A000043 -id:A000043 - cp's OEIS frontend

A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

288, 6272, 1968128, 528515072, 9005000365703168, 590286803193810649088, 151115150991626099228672, 42535295825503226685013029169053827072, 56539106072908298497625662716064949049646203797489239767322203013731319808

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2,(M_p^2-1)^2}.

The first factor of a(n) is A330818. The first element of the spectral basis of a(n) is A330819, and the second element is A330820.

			Since p=2 and M_2=3, then a(1)=2^(2*2+1)*3^3=288, and 288 has spectral basis {15^2, 2^6}, consisting of powers.

Walter Kehowski, Table of n, a(n) for n = 1..12

Cf. A000043, A000668, A132794, A133049, A330818, A330819, A330820.

A330817:=[]:
for www to 1 do
for i from 1 to 31 do
  #ithprime(31)=127
  p:=ithprime(i);
  q:=2^p-1;
  if isprime(q) then x:=2^(2*p+1)*q^2; A330817:=[op(A330817),x]; fi;
od;
od;
A330817;

2^(2 * (p = MersennePrimeExponent[Range[9]]) + 1) * (2^p - 1)^2 (* Amiram Eldar, Jan 03 2020 *)

A065549 a(1) = 1; for n > 1, a(n) = 2^((A000043(n) - 1)/2).

Original entry on oeis.org

2, 4, 8, 64, 256, 512, 32768, 1073741824, 17592186044416, 9007199254740992, 9223372036854775808, 1852673427797059126777135760139006525652319754650249024631321344126610074238976

Offset: 2

Labos Elemer, Nov 13 2001

nonn

Proper subset of A065405.
These values also relate to the sequence of perfect numbers. Every even perfect number except 6 can be written as Sum_{k=1..a(n)} (2*k-1)^3. - Derek Orr, Sep 28 2013
Positive real roots of 2n^4 - n^2 - A000396(n) = 0 for A000396(n) > 6. - César Aguilera, Nov 11 2018

Muniru A Asiru, Table of n, a(n) for n = 2..20

Cf. A000043, A065403, A065404, A065405, A000396.

Array[2^((MersennePrimeExponent@ # - 1)/2) &, 12, 2] (* Michael De Vlieger, Aug 25 2018 *)

lista(nn) = {forprime(p=3, nn, if (isprime(2^p-1), print1(2^((p-1)/2), ", ")););} \\ Michel Marcus, Aug 04 2016

log(n) is approximately log(sqrt(A000668(n)/2)). - César Aguilera, Nov 11 2018

A139307 a(n) = (2^(2*p - 1)) - 1, where p is A000043(n).

Original entry on oeis.org

7, 31, 511, 8191, 33554431, 8589934591, 137438953471, 2305843009213693951, 2658455991569831745807614120560689151, 191561942608236107294793378393788647952342390272950271

Offset: 1

Omar E. Pol, Apr 13 2008, May 08 2008

nonn

Ultraperfect numbers (A139306) minus 1.

			a(5) = 33554431 because A000043(5) = 13 and (2^(2*13 - 1))-1 = 2^25 - 1 = 33554431.

Amiram Eldar, Table of n, a(n) for n = 1..15

Cf. A000043, A139287, A139295, A139306.

2^(2 * MersennePrimeExponent[Range[10]] - 1) - 1 (* Amiram Eldar, Oct 17 2024 *)

a(n) = (2^(2*A000043(n) - 1)) - 1 = A139306(n) - 1.

A139480 a(n) = A000043(n) - 3.

Original entry on oeis.org

0, 2, 4, 10, 14, 16, 28, 58, 86, 104, 124, 518, 604, 1276, 2200, 2278, 3214, 4250, 4420, 9686, 9938, 11210, 19934, 21698, 23206, 44494, 86240, 110500, 132046, 216088, 756836, 859430, 1257784, 1398266, 2976218, 3021374, 6972590, 13466914, 20996008, 24036580, 25964948

Offset: 2

Artur Jasinski, Apr 22 2008

nonn

2^a(n)-1 is divisible by 3. For (2^a(n)-1)/3 see A124477.

For a(n)/2 see A139481.

Amiram Eldar, Table of n, a(n) for n = 2..48

Cf. A000043, A000668, A124477, A139479, A139481.

MersennePrimeExponent[Range[2, 48]] - 3 (* Amiram Eldar, Oct 17 2024 *)

Definition corrected by Omar E. Pol, May 23 2008
Edited by N. J. A. Sloane, May 23 2008
a(40)-a(42) from Amiram Eldar, Oct 17 2024

A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

Original entry on oeis.org

13, 61, 2281, 3217, 23209, 44497, 132049, 13466917, 30402457, 42643801

Offset: 1

Artur Jasinski, Sep 30 2008

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subsequence of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

Select[MersennePrimeExponent[Range[47]], PowerMod[2, #, 6!] == 272 &] (* Amiram Eldar, Mar 22 2020 *)

a(10) from Amiram Eldar, Mar 22 2020

A145045 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 607 mod 6!

