A000043 -id:A000043 - cp's OEIS frontend

A330824 Numbers of the form 2^(2*p), where p is a Mersenne exponent, A000043.

16, 64, 1024, 16384, 67108864, 17179869184, 274877906944, 4611686018427387904, 5316911983139663491615228241121378304

Offset: 1

Walter Kehowski, Jan 06 2020

nonn

Also the second element of the power-spectral basis of A064591. The first element of the power-spectral basis of A064591 is A133049.

			a(1) = 2^(2*2) = 16. Also A133049(1) = 3^2 = 9, and the spectral basis of A064591(1) = 24 is {9, 16}, consisting of primes and powers.

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

Cf. A000043, A000668, A064591, A132794, A133049.

a := proc(n) if isprime(2^n-1) then return 2^(2*n) fi; end;
[seq(a(n),n=1..31)]; # ithprime(31) = 127

2^(2*MersennePrimeExponent[Range[10]]) (* Harvey P. Dale, Jun 27 2023 *)

forprime(p=1,99,isprime(2^p-1)&&print1(4^p",")) \\ or better: {A330824(n)=4^A000043(n)}. - M. F. Hasler, Feb 07 2020

a(n) = 2^(2*A000043(n)) = 4^A000043(n).

A332220 a(n) is the number of Mersenne exponents (terms of A000043) <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11

Offset: 1

Antti Karttunen, Feb 09 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..44497

A left inverse of A000043, partial sums of A332219.

Accumulate@ Array[Boole@ PrimeQ[2^# - 1] &, 107] (* Michael De Vlieger, Feb 09 2020 *)
With[{mpe=MersennePrimeExponent[Range[47]]},Accumulate[Table[If[ MemberQ[ mpe,n],1,0],{n,120}]]] (* Harvey P. Dale, Aug 22 2021 *)

A332219(n) = (isprime(n)&&isprime((2^n)-1));
A332220(n) = if(1==n,0,A332219(n)+A332220(n-1));

a(1) = 0; and for n > 1, a(n) = A332219(n) + a(n-1).

For all n >= 1, a(A000043(n)) = n.

A134458 First differences of A000043.

Original entry on oeis.org

1, 2, 2, 6, 4, 2, 12, 30, 28, 18, 20, 394, 86, 672, 924, 78, 936, 1036, 170, 5266, 252, 1272, 8724, 1764, 1508, 21288, 41746, 24260, 21546, 84042, 540748, 102594, 398354, 140482, 1577952, 45156, 3951216, 6494324, 7529094, 3040572, 1928368, 4437506, 2180200, 4574010, 5487134, 468808, 14772552

Offset: 1

Alessio 'BiGAlex' Periloso (bigalex(AT)bigalex.it), Oct 26 2007

nonn

Differences between exponents of Mersenne primes.

Mersenne primes (of form 2^p - 1 where p is a prime): A000668, A000043.

Checked and extended by N. J. A. Sloane, Oct 29 2007
More terms from Derek Maciel, Jan 07 2018
a(45)-a(46) from Ivan Panchenko, Apr 11 2018
a(47) from Rodolfo Ruiz-Huidobro, May 16 2024

A139096 Infraperfect numbers: a(n) = 2^(2*p - 1) - 2^p, where p is A000043(n).

Original entry on oeis.org

4, 24, 480, 8064, 33546240, 8589803520, 137438429184, 2305843007066210304, 2658455991569831743501771111346995200, 191561942608236107294793377774818628309652252823388160

Offset: 1

Omar E. Pol, Apr 22 2008

nonn

Difference between n-th even perfect number and n-th even superperfect number A061652(n). Difference between n-th ultraperfect number A139306(n) and n-th Mersenne prime A000668(n), minus 1. Also, difference between n-th perfect number A000396(n) and n-th superperfect number A019279(n), if there are no odd perfect and superperfect numbers.

			a(2) = 24 because A000043(2) = 3 then 2^(2*3 - 1) - 2^3 = 2^5 - 2^3 = 32 - 8 = 24.

Amiram Eldar, Table of n, a(n) for n = 1..15
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A072868, A139306.

