cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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.

Original entry on oeis.org

225, 3969, 1046529, 268402689, 4503599493152769, 295147905144993087489, 75557863725364567605249, 21267647932558653957237540927630737409, 28269553036454149273332760011886696242605918383730576346715242738439159809
Offset: 1

Views

Author

Walter Kehowski, Jan 01 2020

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Walter Kehowski, Jan 01 2020

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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

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.

Original entry on oeis.org

288, 6272, 1968128, 528515072, 9005000365703168, 590286803193810649088, 151115150991626099228672, 42535295825503226685013029169053827072, 56539106072908298497625662716064949049646203797489239767322203013731319808
Offset: 1

Views

Author

Walter Kehowski, Jan 01 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

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

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

Views

Author

Walter Kehowski, Jan 06 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    a := proc(n) if isprime(2^n-1) then return 2^(2*n) fi; end;
[seq(a(n),n=1..31)]; # ithprime(31) = 127
  • Mathematica
    2^(2*MersennePrimeExponent[Range[10]]) (* Harvey P. Dale, Jun 27 2023 *)
  • PARI
    forprime(p=1,99,isprime(2^p-1)&&print1(4^p",")) \\ or better: {A330824(n)=4^A000043(n)}. - M. F. Hasler, Feb 07 2020

Formula

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

A132793 Numbers n such that sigma(phi(n))-phi(n) = phi(sigma(n)-n).

Original entry on oeis.org

3, 70, 138, 792, 924, 1692, 1932, 2124, 2250, 2988, 3852, 30936, 112644, 189252, 240120, 261660, 263928, 338760, 364308, 379470, 390432, 504216, 529110, 785568, 862290, 917700, 979596, 1022310, 1124220, 1404270, 1434072, 2004372, 2526000
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 31 2007

Keywords

Comments

Used sigma(n)-n, namely the sum of proper divisors.

Links

Crossrefs

Cf. A000010, A000203, A001229, A018784, A033632, A132794.

Programs

  • Maple
    with(numtheory); P:=proc(n) local i,j,k; for i from 1 by 1 to n do j:=sigma(phi(i))-phi(i); k:=phi(sigma(i)-i); if j=k then print(i); fi; od; end: P(150000);
  • Mathematica
    Select[Range[2600000],DivisorSigma[1,EulerPhi[#]]-EulerPhi[#]==EulerPhi[ DivisorSigma[1,#]-#]&] (* Harvey P. Dale, Mar 24 2016 *)
  • PARI
    isA132793(n)={ if( sigma(eulerphi(n))-eulerphi(n) == eulerphi(sigma(n)-n), 1, 0 ) ; }
{ for(n=2,6000000, if(isA132793(n), print1(n, ", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 11 2007

Extensions

More terms from R. J. Mathar, Nov 11 2007
Invalid first term removed by Donovan Johnson, Sep 11 2013
Showing 1-5 of 5 results.

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