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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A062819 Inverse Moebius transform of perfect numbers, A000396.

Original entry on oeis.org

6, 34, 502, 8162, 33550342, 8589869586, 137438691334, 2305843008139960290, 2658455991569831744654692615953842678, 191561942608236107294793378084303638130997321581719586, 13164036458569648337239753460458722910223472318386943117783728134
Offset: 1

Views

Author

Labos Elemer, Jul 20 2001

Keywords

Links

Crossrefs

Cf. A000396.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, PerfectNumber[#] &]; Array[a, 11] (* Amiram Eldar, Feb 08 2025 *)

Formula

a(n) = Sum_{d|n} A000396(d). - Amiram Eldar, Feb 08 2025

Extensions

Offset corrected and a(11) added by Amiram Eldar, Feb 08 2025

A139246 Triangle read by rows: row n lists the proper divisors of n-th perfect number A000396(n).

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 7, 14, 1, 2, 4, 8, 16, 31, 62, 124, 248, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 1
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008, corrected Apr 25 2008

Keywords

Comments

Rows n has A133033(n) terms.
The n-th row sum is the n-th perfect number A000396(n).

Examples

			Triangle begins:
  1, 2, 3
  1, 2, 4, 7, 14
  1, 2, 4, 8, 16, 31, 62, 124, 248
  1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064
  ...

Links

Crossrefs

Cf. A000396, A018254, A018487, A133024, A133025, A133031, A133033, A135652, A135653, A135654, A135655, A139247.

Programs

  • Mathematica
    Table[Most[Divisors[PerfectNumber[n]]],{n,6}]//Flatten (* Harvey P. Dale, Jul 08 2024 *)

A154896 Sum of proper divisors minus the number of proper divisors of the perfect number A000396(n).

Original entry on oeis.org

3, 23, 487, 8115, 33550311, 8589869023, 137438691291, 2305843008139952067, 2658455991569831744654692615953842055, 191561942608236107294793378084303638130997321548169039
Offset: 1

Views

Author

Omar E. Pol, Jan 25 2009

Keywords

Comments

a(n) is also the difference between the n-th perfect number and its number of proper divisors.

Examples

			a(2) is 23 because the second perfect number is 28 and the sum of proper divisors of 28 is also 28 = 1+2+4+7+14 and the number of proper divisors of 28 is 5, then 28-5 = 23.

Crossrefs

Cf. A000396, A001065, A032741, A133033, A152770.

Formula

a(n) = A152770(A000396(n)) = A000396(n)-A133033(n).

Extensions

More terms from Max Alekseyev, Dec 12 2011

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that k*d*m + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

Original entry on oeis.org

1, 2136, 13494274080, 216818853118725
Offset: 1

Views

Author

Amiram Eldar, Sep 07 2018

Keywords

Comments

Chernick proved that (6m + 1)*(12m + 1)*(18m + 1) is a Carmichael number, if all the 3 factors are primes (A033502, A046025).
Lieuwens generalized it to Product_{i} (k*d(i)*m + 1), for k a perfect number.
a(1) corresponds to 6. It was found by Jack Chernick in 1939.
a(2) corresponds to 28. It was found by Dubner in 1996. Lieuwens evaluated that the least corresponding Carmichael number > 10^27.
a(3) corresponds to 496. It was found by Jim Fougeron in 2002 (Dubner found a larger value: 474382033125).
a(4) corresponds to 8128. It was found by Phil Carmody in 2002.
The corresponding Carmichael numbers are 1729, 599966117492747584686619009, 1.631... * 10^126, 4.559... * 10^260, ...

Examples

			28 = 1 + 2 + 4 + 7 + 14 is the second perfect number. 2136 is the least number m such that 28*1*333 + 1 = 59809, 28*2*2136 + 1 = 119617, 28*4*2136 + 1 = 239233, 28*7*2136 + 1 =  418657 and 28*14*2136 + 1 = 837313 are all primes, therefore 59809*119617*239233*418657*837313 = 599966117492747584686619009 is a Carmichael number.

References

  • Harold Davenport, The Higher Arithmetic, Cambridge University Press, 7th ed., 1999, exercise 8.4.
  • Harvey Dubner, Carmichael numbers and Egyptian fractions, Mathematica japonicae, Vol. 43, No. 2 (1996), pp. 411-419.

Links

Crossrefs

Cf. A000396, A002997, A033502, A046025, A067199.

Programs

  • Mathematica
    ms = {2, 3, 5, 7, 13}; ns = Length[ms]; M[p_] := 2^(p - 1)*(2^p - 1); L[m_] := Module[{}, d = Most[Divisors[m]]*m; aQ[n_] := AllTrue[d*n + 1, PrimeQ]; n=1; While[!aQ[n], n++];n]; s={}; Do[m = M[ms[[k]]]; b = L[m]; AppendTo[s, b], {k, 1, ns}]; s

A330163 Even perfect numbers m from A000396 such that w = (m + 2^(k(m) - 1) - 1) * 2^(2*(k(m) - 1)) is also an even perfect number, where k(m) is the Mersenne exponent A000043(m).

Original entry on oeis.org

6, 28, 8128, 2305843008139952128
Offset: 1

Views

Author

Jaroslav Krizek, Dec 04 2019

Keywords

Comments

Corresponding values of even perfect numbers w: 28, 496, 33550336, 2658455991569831744654692615953842176, ... (A330164).
Corresponding values of Mersenne exponents k(m) and k(w): (2, 3, 7, 31, ...), (3, 5, 13, 61, ...), where k(w) = 2*k(m) - 1.