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

a(n) = 2^(2*A000043(n) - 1) - 2^A000043(n) = A139306(n) - 2^A000043(n) = A139306(n) - A000668(n) - 1 = A139306(n) - (A000668(n)+1) = A139306(n) - 2*A061652(n) = A139306(n) - A072868(n).

More terms from R. J. Mathar, Feb 05 2010

A152923 a(n) = 2^(2*p-1)/8, where p is A000043(n).

Original entry on oeis.org

1, 4, 64, 1024, 4194304, 1073741824, 17179869184, 288230376151711744, 332306998946228968225951765070086144, 23945242826029513411849172299223580994042798784118784, 1645504557321206042154969182557350504982735865633579863348609024

Offset: 1

Omar E. Pol, Dec 15 2008

nonn

Ultraperfect numbers (A139306), divided by 8.

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

Cf. A000043, A139292, A139306, A152921, A152922.

2^(2 * MersennePrimeExponent[Range[10]] - 4) (* Amiram Eldar, Oct 17 2024 *)

a(n) = A139306(n)/8 = A152921(n)/4 = A152922(n)/2.

a(9)-a(11) from Amiram Eldar, Oct 17 2024

A330836 Numbers of the form 2^(2p-1)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

Original entry on oeis.org

4704, 1476096, 396386304, 6753750274277376, 442715102395357986816, 113336363243719574421504, 31901471869127420013759771876790370304, 42404329554681223873219247037048711787234652848116929825491652260298489856

Offset: 1

Walter Kehowski, Jan 12 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}. The first element of the spectral basis of a(n) is A330819(n+1), the second element is A330837(n+1), and the third element is A330820(n+1). Generally, a power-spectral basis is a spectral basis that consists of primes and powers.
The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 2*a(n)+1. In this case, we say that a(n) has index 2.
a(n) is also isospectral with A330838(n), that is, a(n) and A330838(n) have the same spectral basis, but A330838(n) has index 1. Thus, A330838(n) and a(n) form an isospectral pair.
Subsequence of Zumkeller numbers (A083207), since a(n) = 2^r * 3 * s, where s is relatively prime to 6. - Ivan N. Ianakiev, Feb 03 2020

			If p = 3, then a(1) = 2^(2*3-1)*3*7^2 = 4704, and the spectral basis of 4704 is {63^2, 56^2, 48^2}, consisting of powers. The spectral sum of a(1), that is, the sum of the elements of its spectral basis, is 2*4704+1 = 9409. In this case, we say that a(1) has index 2. The number A330838(1) = 9704 has the same spectral basis as a(1), but with index 1. We say that A330838(1) and a(1) are isospectral and form an isospectral pair.

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A139306, A330819, A330820, A330837, A330838.

a := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n+1);
  m:=2^p-1;
  return 2^(2*p-1)*3*m^2;
end:

f[p_] := 2^(2*p - 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)

a(n) = A139306(n+1) * 3 * A133049(n+1).

A330838 Numbers of the form 2^(2p)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

Original entry on oeis.org

9408, 2952192, 792772608, 13507500548554752, 885430204790715973632, 226672726487439148843008, 63802943738254840027519543753580740608, 84808659109362447746438494074097423574469305696233859650983304520596979712

Offset: 1

Walter Kehowski, Jan 17 2020

nonn

a(n) has the same spectral basis as A330836(n), namely {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}, so the two numbers are isospectral as well as power-spectral, that is, they have the same spectral basis and that basis consists of powers. The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 1*a(n)+1, while the spectral sum of A330836(n) is 2*A330836(n)+1. We say that a(n) and A330836(n) form an isospectral pair, with a(n) of index 1 and A330836(n) of index 2.

Subsequence of Zumkeller numbers (A083207), since a(n) = 2^r * 3 * s, where s is relatively prime to 6. - Ivan N. Ianakiev, Feb 03 2020

			If p = 3, then M_3 = 7 and a(1) = 2^(2*3)*3*7^2 = 9408, with spectral basis {63^2, 56^2, 48^2}, and spectral sum equal to 1*9408 + 1 = 9409. However, {63^2, 56^2, 48^2} is also the spectral basis of A330836(1) = 4704, with spectral sum equal to 2*4704+1.

