A133049 -id:A133049 - cp's OEIS frontend

A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

576, 12544, 3936256, 1057030144, 18010000731406336, 1180573606387621298176, 302230301983252198457344, 85070591651006453370026058338107654144, 113078212145816596995251325432129898099292407594978479534644406027462639616

Offset: 1

Walter Kehowski, Jan 23 2020

nonn

Also a(n+1) is the second element of the power-spectral basis of A330839(n), where by power-spectral we mean that the spectral basis consists of primes and powers.

			a(2) = 4*7^2*2^(2*3) = 2^8*7^2 = 112^2, and the spectral basis of A330839(1) = 18816 is {63^2, 112^2, 48^2}, consisting only of powers.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A330818, A330819, A330820, A330839.

A330840 := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n);
  m:=2^p-1;
  return 4*m^2*(m+1)^2;
end:

f[p_] := 2^(2*p + 2)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[9] (* Amiram Eldar, Jan 24 2020 *)

a(n) = 4 * A133049(n) * A330824(n).

A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 8, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 1, 2, 4, 8, 16

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the proper divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Row n has A000043(n) - 1 = A090748(n) terms.

			Triangle begins:
  1
  1, 2
  1, 2, 4, 8
  1, 2, 4, 8, 16, 32
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
  ...

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

Cf. A000043, A000668, A018254, A018487, A019279, A061652, A090748, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655, A139246.

A330839 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

18816, 5904384, 1585545216, 27015001097109504, 1770860409581431947264, 453345452974878297686016, 127605887476509680055039087507161481216, 169617318218724895492876988148194847148938611392467719301966609041193959424

Offset: 1

Walter Kehowski, Jan 21 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2, 4*M_p^2*(M_p+1)^2, (M_p^2-1)^2}, where by power-spectral basis we mean a spectral basis that consists of primes and powers. The first element of the power-spectral basis is A330819(n+1), the second element is A330840(n+1), and the third element is A330820(n+1).

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

			a(1) = 2^(2*3+1) * 3 * 7^2 = 18816, and 18816 has spectral basis {63^2, 112^2, 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, A000668, A133049, A330818, A330819, A330820, A330840.

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^(2p + 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 22 2020 *)

a(n) = A330818(n+1) * 3 * A133049(n+1).

A330841 Numbers of the form 2^(2p-3)9*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

3528, 1107072, 297289728, 5065312705708032, 332036326796518490112, 85002272432789680816128, 23926103901845565010319828907592777728, 31803247166010917904914435277786533840425989636087697369118739195223867392

Offset: 1

Walter Kehowski, Jan 25 2020

nonn

a(1) = 3528 has power-spectral basis {21^2, 28^2, 48^2}, of index 1. If n > 1, then a(n) has power-spectral basis {M^2*(M+2)^2, (1/4)*M^2*(M+1)^2, (M^2-1)^2}, with index 2, where M=A000668(n+1) is the (n+1)-st Mersenne prime. The first element of the spectral basis of a(n), n > 1, is A330819(n+1), the second element is A133051(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 a(1) + 1 whenever n = 1, and 2*a(n)+1 whenever n > 1. In this case, we say that a(n) has index 1 and index 2, respectively.
a(n), n > 1, is also isospectral with 9*A133051(n), that is, a(n) and 9*A133051(n) have the same spectral basis, but 9*A133051(n) has index 1. Thus 9*A133051(n) and a(n) form an isospectral pair.

			a(2) = 2^(2*5-3)*9*31^2 = 2^7*9*31^2 = 1107072 has spectral basis {1023^2, 496^2, 960^2}, consisting of powers. The spectral sum of a(2), that is, the sum of the elements of its spectral basis, is 2*a(2)+1 = 2214145. In this case we say that a(2) has index 2. The number 9 * A330817(2) = 2^(2*5-2)*9*31^2 = 2^8*9*31^2 = 2214144 has the same spectral basis as a(2), but with index 1. We say that 9 * A330817(2) and a(2) 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, A133051, A152921, A152922, A330818, A330819, A330820, 9*A133051, 9*A330817, A330837.

a := proc(n::posint)
local p, m;
p:=NumberTheory[IthMersenne](n+1);
m:=2^p-1;
return 2^(2*p-3)*9*m^2;
end;

f[p_] := 9*2^(2*p - 3)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A152922(n+1) * 9 * A133049(n+1).