Original entry on oeis.org

107, 86243, 756839, 25964951, 37156667

Offset: 1

Artur Jasinski, Sep 30 2008

nonn

Mersenne numbers (with the exception of the first two) are congruent to 31, 127, 271, 607 mod 6!. This sequence is a subset of A000043.

Cf. A000043, A000668, A124477, A139484, A145038, A112633, A145041, A145042, A145044, A145045, A145046.

p = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 43112609}; a = {}; Do[If[Mod[2^p[[n]] - 1, 6! ] == 607, AppendTo[a, p[[n]]]], {n, 1, Length[p]}]; a (*Artur Jasinski*)

Comment rewritten by Harvey P. Dale, Sep 02 2023

A152921 a(n) = 2^(2p-1)/2, where p is A000043(n).

Original entry on oeis.org

4, 16, 256, 4096, 16777216, 4294967296, 68719476736, 1152921504606846976, 1329227995784915872903807060280344576, 95780971304118053647396689196894323976171195136475136, 6582018229284824168619876730229402019930943462534319453394436096

Offset: 1

Omar E. Pol, Dec 15 2008

nonn

Ultraperfect numbers (A139306), divided by 2.
Also, a(n) is the largest proper divisor of the n-th ultraperfect number.
The cototient (A051953) of the even perfect numbers (A000396). - Amiram Eldar, Mar 06 2022
These cototients are squares = (2^(p-1))^2. - Bernard Schott, Mar 14 2022

Cf. A000043, A000396, A019279, A051953, A061652, A139288, A139306, A152922, A152923.

a[n_] := 4^(MersennePrimeExponent[n] - 1); Array[a, 12] (* Amiram Eldar, Mar 06 2022 *)

a(n) = A139306(n)/2.
a(n) = A051953(A000396(n)), if there are no odd perfect numbers. - Amiram Eldar, Mar 06 2022
a(n) = A061652(n)^2. - Bernard Schott, Mar 14 2022

More terms from Amiram Eldar, Mar 06 2022

A152922 a(n) = 2^(2*p-1)/4, where p is A000043(n).

Original entry on oeis.org

2, 8, 128, 2048, 8388608, 2147483648, 34359738368, 576460752303423488, 664613997892457936451903530140172288, 47890485652059026823698344598447161988085597568237568

Offset: 1

Omar E. Pol, Dec 15 2008

nonn

Ultraperfect numbers (A139306), divided by 4.

Amiram Eldar, Table of n, a(n) for n = 1..15

Cf. A000043, A139290, A139306, A152921, A152923.

2^(2 * MersennePrimeExponent[Range[10]] - 3) (* Amiram Eldar, Oct 17 2024 *)

a(n) = A139306(n)/4 = A152921(n)/2.

a(9)-a(10) from Amiram Eldar, Oct 17 2024

A153798 a(n) = A000043(n)-2.

Original entry on oeis.org

0, 1, 3, 5, 11, 15, 17, 29, 59, 87, 105, 125, 519, 605, 1277, 2201, 2279, 3215, 4251, 4421, 9687, 9939, 11211, 19935, 21699, 23207, 44495, 86241, 110501, 132047, 216089, 756837, 859431, 1257785, 1398267, 2976219, 3021375, 6972591, 13466915, 20996009, 24036581

Offset: 1

Omar E. Pol, Jan 20 2009

nonn

Base 2 logarithm of A134708(n).

Amiram Eldar, Table of n, a(n) for n = 1..48

Cf. A000043, A014635, A134708.

MersennePrimeExponent[Range[48]] - 2 (* Amiram Eldar, Oct 17 2024 *)

a(40)-a(41) from Amiram Eldar, Oct 17 2024

A153801 Index of Mersenne number A000225 that is also Mersenne prime A000668, minus n-th prime: a(n) = A000043(n) - A000040(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 4, 2, 12, 38, 60, 76, 90, 480, 564, 1232, 2150, 2222, 3156, 4186, 4352, 9616, 9862, 11130, 19848, 21604, 23108, 44394, 86136, 110394, 131936, 215964, 756708, 859296, 1257648, 1398120, 2976070, 3021220, 6972430, 13466750, 20995838, 24036404, 25964770

Offset: 1

Omar E. Pol, Jan 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..48

Cf. A000040, A000043, A000668, A000225, A000396, A007691, A153800, A153802, A153803.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Prime[Range[max]]] (* Amiram Eldar, Oct 21 2024 *)

More terms from R. J. Mathar, Feb 19 2009

More terms from Jinyuan Wang, Mar 02 2020

cp's OEIS Frontend

A330817 Numbers of the form 2^(2*p+1)*M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

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A065549 a(1) = 1; for n > 1, a(n) = 2^((A000043(n) - 1)/2).

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A139307 a(n) = (2^(2*p - 1)) - 1, where p is A000043(n).

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A139480 a(n) = A000043(n) - 3.

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A145044 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 271 mod 6!.

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A145045 Primes p (A000043) such that 2^p-1 is prime (A000668) and congruent to 607 mod 6!

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A152921 a(n) = 2^(2p-1)/2, where p is A000043(n).

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A152922 a(n) = 2^(2*p-1)/4, where p is A000043(n).

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A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.