the exponents e in the prime factorization of x are in a(1..n-1).

Example

a(2) = 2, since 2 = 2^1 and all {1} are in a(1..1) = [1].
a(3) != 3, since 3 = 2^0 * 3^1 and not all {0,1} are in a(1..2) = [1,2].
a(3) = 4, since 4 = 2^2 and all {2} are in a(1..2) = [1,2].
a(4) != 5, since 5 = 2^0 * 3^0 * 5^1 and not all {0,1} are in a(1..3) = [1,2,4].
a(4) = 6, since 6 = 2^1 * 3^1 and all {1} are in a(1..3) = [1,2,4].

Prog

(Haskell)
wheelSeeds = [2, 3, 5, 7, 11, 13]
wheelOffsets = filter (\c -> all (\s -> mod c s /= 0) wheelSeeds) [1..product wheelSeeds]
restOfWheel = (concat (map (replicate (length wheelOffsets)) (map (* (product wheelSeeds)) [1..])))
wheel = wheelSeeds ++ (tail wheelOffsets) ++ (zipWith (+) (cycle wheelOffsets) restOfWheel)
isPrime n = and [n > 1, all (\c -> mod n c /= 0) (takeWhile (\c -> c * c <= n) wheel)]
primes = filter isPrime wheel
exponents bases acc n =
    if (n == 1)
        then (dropWhile (== 0) acc)
        else if (mod n (head bases) == 0)
            then (exponents bases (((head acc) + 1) : (tail acc)) (div n (head bases)))
            else (exponents (tail bases) (0 : acc) n)
a = filter (\n -> all (\e -> elem e (takeWhile (<= e) a)) (exponents primes [0] n)) [1..]

Keyword

nonn

Author

Chris Murray, Jul 12 2019

Status

approved