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 1-4 of 4 results.

A002659 a(n) = 2*sigma(n) - 1.

Original entry on oeis.org

1, 5, 7, 13, 11, 23, 15, 29, 25, 35, 23, 55, 27, 47, 47, 61, 35, 77, 39, 83, 63, 71, 47, 119, 61, 83, 79, 111, 59, 143, 63, 125, 95, 107, 95, 181, 75, 119, 111, 179, 83, 191, 87, 167, 155, 143, 95, 247, 113, 185, 143, 195, 107, 239, 143, 239, 159, 179, 119, 335, 123, 191
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

References

  • P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000203, A002660, A002791, A074400.
A row of the array in A242639.

Programs

  • Mathematica
    2DivisorSigma[1,Range[70]]-1 (* Harvey P. Dale, Apr 14 2014 *)
  • PARI
    a(n)=if(n<1,0,2*sigma(n)-1)

Formula

G.f. for Moebius transf.: (x + 2x^2 - x^3 ) / (1 - x )^2.
a(n) = A074400(n) - 1. - Filip Zaludek, Oct 30 2016

Extensions

Better definition from Ralf Stephan, Nov 18 2004

A242639 Array read by antidiagonals upwards: A(s,n) (s>=1, n >= 1) = Sum_{d|n, d <= s} d^2 + s*Sum_{d|n, d>s} d.

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 5, 7, 7, 1, 5, 10, 13, 6, 1, 5, 10, 17, 11, 12, 1, 5, 10, 21, 16, 23, 8, 1, 5, 10, 21, 21, 32, 15, 15, 1, 5, 10, 21, 26, 38, 22, 29, 13, 1, 5, 10, 21, 26, 44, 29, 41, 25, 18, 1, 5, 10, 21, 26, 50, 36, 53, 37, 35, 12, 1, 5, 10, 21, 26, 50, 43, 61, 46, 50, 23, 28
Offset: 1

Views

Author

N. J. A. Sloane, May 21 2014

Keywords

Examples

			The array begins:
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...
1, 5, 7, 13, 11, 23, 15, 29, 25, 35, 23, 55, ...
1, 5, 10, 17, 16, 32, 22, 41, 37, 50, 34, 80, ...
1, 5, 10, 21, 21, 38, 29, 53, 46, 65, 45, 102, ...
1, 5, 10, 21, 26, 44, 36, 61, 55, 80, 56, 120, ...
1, 5, 10, 21, 26, 50, 43, 69, 64, 90, 67, 138, ...
1, 5, 10, 21, 26, 50, 50, 77, 73, 100, 78, 150, ...
1, 5, 10, 21, 26, 50, 50, 85, 82, 110, 89, 162, ...
1, 5, 10, 21, 26, 50, 50, 85, 91, 120, 100, 174, ...
1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 111, 186, ...
...

References

  • P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. Note that the entry 53 should be 50.

Crossrefs

Rows give A000203, A002659, A002660, A002791, A241603, A242643.
Main diagonal is A001157.
See A242640 for the upper triangle of this array.

Programs

  • Maple
    # Produces the square array:
with(numtheory):
A:=proc(s,n) local d,s1,s2;
s1:=0; s2:=0;
for d in divisors(n) do
if d <= s then s1:=s1+d^2 else s2:=s2+d; fi;  od:
s1+s*s2; end;
for s from 1 to 12 do lprint([seq(A(s,n),n=1..12)]); od:
  • Mathematica
    A[s_, n_] := DivisorSum[n, If[#<=s, #^2, 0]+If[#>s, s*#, 0]&];
Table[A[s-n+1, n], {s, 1, 12}, {n, 1, s}] // Flatten (* Jean-François Alcover, Mar 07 2023 *)

A002791 a(n) = Sum_{d|n, d <= 4} d^2 + 4*Sum_{d|n, d>4} d.

Original entry on oeis.org

1, 5, 10, 21, 21, 38, 29, 53, 46, 65, 45, 102, 53, 89, 90, 117, 69, 146, 77, 161, 122, 137, 93, 230, 121, 161, 154, 217, 117, 278, 125, 245, 186, 209, 189, 354, 149, 233, 218, 353, 165, 374, 173, 329, 306, 281, 189, 486, 225, 365, 282, 385, 213, 470, 285, 473, 314, 353, 237, 662, 245, 377, 410, 501, 333, 566, 269, 497
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002659, A002660.
A row of the array in A242639.

Programs

  • Maple
    with(numtheory):
A:=proc(s,n) local d,s1,s2;
s1:=0; s2:=0;
for d in divisors(n) do
if d <= s then s1:=s1+d^2 else s2:=s2+d; fi;  od:
s1+s*s2; end;
f:=s->[seq(A(s,n),n=1..80)];
f(4);
  • Mathematica
    a[n_] := DivisorSum[n, #^2 &, # < 5 &] + 4 * DivisorSum[n, # &, # > 4 &]; Array[a, 70] (* Amiram Eldar, Aug 17 2019 *)

Formula

Conjectured: Inverse Moebius transform of g.f.: (x + 2x^2 + 2x^3 + 2x^4 - 3x^4) / (1 - x)^2. - Sean A. Irvine, May 16 2014
Conjectured: a(n) = 4 * sigma(n) - f(n mod 6) where f(0) = 10, f(1) = 3, f(2) = 7, f(3) = 6, f(4) = 7, f(5) = 3. - Sean A. Irvine, May 17 2014

Extensions

Edited by N. J. A. Sloane, May 21 2014

A241603 a(n) = Sum_{d|n, d <= 5} d^2 + 5*Sum_{d|n, d>5} d.

Original entry on oeis.org

1, 5, 10, 21, 26, 44, 36, 61, 55, 80, 56, 120, 66, 110, 110, 141, 86, 179, 96, 196, 150, 170, 116, 280, 151, 200, 190, 266, 146, 344, 156, 301, 230, 260, 236, 435, 186, 290, 270, 436, 206, 464, 216, 406, 380, 350, 236, 600, 281, 455, 350, 476, 266, 584, 356, 586, 390, 440, 296, 820, 306, 470, 510, 621, 416, 704, 336
Offset: 1

Views

Author

N. J. A. Sloane, May 21 2014

Keywords

References

  • P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367.

Links

Crossrefs

Cf. A002659, A002660.
A row of the array in A242639.

Programs

  • Maple
    with(numtheory):
A:=proc(s,n) local d,s1,s2;
s1:=0; s2:=0;
for d in divisors(n) do
if d <= s then s1:=s1+d^2 else s2:=s2+d; fi; od:
s1+s*s2; end;
f:=s->[seq(A(s,n),n=1..80)];
f(5);
  • Mathematica
    sd5[n_]:=Module[{d=Divisors[n]},Total[Select[d,#<6&]^2]+5Total[Select[ d,#>5&]]]; Array[sd5,70] (* Harvey P. Dale, Mar 18 2015 *)

Extensions

Typo in definition corrected by N. J. A. Sloane, Mar 18 2015
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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