A174744 (2^p-1)^2 where p is prime.

Original entry on oeis.org

9, 49, 961, 16129, 4190209, 67092481, 17179607041, 274876858369, 70368727400449, 288230375077969921, 4611686014132420609, 18889465931203702947841, 4835703278454118652313601, 77371252455318674995150849, 19807040628565802923409276929

Offset: 1

Vincenzo Librandi, Apr 11 2010

nonn

			For p=2, (2^2-1)^2=9; for p=3, (2^3-1)^2=49; for p=5, (2^5-1)^2=961.

Cf. A133049.

[(2^p-1)^2: p in [0..1000] |IsPrime(p)];

Table[(2^Prime[n] - 1)^2, {n, 20}] (* Vincenzo Librandi, May 01 2015 *)

Definition clarified by Jon E. Schoenfield, Jun 18 2010

More terms from Vincenzo Librandi, May 01 2015

A228018 Prime powers p (A025475) such that the sum of the proper divisors of p is also a prime power.

Original entry on oeis.org

9, 49, 243, 961, 16129, 67092481, 17179607041, 274876858369, 4611686014132420609

Offset: 1

Alex Ratushnyak, Aug 02 2013

Numbers k such that both k and A001065(k) are in A025475.
Eight of the first nine terms are squares of Mersenne primes (A133049).
From Pontus von Brömssen, Sep 17 2024: (Start)
All squares of Mersenne primes are terms.
10^20 < a(10) <= A133049(9) = (2^61-1)^2.
All terms are odd. Otherwise, 2^k would be a term for some positive k >= 2 and then A001065(2^k) = 2^k-1 would be a prime power in A025475, which is impossible by Catalan's conjecture (Mihăilescu's theorem).
(End)

			Proper divisors of 243 are 1, 3, 9, 27, 81, their sum is 121 = 11^2, so 243 is in the sequence.

Cf. A025475, A001065, A133049.

#include 
#include 
#define TOP (1ULL<<32)
typedef unsigned long long U64;
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1== *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
U64 findElement(U64 *a, U64 start, U64 end, U64 element) {
  if (start+1==end)  return (a[start]==element);
  U64 mid = (start+end)/2;
  if (a[mid] > element)
    return findElement(a, start, mid, element);
  return findElement(a, mid, end, element);
}
int main() {
  U64 i, j, p, n=0, *pp = (U64*)malloc(TOP/2), sum;
  unsigned char *c = (unsigned char *)malloc(TOP/16);
  if (!c || !pp) exit(1);
  memset(c, 0, TOP/16);
  pp[n++] = 1;
  for (i=1; i < TOP; i+=2) {
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      for (p=i+(i==1), j = p*p; ; j*=p) {
        pp[n++] = j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<60)*16.0)) break;
      }
      if (i>1)
        for (j=i*i>>1; j>3] |= 1<<(j&7);
    }
    if ((i&(i-2))==1)  printf("%llu ", i);
  }
  printf("// %llu\n\n", n);
  qsort(pp, n, 8, compare64);
  for (i=1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0)
      for (p=i+(i==1), sum=1+p, j = p*p; ; j*=p) {
        if (findElement(pp, 0, n, sum)) printf("%llu, ", j);
        sum += j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<60)*16.0)) break;
      }
  return 0;
}

for(n=1,10^6,if(!isprime(n),v=factor(n);if(matsize(v)[1]==1,s=sumdiv(n,d,d)-n;if(!isprime(s),vv=factor(s);if(matsize(vv)[1]==1,print(n)))))) /* Ralf Stephan, Aug 05 2013 */

cp's OEIS Frontend

A330840 a(n) = 4*M(n)^2*(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

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A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(n).

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A330839 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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A330841 Numbers of the form 2^(2*p-3)*9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

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A174744 (2^p-1)^2 where p is prime.

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A228018 Prime powers p (A025475) such that the sum of the proper divisors of p is also a prime power.

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C

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A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

A330839 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.

A330841 Numbers of the form 2^(2p-3)9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.