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.

A175807 A007318-perfect numbers.

Original entry on oeis.org

2, 3, 4, 5, 12, 22, 26, 154
Offset: 1

Views

Author

Vladimir Shevelev, Dec 05 2010

Keywords

Comments

See definition in comment to A175522. The definition is applied to the flattened view of the binomial coefficients with a single index, without regard to fact that A007318 is a triangle.
No more terms up to 10^6. - Michel Marcus, Feb 07 2016

Examples

			Since A007318(1)+ A007318(2)+ A007318(3)+ A007318(4)+ A007318(6)=6= A007318(12), then 12 is in the sequence.

Crossrefs

Cf. A175811, A175522, A000396.

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:
isA175807 := 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 do if isA175807(n) then printf("%d,\n",n); end if; end do: # R. J. Mathar, Dec 05 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

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

A177052 Ceiling(n/2)-abundant numbers.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228
Offset: 1

Views

Author

Vladimir Shevelev, Dec 09 2010

Keywords

Comments

For definition, see A175522.
All positive numbers == 0 (mod 6) are in the sequence (basically A008588). In addition, note that all odd primes are ceiling(n/2)-deficient numbers. The first odd term of the sequence is 315.

Links

Crossrefs

Cf. A177050 (perfect version), A005100, A005101, A175524, A175526, A175811, A175821, A175853.

Programs

  • PARI
    isok(n) = sumdiv(n, d, (d ceil(n/2); \\ Michel Marcus, Feb 08 2016
  • Sage
    is_A177052 = lambda n: sum(ceil(d/2) for d in divisors(n)) > 2*ceil(n/2) # D. S. McNeil, Dec 10 2010

Formula

{n : Sum_{d|n, dA004526(1+d) > A004526(1+n)}. [R. J. Mathar, Dec 11 2010]

A177085 Ceiling(n/3)-abundant numbers.

Original entry on oeis.org

6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 110, 112, 114, 120, 126, 128, 132, 136, 138, 140, 144, 150, 152, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200
Offset: 1

Views

Author

Vladimir Shevelev, Dec 09 2010

Keywords

Comments

For definition, see comment to A175522.
It appears most terms are even, but there exist odd terms: 945, 1155, 1575, etc. For n < 10^5 there are 211 odd terms of the sequence and 24628 even ones.

Links

Crossrefs

Cf. A177084 (perfect version), A005100, A005101, A175524, A175526, A175811, A175821, A175837, A074206, A177052.

Programs

  • PARI
    isok(n) = sumdiv(n, d, (d ceil(n/3); \\ Michel Marcus, Feb 08 2016
  • Sage
    is_A177085 = lambda n: sum(ceil(d/3) for d in divisors(n)) > 2*ceil(n/3) # D. S. McNeil, Dec 10 2010

A177512 A053735-deficient numbers.

Original entry on oeis.org

1, 2, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251
Offset: 1

Views

Author

Vladimir Shevelev, Dec 11 2010

Keywords

Comments

For definition, see A175522.
All primes, except for 3, are in the sequence.
It also contains squared primes p^2 for p = 5, 7, 11, 13, 19, 23, 29, 31, 37.. (not matching current OEIS sequences). What characterizes these p?

Links

Crossrefs

Cf. A177511 (perfect version), A175524, A175811, A000040, A005100, A005101.

Programs

  • PARI
    isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 06 2016
  • Sage
    is_A177512 = 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)}.
Showing 1-5 of 5 results.

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