A133049 -id:A133049 - cp's OEIS frontend

A330819 Numbers of the form M_p^2(M^p+2)^2, where M_p is a Mersenne prime (A000668) and p is a Mersenne exponent (A000043). Also the first element of the spectral basis of A330817.

225, 3969, 1046529, 268402689, 4503599493152769, 295147905144993087489, 75557863725364567605249, 21267647932558653957237540927630737409, 28269553036454149273332760011886696242605918383730576346715242738439159809

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

The second element of the spectral basis of A330817 is A330820.

			If p=2, then M_2=3, and a(1) = 3^2(3+2)^2 = 15^2 = 225.

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

Cf. A000043, A000668, A132794, A133049, A330817, A330818, A330820.

A330819:=[]:
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; A330819:=[op(A330819),x]; fi;
od;
od;
A330819;

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

a(n) = A000668(n)^2*(A000668(n)+2)^2.

A330818 Numbers of the form 2^(2*p+1), where p is a Mersenne exponent, A000043.

Original entry on oeis.org

32, 128, 2048, 32768, 134217728, 34359738368, 549755813888, 9223372036854775808, 10633823966279326983230456482242756608, 766247770432944429179173513575154591809369561091801088

Offset: 1

Walter Kehowski, Jan 01 2020

nonn

Also the first factor of A330817, 2^(2*p+1)*M_p^2. The second factor of A330817 is A133049.

			a(1) = 2^(2*2+1) = 32. Since M_2=3, the number 2^5*3^2 has power-spectral basis {225,64}.

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

Cf. A000043, A000668, A132794, A133049, A330817, A330819, A330820.

A330818:=[]:
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); A330818:=[op(A330818),x]; fi;
od;
od;
A330818;

2^(2 * MersennePrimeExponent[Range[10]] + 1) (* Amiram Eldar, Jan 03 2020 *)

a(n) = 2^(2*A000043(n)+1).

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

Original entry on oeis.org

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 *)

A379123 a(n) = A379113(A379121(n)), where A379121 gives those odd squares k for which A379113(k) > 1.

Original entry on oeis.org

9, 121, 9, 9, 81, 1521, 9, 9, 49, 49, 81, 9, 9, 625, 49, 49, 9, 961, 9, 9, 9, 961, 961, 49, 9, 961, 961, 169, 961, 961, 16129, 49, 49, 961, 961, 961, 961, 961, 49, 9, 9, 9, 9, 625, 961, 16129, 16129, 961, 961, 961, 49, 9, 49, 16129, 961, 49, 961, 9, 49, 49, 49, 49, 9, 9, 9, 9, 49, 9, 16129, 9, 9, 49, 49, 9, 49, 9

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

All terms are odd squares (A016754) by definition.
Among the initial 2025 terms, only the following 12 terms occur:
      Term    Occurs     Where
              n times
---------------------------------------------------------------
         9      699
        49      665
        81        2      a(5) and a(11)
       121        1      a(2)
       169        2      a(28) and a(926)
       625        9      at n=14, 44, 85, 110, 155, 447, 654, 896, 1217.
       961      390
      1521        1      a(6)     NB: 1521 = 9*169.
      8649        1      a(1087). NB: 8649 = 9*961.
     16129      246
  67092481        8      First occurrence at a(1120)
3287531569        1      a(1636). NB: 3287531569 = 49*67092481.
Questions: Is this sequence infinite? Do all terms of A133049 eventually appear here? Or any 4th or higher powers of Mersenne and other primes, apart from 81 and 625?

			See examples in A379121.

Cf. A016754, A114390, A133049, A379113, A379121, A379124, A379125.

forstep(n=1,2^18,2,d=A379113(n^2); if(d>1, print1(d,", ")));

a(n) = A379121(n) / A379124(n).

A379124 a(n) = A379119(A379121(n)), where A379121 gives those odd squares k for which A379113(k) > 1 and A379119(n) = n/A379113(n).

Original entry on oeis.org

25, 25, 361, 1369, 361, 25, 10201, 24025, 5041, 7225, 5041, 83521, 85849, 1369, 18769, 27889, 177241, 3969, 1092025, 1243225, 1352569, 13225, 30625, 978121, 5822569, 69169, 70225, 458329, 83521, 97969, 6241, 2253001, 2582449, 212521, 275625, 342225, 358801, 363609, 7502121, 45657049, 63696361, 65626201, 78659161

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

All terms are odd squares (A016754) by definition.

Cf. A016754, A114390, A133049, A379113, A379121, A379123, A379125.

forstep(n=1,2^18,2,d=A379113(n^2); if(d>1, print1((n^2)/d,", ")));

a(n) = A379121(n) / A379123(n).

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

Original entry on oeis.org

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).

A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 248, 496, 127, 254, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336, 131071, 262142, 524284, 1048568, 2097136, 4193792

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the divisors of n-th perfect number that are not powers of 2.

First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

			Triangle begins:
  3, 6,
  7, 14, 28
  31, 62, 124, 248, 496
  127, 254, 508, 1016, 2032, 4064, 8128
  ...
==========================================================
Row .... First term ..... Last term ....... Row sum ......
n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .
==========================================================
1 ............ 3 .............. 6 ......... 3^2 = 9
2 ............ 7 ............. 28 ......... 7^2 = 49
3 ........... 31 ............ 496 ........ 31^2 = 961
4 .......... 127 ........... 8128 ....... 127^2 = 16129
5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481

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

Cf. A000043, A000396, A000668, A018254, A018487, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655.

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).

A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

144, 3136, 984064, 264257536, 4502500182851584, 295143401596905324544, 75557575495813049614336, 21267647912751613342506514584526913536, 28269553036454149248812831358032474524823101898744619883661101506865659904

Offset: 1

Walter Kehowski, Jan 12 2020

nonn

a(n+1) is the second element of the power-spectral basis of both A330836(n) and A330838(n). Also, a(n) = A139256(n)^2, where A139256(n) is the sum of the divisors of the n-th perfect number, A000396(n).

Also: squares of twice the perfect numbers. - M. F. Hasler, Feb 07 2020

			If p=3, then a(2) = (7*2^3)^2 = 56^2, and the spectral basis of A330836(1) = 4704 and A330838(1) = 9408 is {63^2, 56^2, 48^2}, consisting of powers.

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, A000396, A000668, A133049, A139306, A139256, A330819, A330820, A330836.

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

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

forprime(p=1,999,isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)",")) \\ M. F. Hasler, Feb 07 2020

a(n) = A330824(n) * A133049(n).

a(n) = (2*A000396(n))^2 = (2^p-1)^2*4^p with p = A000043(n). - M. F. Hasler, Feb 07 2020

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).

cp's OEIS Frontend

A330819 Numbers of the form M_p^2(M^p+2)^2, where M_p is a Mersenne prime (A000668) and p is a Mersenne exponent (A000043). Also the first element of the spectral basis of A330817.

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A330818 Numbers of the form 2^(2*p+1), where p is a Mersenne exponent, A000043.

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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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A379123 a(n) = A379113(A379121(n)), where A379121 gives those odd squares k for which A379113(k) > 1.

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A379124 a(n) = A379119(A379121(n)), where A379121 gives those odd squares k for which A379113(k) > 1 and A379119(n) = n/A379113(n).

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

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A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(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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A330817 Numbers of the form 2^(2p+1)M_p^2, where M_p is a Mersenne prime, A000668, with Mersenne exponent p, A000043.

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.