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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A185811 a(n) is the k value that corresponds to A185729(n).

Original entry on oeis.org

2, 2, 2, 8, 7, 11, 11, 11, 11, 15, 16, 20, 17, 23, 19, 24, 23, 24, 23, 24, 27, 26, 31, 32, 32, 31, 32, 32, 35, 35, 35, 36, 40, 42, 42, 44, 41, 47, 48, 48, 47, 48, 48, 51, 52, 51, 53, 56, 50, 56, 55, 53, 57, 59, 60, 64, 61, 64, 64, 68, 68, 68, 68, 64
Offset: 1

Views

Author

Andrew Weimholt, Feb 05 2011

Keywords

Examples

			a(4) = 8 because A185729(4) = 51 which is the sum of its first 8 non-divisors.

Links

Crossrefs

Cf. A185729, A064510.

A185961 Let d_1=1 < d_2 < d_3 < ... be the divisors of n; sequence lists positive numbers n such that for some k, n = 2(d_1 + ... + d_k).

Original entry on oeis.org

2, 6, 12, 28, 40, 48, 224, 234, 496, 960, 8128, 47616, 174592, 10371840, 15037440, 28090368, 33550336, 134209536, 207516672, 492101632, 1150402560, 8589869056, 59205411720, 137438691328
Offset: 1

Views

Author

N. J. A. Sloane, Feb 07 2011

Keywords

References

  • Arie Groeneveld, Posting to Sequence Fans Mailing List, Feb 06, 2011

Crossrefs

Supersequence of A000396.
Cf. A064510, A185584, A185960.

Programs

  • PARI
    forstep(n=2, 33550336, 2, d=divisors(n); s=0; for(j=1, numdiv(n), s=s+2*d[j]; if(s>=n, if(s==n, print1(n ", ")); next(2)))) \\ Donovan Johnson, Jan 24 2014

Extensions

a(14)-a(22) from Donovan Johnson, Feb 10 2011
a(23)-a(24) from Donovan Johnson, Jan 24 2014

A295265 Numbers m such that sum of its i first divisors equals the sum of its j first non-divisors for some i, j.

Original entry on oeis.org

4, 8, 10, 13, 14, 16, 19, 20, 21, 22, 26, 28, 30, 32, 34, 38, 39, 40, 43, 44, 46, 50, 52, 53, 56, 58, 60, 62, 63, 64, 68, 70, 72, 74, 76, 80, 82, 86, 88, 89, 90, 92, 94, 98, 99, 100, 103, 104, 106, 110, 111, 112, 116, 117, 118, 122, 124, 128, 130, 132, 134, 135
Offset: 1

Views

Author

Michel Lagneau, Feb 22 2018

Keywords

Comments

Or numbers m such that Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) for some i, j where d(k) are the i first divisors and nd(k) the j non-divisors of m.
The corresponding sums are 3, 3, 3, 14, 3, 3, 20, 3, 11, 3, 3, (3 or 14), 11, 3, 3, 3, 17, 3, 44, 3, 3, 3, 3, 54, 3, 3, 15, 3, 11, 3, 3, 3, 33, 3, 3, 3, ... containing the set of primes {3, 11, 17, 23, 29, 37, 41, 43, 53, 59, 61, 71, 79, ...}.
The equality Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) is not always unique, for instance for a(12) = 28, we find 1 + 2 = 3 and 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14.
The primes of the sequence are 13, 19, 43, 53, 89, 103, 151, 229, 251, 349, 433, ... (primes of the form k(k+1)/2 - 2; see A124199).
+-----+-----+-----+------+-----------------------------------------+
| n | i | j | a(n) | Sum_{k=1..i} d(k) = Sum_{k=1..j} nd(k) |
+-----+-----+-----+------+-----------------------------------------+
| 1 | 2 | 1 | 4 | 1 + 2 = 3 |
| 2 | 2 | 1 | 8 | 1 + 2 = 3 |
| 3 | 2 | 1 | 10 | 1 + 2 = 3 |
| 4 | 2 | 4 | 13 | 1 + 13 = 2 + 3 + 4 + 5 = 14 |
| 5 | 2 | 1 | 14 | 1 + 2 = 3 |
| 6 | 2 | 1 | 16 | 1 + 2 = 3 |
| 7 | 2 | 5 | 19 | 1 + 19 = 2 + 3 + 4 + 5 + 6 = 20 |
| 8 | 2 | 1 | 20 | 1 + 2 = 3 |
| 9 | 3 | 3 | 21 | 1 + 3 + 7 = 2 + 4 + 5 = 11 |
| 10 | 2 | 1 | 22 | 1 + 2 = 3 |
| 11 | 2 | 1 | 26 | 1 + 2 = 3 |
| 12 | 2 | 1 | 28 | 1 + 2 = 3 |
| | 4 | 3 | 28 | 1 + 2 + 4 + 7 = 3 + 5 + 6 = 14 |
| 13 | 4 | 2 | 30 | 1 + 2 + 3 + 5 = 4 + 7 = 11 |
| 14 | 2 | 1 | 32 | 1 + 2 = 3 |

