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 1-6 of 6 results.

A175811 A007318-deficient numbers.

Original entry on oeis.org

1, 7, 11, 13, 17, 18, 19, 23, 24, 25, 29, 30, 31, 32, 33, 37, 38, 39, 40, 41, 42, 43, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117
Offset: 1

Views

Author

Vladimir Shevelev, Dec 05 2010

Keywords

Comments

Definition see in comment to A175522. The same criticism on index-selection as in A175807 applies. All primes greater than 5 are in the sequence.

Links

Crossrefs

Cf. A007318, A175522, A175807 (perfect version), A005100, A005101.

Programs

  • Maple
    A007318 := proc(n) option remember; local t, r; t := 0 ; for r from 0 do if t+r+1 > n then return binomial(r, n-t) ; end if; t := t+r+1 ; end do: end proc:
isA175811 := proc(n) m := 0 ; for d in numtheory[divisors](n) minus {n} do m := m+A007318(d) ; end do; m < A007318(n) ; end proc:
for n from 1 to 120 do if isA175811(n) then printf("%d,", n); end if; end do: # R. J. Mathar, Dec 06 2010
  • PARI
    b(n) = {my(m = 1); while (m*(m+1)/2 < n, m++); if (! ispolygonal(n, 3), m--); binomial(m, n - m*(m+1)/2);}
isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 07 2016

Formula

{n: sum_{d|n, dA007318(d) < A007318(n)}.

Extensions

Terms >25 from R. J. Mathar, Dec 06 2010

A177050 Ceiling(n/2)-perfect numbers.

Original entry on oeis.org

2, 4, 8, 10, 16, 32, 64, 110, 128, 136, 256, 512, 884, 1024, 2048, 4096, 8192, 16384, 18632, 32768, 32896, 65536, 70564, 100804, 116624, 131072, 262144, 391612, 449295, 524288, 1048576, 2097152, 4194304, 8388608, 15370304, 16777216, 33554432, 67108864, 73995392
Offset: 1

Views

Author

Vladimir Shevelev, Dec 09 2010

Keywords

Comments

All powers of 2 except for 1 are terms of the sequence. All numbers of the form 2^(2^k-1)*p, where p=2^(2^k)+1 is a Fermat prime (k >= 1) are in the sequence. Thus numbers 136, 32896, 2147516416 are in the sequence. It is interesting that in this construction Fermat primes play the same role that Mersenne primes in construction of usual even perfect numbers. Unfortunately, the conversion for even ceiling(n/2)-perfect numbers is false: the first counterexample, found by D. S. McNeil, is 110 = 2*5*11. Besides, the first odd term, found by D. S. McNeil, is 449295 = 3*5*7*11*389.

Links

Crossrefs

Cf. A000396, A175522, A175807, A175853.

Programs

  • PARI
    isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 08 2016
  • Sage
    is_A177050 = lambda n: sum(ceil(d/2) for d in divisors(n)) == 2*ceil(n/2) # D. S. McNeil, Dec 10 2010

Extensions

a(31)-a(39) from Michel Marcus, Feb 08 2016

A175821 A007318-abundant numbers.

Original entry on oeis.org

6, 8, 9, 10, 14, 15, 16, 20, 21, 27, 28, 34, 35, 36, 44, 45, 46, 54, 55, 56, 64, 65, 66, 76, 77, 78, 80, 90, 91, 92, 102, 104, 105, 118, 119, 120, 122, 135, 136, 138, 150, 152, 153, 168, 170, 171, 172, 188, 189, 190, 192, 207, 208, 209, 210, 228, 230, 231, 232, 250, 252, 253, 254, 255, 256, 275, 276, 278, 296, 297, 298, 299, 300, 320, 322, 324, 325, 326, 327, 328, 348, 350, 351, 352, 354, 375, 376, 377, 378, 380, 381, 400, 402, 404, 405, 406, 408
Offset: 1

Views

Author

Vladimir Shevelev, Dec 05 2010

Keywords

Comments

The comment in A175522 contains a definition.

Links

Crossrefs

Cf. A175807 (perfect version), A175811 (deficient version), A007318, A005100, A005101.

