A325459 -id:A325459 - cp's OEIS frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.

1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 3, 3, 2, 3, 4, 2, 3, 2, 2, 3, 3, 3, 5, 2, 2, 3, 3, 2, 3, 2, 2, 5, 3, 2, 3, 3, 2, 4, 3, 2, 3, 4, 2, 3, 3, 2, 3, 2, 2, 3, 4, 3, 5, 2, 2, 3, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 2, 3, 3, 2, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 2, 2, 3, 4, 2, 3, 2, 2, 5, 3

Offset: 2

Alexander Adamchuk, Apr 27 2007

nonn

The indices k of the first appearance of number n in a(k) are listed in A063778(n) = {2,3,6,15,36,225,...} = Least number k>1 such that k could be represented in n different ways as general m-gonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)).
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n whose augmented differences are all equal, where the augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k; for example aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Equivalently, a(n) is the number of integer partitions of n whose differences are all equal to the last part minus one. The Heinz numbers of these partitions are given by A307824. For example, the a(35) = 5 partitions are:
  (35)
  (23,12)
  (11,9,7,5,3)
  (8,7,6,5,4,3,2)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
(End)

			a(6) = 3 because 6 = P(2,6) = P(3,3) = P(6,2).

Alois P. Heinz, Table of n, a(n) for n = 2..10000
E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A063778, A177025.
Column k=0 of A239550.
Cf. A007862, A049988, A307824, A325349, A325350, A325356, A325357, A325358, A325458, A325459.

A129654 := proc(n) local resul, dvs, i, r, m ;
   dvs := numtheory[divisors](2*n) ;
   resul := 0 ;
   for i from 1 to nops(dvs) do
      r := op(i, dvs) ;
      if r > 1 then
         m := (2*n/r-4+2*r)/(r-1) ;
         if is(m, integer) then
            resul := resul+1 ;
         fi ;
      fi ;
   od ;
   RETURN(resul) ;
end: # R. J. Mathar, May 14 2007

a[n_] := (dvs = Divisors[2*n]; resul = 0; For[i = 1, i <= Length[dvs], i++, r = dvs[[i]]; If[r > 1, m = (2*n/r-4+2*r)/(r-1); If[IntegerQ[m], resul = resul+1 ] ] ]; resul); Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Sep 13 2012, translated from R. J. Mathar's Maple program *)
Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]], {n, 2, 106}] (* Jonathan Sondow, May 09 2014 *)
atpms[n_]:=Select[Join@@Table[i*Range[k,1,-1],{k,n},{i,0,n}],Total[#+1]==n&];
Table[Length[atpms[n]],{n,100}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(2*n, d, (d>1) && (2*n/d + 2*d - 4) % (d-1) == 0); \\ Daniel Suteu, Dec 22 2018

a(n) = A177025(n) + 1.

G.f.: x * Sum_{k>=1} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020

A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

Original entry on oeis.org

9, 18, 25, 27, 36, 49, 50, 54, 72, 81, 98, 100, 108, 121, 125, 144, 162, 169, 196, 200, 216, 242, 243, 250, 288, 289, 324, 338, 343, 361, 392, 400, 432, 484, 486, 500, 529, 576, 578, 625, 648, 676, 686, 722, 729, 784, 800, 841, 864, 961, 968, 972, 1000, 1058

Offset: 1

Gus Wiseman, May 02 2019

nonn

Also Heinz numbers of integer partitions that are not hooks but whose augmented differences are hooks, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k), and a hook is a partition of the form (n,1,1,...,1). The enumeration of these partitions by sum is given by A325459.

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    18: {1,2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    50: {1,3,3}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    81: {2,2,2,2}
    98: {1,4,4}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   169: {6,6}
   196: {1,1,4,4}
   200: {1,1,1,3,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 2's in A325355.
Numbers n such that n does not belong to A093641 but A325351(n) does.
Cf. A056239, A112798, A136141, A325366, A325389, A325394, A325395, A325396.

N:= 1000: # to get terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)),2)]):
B:= map(proc(p) local y;  seq(p^y, y=2..floor(log[p](N))) end proc, P):
sort(map(proc(t) local z;  seq(2^z*t, z=0..ilog2(N/t)) end proc, B)); # Robert Israel, May 03 2019

Select[Range[1000],MatchQ[FactorInteger[2*#],{{2,},{?(#>2&),_?(#>1&)}}]&]

Sum_{n>=1} 1/a(n) = 2 * Sum_{p prime} 1/(p*(p-1)) - 1 = 2 * A136141 - 1 = 0.54631333809959025572... - Amiram Eldar, Sep 30 2020

A325458 Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 2, 3, 6, 0, 0, 0, 0, 0, 4, 4, 7, 0, 0, 0, 0, 0, 3, 6, 5, 8, 0, 0, 0, 0, 0, 1, 6, 8, 6, 9, 0, 0, 0, 0, 0, 0, 6, 9, 10, 7, 10, 0, 0, 0, 0, 0, 0, 2, 11, 12, 12, 8, 11

Offset: 0

Gus Wiseman, May 04 2019

Conjectured to be equal to A049597.

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  0  3
  0  0  0  1  4
  0  0  0  0  2  5
  0  0  0  0  2  3  6
  0  0  0  0  0  4  4  7
  0  0  0  0  0  3  6  5  8
  0  0  0  0  0  1  6  8  6  9
  0  0  0  0  0  0  6  9 10  7 10
  0  0  0  0  0  0  2 11 12 12  8 11
  0  0  0  0  0  0  2  9 16 15 14  9 12
  0  0  0  0  0  0  0  7 16 21 18 16 10 13
  0  0  0  0  0  0  0  4 18 23 26 21 18 11 14
  0  0  0  0  0  0  0  3 12 29 30 31 24 20 12 15
  0  0  0  0  0  0  0  1 12 27 40 37 36 27 22 13 16
  0  0  0  0  0  0  0  0  8 26 42 51 44 41 30 24 14 17
  0  0  0  0  0  0  0  0  6 23 48 57 62 51 46 33 26 15 18
  0  0  0  0  0  0  0  0  2 21 44 70 72 73 58 51 36 28 16 19
Row n = 9 counts the following partitions:
  (333)  (54)     (63)      (72)       (9)
         (432)    (522)     (621)      (81)
         (441)    (531)     (5211)     (711)
         (3222)   (4221)    (42111)    (6111)
         (3321)   (4311)    (321111)   (51111)
         (22221)  (32211)   (2211111)  (411111)
                  (33111)              (3111111)
                  (222111)             (21111111)
                                       (111111111)

Row sums are A000041.
Column sums are 2^(k - 1) for k > 0.
Cf. A008284, A049597, A093641, A116608, A325242, A325349, A325351, A325355, A325359, A325459.

Table[Length[Select[IntegerPartitions[n],If[n==0,k==0,First[#]+Length[#]-1==k]&]],{n,0,19},{k,0,n}]

Franklin T. Adams-Watters has conjectured at A049597 that the k-th column gives the coefficients of the sum of Gaussian polynomials [k,m] for m = 0..k.

cp's OEIS Frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.

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A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

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A325458 Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.

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cp's OEIS Frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)) = n>1, for m,r>1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A325458 Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.