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.

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2
Offset: 1

Views

Author

Gus Wiseman, Oct 22 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239(n).

Examples

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Crossrefs

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Programs

  • Mathematica
    Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

Formula

a(n) = A056239(n) - A366531(n).

A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 70, 0, 0, 0, 165, 0, 0, 0, 330, 0, 0, 0, 704, 0, 0, 0, 1380, 0, 0, 0, 2688, 0, 0, 0, 4984, 0, 0, 0, 9394, 0, 0, 0, 16665, 0, 0, 0, 29970, 0, 0, 0, 52096, 0, 0, 0, 90090, 0, 0, 0, 152064, 0, 0, 0
Offset: 0

Views

Author

Clark Kimberling, Mar 13 2014

Keywords

Examples

			a(8) counts these 4 partitions:  431, 41111, 3221, 221111.
From _Gus Wiseman_, Oct 24 2023: (Start)
The a(0) = 1 through a(12) = 12 partitions:
  ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)
                               (3221)             (651)
                               (41111)            (4332)
                               (221111)           (5421)
                                                  (33222)
                                                  (52221)
                                                  (63111)
                                                  (432111)
                                                  (3222111)
                                                  (6111111)
                                                  (42111111)
                                                  (222111111)
(End)

Links

Crossrefs

Cf. A239259, A239260, A239262, A239263, A000041.
The LHS (sum of odd parts) is counted by A113685.
The RHS (sum of even parts) is counted by A113686.
Without all the zeros we have a(4n) = A249914(n).
The strict case (without zeros) is A255001.
The Heinz numbers of these partitions are A366748, see also A019507.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.

Programs

  • Mathematica
    z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
(* Peter J. C. Moses, Mar 12 2014 *)

Formula

A239260(n) + a(n) + A239262(n) = A000041(n).
From David A. Corneth, Oct 25 2023: (Start)
a(4*n) = A000009(2*n) * A000041(n) for n >= 0.
a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)

Extensions

More terms from Alois P. Heinz, Mar 15 2014

A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

Original entry on oeis.org

72, 240, 672, 800, 2240, 4224, 5184, 6272, 9984, 14080, 17280, 33280, 39424, 48384, 52224, 57600, 93184, 116736, 161280, 174080, 192000, 247808, 304128, 373248, 389120, 451584, 487424, 537600, 565248, 585728, 640000, 718848, 1013760, 1089536, 1244160, 1384448
Offset: 1

Views

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Keywords

Examples

			6272 = 2*2*2*2*2*2*2*7*7 is droll since 2+2+2+2+2+2+2 = 14 = 7+7.

Links

Crossrefs

For count instead of sum we have A072978.
Partitions of this type are counted by A239261, without zero terms A249914.
For prime indices instead of factors we have A366748, zeros of A366749.
The LHS is A366839 with alternating zeros, for indices A366531, triangle A113686.
The RHS is A366840, for indices A366528, triangle A113685.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A100367, A106529, A195017, A239241, A241638, A325698, A325700, A366533.

Programs

  • Maple
    f:= proc(k, m) # numbers whose sum of prime factors >= m is k; m is prime
   local S,p,j;
   option remember;
   if k = 0 then return [1]
   elif m > k then return []
   fi;
   S:= NULL:
   p:= nextprime(m);
   for j from k by -m to 0 do
     S:= S, op(map(`*`,  procname(j,p) , m^((k-j)/m)))
   od;
   [S]
end proc:
g:= proc(N) local m,R;
  R:= NULL;
  for m from 1 while 2^m < N do
   R:= R, op(map(`*`,select(`<=`,f(2*m,3), N/2^m),2^m));
  od;
  sort([R])
end proc:
g(10^8); # Robert Israel, Feb 20 2025
  • Mathematica
    Select[Range[2, 2*10^6, 2], First[#] == Total[Rest[#]] & [Times @@@ FactorInteger[#]] &] (* Paolo Xausa, Feb 19 2025 *)
  • PARI
    isok(n) = {if (n % 2, return (0)); f = factor(n); return (2*f[1,2] == sum(i=2, #f~, f[i,1]*f[i,2]));} \\ Michel Marcus, Jun 21 2013

Formula

These are even numbers k such that A366839(k/2) = A366840(k). - Gus Wiseman, Oct 25 2023 (corrected Feb 19 2025)

Extensions

Name edited by Paolo Xausa, Feb 19 2025

A366749 Self-signed alternating sum of the prime indices of n.

Original entry on oeis.org

0, -1, 2, -2, -3, 1, 4, -3, 4, -4, -5, 0, 6, 3, -1, -4, -7, 3, 8, -5, 6, -6, -9, -1, -6, 5, 6, 2, 10, -2, -11, -5, -3, -8, 1, 2, 12, 7, 8, -6, -13, 5, 14, -7, 1, -10, -15, -2, 8, -7, -5, 4, 16, 5, -8, 1, 10, 9, -17, -3, 18, -12, 8, -6, 3, -4, -19, -9, -7, 0
Offset: 1

Views

Author

Gus Wiseman, Oct 23 2023

Keywords

Comments

We define the self-signed alternating sum of a multiset y to be Sum_{k in y} k*(-1)^k.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Crossrefs

With summands of 2^(n-1) we get A048675.
With summands of (-1)^k we get A195017.
The version for alternating prime indices is A346697 - A346698 = A316524.
Positions of zeros are A366748, counted by A239261.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A300061 lists numbers with even sum of prime indices, odd A300063.
A366528 adds up odd prime indices, counted by A113685.
A366531 adds up even prime indices, counted by A113686.
Cf. A000720, A019507, A045931, A055396, A061395, A325698, A325700, A366533.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
asum[y_]:=Sum[x*(-1)^x,{x,y}];
Table[asum[prix[n]],{n,100}]

Formula

a(n) = Sum_{k in A112798(n)} k*(-1)^k.
a(n) = A366531(n) - A366528(n).
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.