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A330819, A330820, A330824, A330825, A330826, A330836, A330837.

a := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n+1);
  m:=2^p-1;
  return 2^(2*p)*3*m^2;
end:

f[p_] := 2^(2p)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9]  (* Amiram Eldar, Jan 17 2020 *)

a(n) = A330824(n+1) * 3 * A133049(n+1).

A131960 a(n) = A000043(n) * A000668(n).

Original entry on oeis.org

6, 21, 155, 889, 106483, 2228207, 9961453, 66571993057, 140656423562035331011, 55088331748199422233011027879, 17361742620725829882898847100829589, 21607930299479592429924287571917281427329

Offset: 1

A.K. Devaraj, Aug 02 2007, Aug 06 2007

Note that a(3) = 155 and a(4) = 889 also belong to A119691.
Old name was: Composite numbers such that the first factor is the relevant exponent of Mersenne prime (the second factor).
a(n) is the smallest k > 0 such that A000668(n)^2 divides 2^k-1. Then lpf(2^k-1) = A000668(n), where lpf(m) = A020639(m). - Thomas Ordowski, Feb 03 2019

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A000043, A000668, A002326, A020639, A119691.

Map[# * (2^#-1) &, MersennePrimeExponent[Range[12]]] (* Amiram Eldar, Oct 21 2024 *)

expm = [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]; vector(#expm, i, expm[i]*(2^expm[i]-1)) \\ where expm comes from A000043; Michel Marcus, Feb 07 2019

a(n) = A000043(n)*A000668(n). - R. J. Mathar, Oct 18 2007

a(n) = A002326((A000668(n)^2 - 1)/2). - Thomas Ordowski, Feb 03 2019

Terms corrected, edited, and new name by Michel Marcus, Apr 30 2013

a(12) from Michel Marcus, Feb 07 2019

A153803 a(n) = A153800(n) - A000043(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 6, 10, 49, 103, 122, 167

Offset: 1

Omar E. Pol, Jan 13 2009

Cf. A000040, A000043, A000668, A000225, A000396, A007691, A153800, A153801, A153802.

a007691[m_] := Block[{f = 0, i = 1}, While[f < m, If[Divisible[DivisorSigma[1, i], i], ++f]; ++i;]; Return[i - 1]]
a153800[n_] := Block[{i = 2, j, f = 0}, While[f < n, j = a007691[i]; If[PerfectNumberQ[j], ++f;]; ++i]; Return[i - 1]]
a[n_] := a153800[n] - MersennePrimeExponent[n]
a /@ Range[1, 4] (* Julien Kluge, Dec 15 2016 *)

A161322 a(n) = A000043(n) written in base 4.

Original entry on oeis.org

2, 3, 11, 13, 31, 101, 103, 133, 331, 1121, 1223, 1333, 20021, 21133, 103333, 202123, 203221, 302101, 1002131, 1011013, 2113121, 2123111, 2233031, 10313201, 11103011, 11222221, 22313101, 111003203, 122332213, 200033101, 310300123, 2320301213, 3101310221, 10303010323

Offset: 1

Juri-Stepan Gerasimov, Jun 07 2009

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

Cf. A000043, A007090, A160962.

Map[FromDigits[IntegerDigits[#, 4]] &, MersennePrimeExponent[Range[48]]] (* Amiram Eldar, Oct 16 2024 *)

a(n) = A007090(A000043(n)). - Amiram Eldar, Oct 16 2024

Corrected (1301 replaced by 1223, 2021 replaced by 20021, 22311313 by 22313101) by R. J. Mathar, May 21 2010

a(32)-a(34) from Amiram Eldar, Oct 16 2024

cp's OEIS Frontend

A330824 Numbers of the form 2^(2*p), where p is a Mersenne exponent, A000043.

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A332220 a(n) is the number of Mersenne exponents (terms of A000043) <= n.

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A134458 First differences of A000043.

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A139096 Infraperfect numbers: a(n) = 2^(2*p - 1) - 2^p, where p is A000043(n).

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A152923 a(n) = 2^(2*p-1)/8, where p is A000043(n).

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A330836 Numbers of the form 2^(2*p-1)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

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A330838 Numbers of the form 2^(2*p)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

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A131960 a(n) = A000043(n) * A000668(n).

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A330836 Numbers of the form 2^(2p-1)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

A330838 Numbers of the form 2^(2p)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.