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.

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A138867 First two digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 16, 64, 40, 65, 26, 10, 11, 30, 81, 85, 34, 26, 52, 73, 22, 12, 95, 14, 23, 17, 14, 21, 22, 20, 42, 26, 26, 25, 37, 87, 64, 20, 40, 31, 63, 21, 46, 62, 14, 61, 15, 62, 10, 84, 15
Offset: 1

Views

Author

Omar E. Pol, Apr 02 2008

Keywords

Comments

Also, first two digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Crossrefs

Cf. A019279, A061652, A138862, A138868, A138869, A138873.

Extensions

a(12)-a(31) from R. J. Mathar, Feb 05 2010
a(32)-a(47) from Jinyuan Wang, Mar 14 2020

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

Views

Author

Omar E. Pol, Dec 15 2008

Keywords

Comments

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

Crossrefs

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

Programs

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

Formula

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

Extensions

More terms from Amiram Eldar, Mar 06 2022

A228567 Primes expressible as sigma(sigma(n)) - sigma(n), in order of their occurrence.

Original entry on oeis.org

3, 7, 17, 31, 31, 41, 23, 73, 127, 73, 89, 127, 463, 523, 241, 523, 157, 241, 523, 463, 211, 257, 131, 983, 379, 1153, 311, 1153, 83, 983, 521, 4339, 4339, 113, 8893, 4339, 4339, 1093, 4339, 769, 2851, 8893, 4339, 1429, 1097, 4339, 1093, 4339, 8893, 4339, 8893
Offset: 1

Views

Author

K. D. Bajpai, Nov 10 2013

Keywords

Examples

			a(9)= 127: sigma(sigma(93))-sigma(93)= 255-128= 127, which is prime.
a(11)= 89: sigma(sigma(98))-sigma(98)= 260-171= 89, which is prime.

Links

Crossrefs

Cf. A000203 (sigma(n): sum of divisors of n).
Cf. A019279 (superperfect numbers: sigma(sigma(n))=2n).
Cf. A033632 (numbers n: sigma(n)is prime).
Cf. A051027 (sigma(sigma(n))).

Programs

  • Maple
    with(numtheory):KD := proc() local a; a:= sigma(sigma(n))-sigma(n);if isprime(a) then RETURN (a); fi; end: seq(KD(),n=1..5000);

A138870 First 3 digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 16, 64, 409, 655, 262, 107, 115, 309, 811, 850, 343, 265, 520, 737, 223, 129, 953, 142, 239, 173, 140, 215, 224, 201, 427, 268, 260, 256, 373, 870, 647, 206, 407, 311, 637, 218, 462, 629, 149, 610, 157, 622, 101, 849, 158
Offset: 1

Views

Author

Omar E. Pol, Apr 02 2008

Keywords

Comments

Also, first 3 digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Crossrefs

Cf. A019279, A061652, A138864, A138867, A138871, A138872, A138876.

Extensions

a(12)-a(31) from R. J. Mathar, Feb 05 2010
a(32)-a(41) from Max Alekseyev, Feb 11 2012
a(42)-a(47) from Jinyuan Wang, Mar 14 2020

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

Views

Author

Omar E. Pol, Apr 22 2008

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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).

Extensions

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

A153475 Sum of the first n even superperfect numbers (A061652).

Original entry on oeis.org

2, 6, 22, 86, 4182, 69718, 331862, 1074073686, 1152921505680920662, 309485010974266574405701718, 81129947899617655962363410845782, 85070672860182515483499614221352898646
Offset: 1

Views

Author

Omar E. Pol, Dec 27 2008

Keywords

Comments

Also, sum of first n superperfect numbers A019279, if there are no odd superperfect numbers.

Crossrefs

Cf. A092336, A019279, A061652, A153474, A153476.

Formula

a(n) = A061652(1) + A061652(2) + ... + A061652(n). - Max Alekseyev, Jul 27 2009

Extensions

More terms from Max Alekseyev, Jul 27 2009

A249902 Numbers n such that 2n-1 and sigma(n) are both primes.

Original entry on oeis.org

2, 4, 9, 16, 64, 289, 1681, 2401, 3481, 4096, 15625, 65536, 85849, 262144, 491401, 531441, 552049, 683929, 703921, 734449, 1352569, 1885129, 3411409, 3892729, 5470921, 7091569, 7778521, 9247681, 10374841, 12652249, 18139081, 19439281, 22287841, 23902321
Offset: 1

Views

Author

Jaroslav Krizek, Nov 14 2014

Keywords

Comments

Intersection of A006254 and A023194.
Sequence is a supersequence of the even superperfect numbers m_k (A061652 or even terms from A019279) because sigma(m_k) = 2*(m_k)-1 = k-th Mersenne prime A000668(k) for k>=1.
Conjecture: 2 and 9 are the only numbers n such that 2n - 1, 2n + 1 and sigma(n) are all primes.

Examples

			289 is in the sequence because 2*289 - 1 = 577 and sigma(289) = 307 (both primes).

Links

Crossrefs

Cf. A000203, A006254, A023194, A019279, A249903.

Programs

  • Magma
    [n: n in [2..10000000] | IsPrime(2*n-1) and IsPrime(SumOfDivisors(n))];
  • Mathematica
    Select[Range[10^7], PrimeQ[2 # - 1] && PrimeQ[DivisorSigma[1, #]] &] (* Vincenzo Librandi, Nov 15 2014 *)
  • PARI
    for(n=1,10^6,if(isprime(2*n-1)&&isprime(sigma(n)),print1(n,", "))) \\ Derek Orr, Nov 14 2014
  • Python
    from sympy import isprime, divisor_sigma
A249902_list = [2]+[n for n in (d**2 for d in range(1,10**3)) if isprime(2*n-1) and isprime(divisor_sigma(n))] # Chai Wah Wu, Jul 23 2016

A138820 Concatenation of n-th even superperfect number, n-th Mersenne prime A000668(n) and n-th perfect number.

Original entry on oeis.org

236, 4728, 1631496, 641278128, 4096819133550336, 655361310718589869056, 262144524287137438691328, 107374182421474836472305843008139952128, 115292150460684697623058430092136939512658455991569831744654692615953842176
Offset: 1

Views

Author

Omar E. Pol, Apr 05 2008

Keywords

Comments

Concatenation of A061652(n), A000668(n) and A000396(n).
Also concatenation of n-th superperfect number, n-th Mersenne prime A000668(n) and n-th perfect number, if there are no odd superperfect numbers.

Crossrefs

Cf. A000396, A000668, A019279, A061652, A138818, A138819.

A138834 Bisection of even superperfect numbers A061652.

Original entry on oeis.org

2, 16, 4096, 262144, 1152921504606846976, 81129638414606681695789005144064
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

Also, bisection of superperfect numbers A019279, if there are no odd superperfect numbers.

Links

Crossrefs

Cf. A019279, A061652, A099057, A099058, A134721, A134722.

Formula

a(n) = A061652(2*n-1). - Jinyuan Wang, Mar 14 2020

A138868 Last two digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 16, 64, 96, 36, 44, 24, 76, 56, 64, 64, 76, 64, 44, 4, 76, 36, 96, 4, 56, 76, 96, 36, 76, 56, 36, 4, 4, 56, 24, 44, 96, 64, 56, 76, 36, 96, 36, 24, 4, 24, 36, 36, 64, 76, 56
Offset: 1

Views

Author

Omar E. Pol, Apr 02 2008

Keywords

Comments

Also, last two digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Crossrefs

Cf. A019279, A061652, A138867, A138869, A138870, A138874.

Extensions

More terms from Jinyuan Wang, Mar 14 2020
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