Formula

A000027 \ { A175807 U A175811}. [R. J. Mathar, Dec 06 2010]

Extensions

Terms beyond 27 from R. J. Mathar, Dec 06 2010

A176234 Floor(sqrt(n))-perfect numbers.

Original entry on oeis.org

2, 3, 4, 21, 26, 27, 33, 35, 38, 46, 58, 62, 74, 475, 605, 1083, 1719, 2007, 2151, 2169, 2259, 2313, 2421, 2431, 2439, 2493, 2529, 2547, 2637, 2737, 2763, 2799, 2979, 3123, 3303, 3357, 3367, 3451, 3619, 3681, 3698, 4255, 4465, 4625, 5035, 5125, 5185, 5695, 6205
Offset: 1

Views

Author

Vladimir Shevelev, Dec 07 2010

Keywords

Comments

See definition in comment to A175522.
The even terms begin: 2, 4, 26, 38, 46, 58, 62, 74, 3698, 34226, 34726, ... - Michel Marcus, Feb 08 2016

Examples

			floor(sqrt(35))=5; floor(sqrt(1))+floor(sqrt(5))+floor(sqrt(7))=5. Therefore, 35 is in the sequence.

Links

Crossrefs

Cf. A175522, A000396, A175807.

Programs

  • Mathematica
    f[n_] := Sum[Floor[Sqrt[Divisors[n][[i]]]], {i, 1, Length[Divisors[n]] - 1}]; Select[Range[3000], f[#] == Floor[Sqrt[#]] &]
  • PARI
    isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 08 2016

Extensions

More terms from Michel Marcus, Feb 08 2016

A177084 Ceiling(n/3)-perfect numbers.

Original entry on oeis.org

2, 3, 4, 10, 14, 50, 52, 130, 184, 315, 688, 988, 2528, 6490, 35456, 396916, 537088, 538112, 801376, 1297312, 8452096, 8456192, 35221184, 53996590, 134520832, 222469702
Offset: 1

Views

Author

Vladimir Shevelev, Dec 09 2010

Keywords

Comments

For definition, see comment of A175522.

Crossrefs

Cf. A000396, A002264, A175522, A175807, A175853, A177050.

Programs

  • Mathematica
    aQ[n_] := DivisorSum[n, Ceiling[#/3] &, # < n &] == Ceiling[n/3]; Select[Range[10^6], aQ] (* Amiram Eldar, Jul 20 2019 *)
  • Sage
    is_A177084 = lambda n: sum(ceil(d/3) for d in divisors(n)) == 2*ceil(n/3) # D. S. McNeil, Dec 10 2010

Formula

{n: Sum_{d|n, dA002264(2+d) = A002264(2+n)}. - R. J. Mathar, Dec 11 2010

Extensions

a(21)-a(26) from Amiram Eldar, Jul 20 2019

A177511 A053735-perfect numbers.

Original entry on oeis.org

3, 26, 62, 74, 77, 133, 134, 143, 155, 161, 185, 203, 206, 209, 215, 218, 319, 323, 341, 386, 398, 458, 473, 542, 545, 551, 554, 562, 565, 581, 589, 611, 614, 629, 635, 662, 671, 695, 698, 703, 706, 707, 713, 718, 721, 889, 899, 913, 959, 965, 998
Offset: 1

Views

Author

Vladimir Shevelev, Dec 11 2010

Keywords

Comments

For definition, see A175522.

Links

Crossrefs

Cf. A175522, A000396, A175807, A175853, A176234, A177084, A177050.

Programs

  • Maple
    A053735 := proc(n) add(d, d=convert(n,base,3)) ;end proc:
isA177511 := proc(n) local a,d ; a := 0 ; for d in numtheory[divisors](n) minus {n} do a := a+A053735(d) ; end do: a = A053735(n) ;end proc:
for n from 1 to 1000 do if isA177511(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar
  • PARI
    isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 06 2016
  • Sage
    A053735 = lambda n: sum(n.digits(base=3))
is_A177511 = lambda n: sum(A053735(d) for d in divisors(n)) == 2*A053735(n)
# D. S. McNeil, Dec 11 2010

Formula

{n : sum_{d|n, dA053735(d) = A053735(n)}.

Extensions

Extended by D. S. McNeil, Dec 11 2010
Showing 1-6 of 6 results.

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