Examples

			30 is in the sequence because d(1) + d(2) + d(3) + d(4) = 1 + 2 + 3 + 5 = 11 and nd(1) + nd(2) = 4 + 7 = 11.

Links

Crossrefs

Cf. A064510, A124199, A185729, A240698.

Programs

  • Maple
    with(numtheory):nn:=300:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):lst:={}:ii:=0:
  for i from 1 to n do:
   lst:=lst union {i}:
  od:
    lst:=lst minus d:n1:=nops(lst):
     for m from 1 to n0 while(ii=0) do:
      s1:=sum(‘d[i]’, ‘i’=1..m):
       for j from 1 to n1 while(ii=0) do:
        s2:=sum(‘lst[i]’, ‘i’=1..j):
         if s1=s2
          then
          ii:=1:printf(`%d, `,n):
         else
         fi:
        od:
     od:
  od:
  • Mathematica
    fQ[n_] := Block[{d = Divisors@ n}, nd = nd = Complement[Range@ n, d]; Intersection[Accumulate@ d, Accumulate@ nd] != {}]; Select[ Range@135, fQ] (* Robert G. Wilson v, Mar 06 2018 *)
  • PARI
    isok(n) = {d = divisors(n); psd = vector(#d, k, sum(j=1, k, d[j])); nd = setminus([1..n], d); psnd = vector(#nd, k, sum(j=1, k, nd[j])); #setintersect(psd, psnd) != 0;} \\ Michel Marcus, May 05 2018

A306373 Integers m such that the sum of the first k divisors is equal to 2*m for some k less than the number of divisors of m.

Original entry on oeis.org

120, 672, 4320, 4680, 26208, 523776, 20427264, 29795040, 34369920, 96445440, 197064960, 459818240, 557107200
Offset: 1

Views

Author

Michel Marcus, Feb 11 2019

Keywords

Comments

3-perfect numbers (A005820) are terms.
All known terms of A055153 (abundancy 7/2) are terms.
1907020800 (with abundancy 23/6) is a term too.
A055153 is a subsequence, because no term of that sequence may be odd and so for each k in A055153 we have 2*k = sigma(k) - k - k/2. - Charlie Neder, Feb 12 2019

Crossrefs

Cf. A005820 (3-perfect numbers), A055153 (abundancy 7/2).
Cf. A064510, A194472 (both with equal to m rather than to 2*m).

Programs

  • PARI
    isok(n) = {if (sigma(n) < 2*n, return (0)); my(d = divisors(n), s = 0); for (k=1, #d-1, s += d[k]; if (s == 2*n, return (1)); if (s > 2*n, break);); return (0);}
  • PARI
    is(n) = my(d = divisors(n), s = vecsum(d) - d[#d]); forstep(i = #d-1, 1, -1, if(s <= 2*n, return(s == 2*n)); s-=d[i]); 0 \\ David A. Corneth, Feb 11 2019

Extensions

a(11)-a(13) from Jinyuan Wang, Feb 11 2019
Previous Showing 11-14 of 14 results.

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

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