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 1-3 of 3 results.

A320142 Numbers that have exactly two middle divisors.

Original entry on oeis.org

6, 12, 15, 20, 24, 28, 30, 35, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 77, 80, 84, 88, 90, 91, 96, 99, 104, 108, 110, 112, 117, 126, 130, 132, 135, 140, 143, 150, 153, 154, 156, 160, 165, 168, 170, 176, 182, 187, 190, 192, 195, 198, 204, 208, 209, 210, 216, 220, 221, 224, 228, 231, 234, 238, 247, 255, 260
Offset: 1

Views

Author

Omar E. Pol, Oct 06 2018

Keywords

Comments

Conjecture 1: numbers k with the property that the difference between the number of partitions of k into an odd number of consecutive parts and the number of partitions of k into an even number of consecutive parts is equal to 2.
Conjecture 2: numbers k with the property that symmetric representation of sigma(k) has width 2 on the main diagonal.
By the theorem in A067742 conjecture 2 is true. - Hartmut F. W. Hoft, Aug 18 2024

Examples

			15 is in the sequence because 15 has two middle divisors: 3 and 5.
On the other hand, in accordance with the first conjecture, 15 is in the sequence because there are three partitions of 15 into an odd number of consecutive parts: [15], [8, 7], [5, 4, 3, 2, 1], and there is only one partition of 15 into an even number of consecutive parts: [8, 7], therefore the difference of the number of those partitions is 3 - 1 = 2.
On the other hand, in accordance with the second conjecture, 15 is in the sequence because the symmetric representation of sigma(15) = 24 has width 2 on the main diagonal, as shown below in the fourth quadrant:
.                                _
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                          _ _ _|_|
.                      _ _| |      8
.                     |    _|
.                    _|  _|
.                   |_ _|  8
.                   |
.    _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|
.                    8
.

Crossrefs

Column 2 of A320051.
First differs from A001284 at a(19).
For the definition of middle divisors see A067742.
Cf. A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320137.

Programs

  • Mathematica
    a320142Q[k_] := Length[Select[Divisors[k], k/2<=#^2<2k&]]==2
a320142[n_] := Select[Range[n], a320142Q]
a320142[260] (* Hartmut F. W. Hoft, Aug 20 2024 *)

A278638 Numbers n such that 1/n is a difference of Egyptian fractions with all denominators < n.

Original entry on oeis.org

6, 12, 15, 18, 20, 21, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120, 126, 130, 132, 133, 135, 136, 138, 140, 143, 144, 145, 147, 150, 152, 153
Offset: 1

Views

Author

Robert Israel, Nov 24 2016

Keywords

Comments

Numbers n such that we can write 1/n = Sum_{1<=k
Numbers n such that A072207(n) < 2*A072207(n-1).
If n is in the sequence, so is k*n for all k>1 (cf. A328226).
Contains A001284, because 1/(m*k) = 1/(m*(k-m))-1/(k*(k-m)).
Disjoint from A000961.
2*p^k with p prime is in the sequence if and only if p=3.
3*p^k with p prime is in the sequence if and only if p=2,5,7 or 11.
4*p^k with p prime is in the sequence if and only if p=3,5,7,11,13,17 or 19.
For each m that is not a term, there are only finitely many primes p such that some m*p^k is a term. [Corrected by Max Alekseyev, Oct 08 2019]

Examples

			44 is in the sequence because 1/44 = (1/12 + 1/33) - 1/11.
4 is not in the sequence because 1/4 can't be written as the difference of sums of two subsets of {1, 1/2, 1/3}.

Links

Crossrefs

Cf. A000961, A001284, A072207.
Contains A005279. - Robert G. Wilson v, Nov 27 2016

Programs

  • Maple
    N:= 200: # to get all terms <= N
V:= Vector(N):
f:= proc(n)  option remember;
local F,E,p,e,k,m,L,L1,i,s,t,sg,Maybe;
global Rep;
  F:= numtheory:-factorset(n);
  if nops(F) = 1 then return false fi;
  if ormap(m -> n < m^2 and m^2 < 2*n, numtheory:-divisors(n)) then
    for m in numtheory:-divisors(n) do
      if n < m^2 and m^2 < 2*n then
        k:= n/m; Rep[n]:= [m*(k-m),-k*(k-m)]; return true
      fi
    od
  fi;
  F:= convert(F,list);
  E:= map(p -> padic:-ordp(n,p), F);
  i:= max[index](zip(`^`,F,E));
  p:= F[i];
  e:= E[i];
  k:= n/p^e;
  Maybe:= false;
  for i from 3^(k-1) to 2*3^(k-1)-1 do
    L:= (-1) +~ convert(i,base,3);
    s:= 1/k - add(L[i]/i,i=1..k-1);
    if numer(s) mod p = 0 then
    Maybe:= true;
      t:= abs(s/p^e); sg:= signum(s);
      if  (numer(t) <= 1 and (denom(t) < n or (denom(t) < N and V[denom(t)] = 1))) or (numer(t) = 2 and denom(t) < N and V[denom(t)] = 1) then
         L1:= subs(0=NULL, [seq(L[i]*i*p^e,i=1..k-1)]);
         if t = 0 then ;
         elif numer(t) = 1 and denom(t) < n then L1:= [op(L1),sg/t]
         elif procname(2/t) then
            L1:= ([op(L1), 2*sg/t, op(expand(sg*Rep[2/t]))])
         else next
         fi;
         if max(abs~(L1)) < n then Rep[n]:= L1; return true fi;
      fi;
    fi
  od:
  if Maybe then printf("Warning: %d is uncertain\n",n)
else false
fi;
end proc:
for n from 6 to N do
  if V[n] = 0 and f(n) then
    V[n] := 1;
    for j from 2*n to N by n do
      if not assigned(Rep[j]) then
        V[j]:= 1;
        Rep[j] := map(`*`,Rep[n],j/n);
        f(j):= true;
      fi
    od;
  fi;
od:
select(t -> V[t]=1,[$6..N]);
  • Mathematica
    sol[n_] := Module[{c, cc}, cc = Array[c, n-1]; FindInstance[AllTrue[cc, -1 <= # <= 1&] && 1/n == Total[cc/Range[n-1]], cc, Integers, 1]];
Reap[For[n = 6, n <= 200, n++, If[sol[n] != {}, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 29 2020 *)

A328226 Primitive elements of A278638.

Original entry on oeis.org

6, 15, 20, 21, 28, 33, 35, 44, 52, 55, 65, 68, 76, 77, 85, 91, 95, 115, 117, 119, 133, 143, 145, 153, 155, 161, 171, 184, 185, 187, 203, 205, 207, 209, 215, 217, 221, 232, 235, 247, 248, 253, 259, 261, 265, 279, 287, 296, 299, 301, 305, 319, 323, 328, 329, 333, 335, 341
Offset: 1

Views

Author

Max Alekseyev, Oct 08 2019

Keywords

Comments

A number k belongs to this sequence if k is a term of A278638, while its proper divisors are not. It follows that A278638 is formed by the multiples of the terms of this sequence.

Crossrefs

Cf. A000961, A001284, A072207, A278638.
Showing 1-3 of 3 results.

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