Crossrefs

Cf. A000043, A000396, A330164.

Programs

  • Magma
    [(2^k - 1) * (2^(k - 1)): k in [1..100] | SumOfDivisors((2^k - 1) * (2^(k - 1))) / ( (2^k - 1) * (2^(k - 1))) eq 2 and SumOfDivisors(((2^k - 1) * (2^(k - 1)) + (2^(k - 1) - 1)) * (2^(2*(k - 1)))) / (((2^k - 1) * (2^(k - 1)) + (2^(k - 1) - 1)) * (2^(2*(k - 1)))) eq 2]
  • Mathematica
    f[n_] := 2^(n - 1)*(2^n - 1); g[n_] := 2^n - 2^((n - 1)/2); mers = MersennePrimeExponent[Range[10]]; g /@ Select[mers, MemberQ[f /@ mers, g[#]] &] (* Amiram Eldar, Dec 06 2019 *)

A330164 Even perfect numbers w from A000396 such that number m = w / 2^(k(w) - 1) - 2^((k(w) - 1)/2) + 1 = 2^k(w) - 2^((k(w) - 1)/2) is also an even perfect number, where k(w) is the Mersenne exponent (A000043) for number w.

Original entry on oeis.org

28, 496, 33550336, 2658455991569831744654692615953842176
Offset: 1

Views

Author

Jaroslav Krizek, Dec 04 2019

Keywords

Comments

Corresponding values of even perfect numbers m: 6, 28, 8128, 2305843008139952128, ... (A330163).
Corresponding values of Mersenne exponents k(w) and k(m): (3, 5, 13, 61, ...), (2, 3, 7, 31, ...), where k(m) = (k(w) + 1)/2.

Crossrefs

Cf. A000043, A000396, A330163.

Programs

  • Magma
    [(2^k - 1) * 2^(k - 1): k in [1..100] | SumOfDivisors((2^k - 1) * 2^(k - 1)) / ((2^k - 1) * 2^(k - 1)) eq 2 and SumOfDivisors(2^k - 2^((k-1) div 2)) / (2^k - 2^((k-1) div 2) ) eq 2]
  • Mathematica
    f[n_] := 2^(n - 1)*(2^n - 1); g[n_] := 2^n - 2^((n - 1)/2); mers = MersennePrimeExponent[Range[10]]; f /@ Select[mers, MemberQ[f /@ mers, g[#]] &] (* Amiram Eldar, Dec 06 2019 *)

A379725 a(n) = A023900(n*A000396(n)).

Original entry on oeis.org

2, 6, -60, 126, -32760, -262140, -3145716, 2147483646, -4611686018427387900, -2475880078570760549798248440, -1622592768292133633915780102881260, -340282366920938463463374607431768211452
Offset: 1

Views

Author

Mats Granvik, Antti Karttunen, Dec 31 2024

Keywords

Crossrefs

Cf. A379724, A000396, A023900, A000668.

Programs

  • Mathematica
    nn = 12; A023900[n_] := DivisorSum[n, MoebiusMu[#]  # &]; Table[A023900[PerfectNumber[n]*n], {n, 1, nn}]

Formula

Conjecture: a(n) = (-1)^(n+1)*A023900(n)*A023900(A000396(n)).

A062818 Values of Moebius transform of perfect numbers, A000396.

Original entry on oeis.org

6, 22, 490, 8100, 33550330, 8589868538, 137438691322, 2305843008139944000, 2658455991569831744654692615953841680, 191561942608236107294793378084303638130997321514618858
Offset: 1

Views

Author

Labos Elemer, Jul 20 2001

Keywords

Links

Crossrefs

Cf. A000396.

Extensions

Offset 1 from Michel Marcus, Nov 04 2018

A096360 Divisors of perfect numbers (A000396), sorted.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 14, 16, 28, 31, 32, 62, 64, 124, 127, 128, 248, 254, 256, 496, 508, 512, 1016, 1024, 2032, 2048, 4064, 4096, 8128, 8191, 8192, 16382, 16384, 32764, 32768, 65528, 65536, 131056, 131071, 131072, 262112, 262142, 262144, 524224, 524284
Offset: 1

Views

Author

Lekraj Beedassy, Jun 30 2004

Keywords

Crossrefs

Cf. A000396, A000043.

Programs

  • Mathematica
    a = {}; Do[p = 2^Prime[n] - 1; If[ PrimeQ[p], AppendTo[a, Divisors[ p*(p + 1)/2]]], {n, 17}]; a = Take[ Union[ Flatten[a]], 46] (* Robert G. Wilson v, Jul 14 2004 *)

Extensions

Edited and extended by Robert G. Wilson v and Ray Chandler, Jul 14 2004

A138728 Array read by rows: row n lists n-th even superperfect number A061652(n), n-th Mersenne prime A000668(n) and n-th perfect number A000396(n).

Original entry on oeis.org

2, 3, 6, 4, 7, 28, 16, 31, 496, 64, 127, 8128, 4096, 8191, 33550336, 65536, 131071, 8589869056, 262144, 524287, 137438691328, 1073741824, 2147483647, 2305843008139952128, 1152921504606846976, 2305843009213693951
Offset: 1

Views

Author

Omar E. Pol, Apr 07 2008

Keywords

Examples

			Array begins:
2, 3, 6
4, 7, 28
16, 31, 496

Links

Crossrefs

Cf. A000396, A000668, A019279, A061652, A138818, A138819.
Previous Showing 21-30 of 689 results. Next

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