A307516 -id:A307516 - cp's OEIS frontend

A000094 Number of trees of diameter 4.

0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 65, 88, 121, 161, 215, 280, 367, 471, 607, 771, 980, 1232, 1551, 1933, 2410, 2983, 3690, 4536, 5574, 6811, 8317, 10110, 12276, 14848, 17941, 21600, 25977, 31146, 37298, 44542, 53132, 63218, 75131, 89089

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n-1 with at least two parts of size 2 or larger. - Franklin T. Adams-Watters, Jan 13 2006
Also equal to the number of partitions p of n-1 such that max(p)-min(p) > 1. Example: a(7)=5 because we have [5,1],[4,2],[4,1,1],[3,2,1] and [3,1,1,1]. - Giovanni Resta, Feb 06 2006
Also number of partitions of n-1 with at least two parts that are smaller than the largest part. Example: a(7)=5 because we have [4,1,1],[3,2,1],[3,1,1,1],[2,2,1,1,1] and [2,1,1,1,1]. - Emeric Deutsch, May 01 2006
Also number of regions of n-1 that do not contain 1 as a part, n >= 2 (cf. A186114, A206437). - Omar E. Pol, Dec 01 2011
Also rank of the last region of n-1 multiplied by -1, n >= 2 (cf. A194447). - Omar E. Pol, Feb 11 2012
Also sum of ranks of the regions of n-1 that contain emergent parts, n >= 2 (cf. A182699). For the definition of "regions of n" see A206437. - Omar E. Pol, Feb 21 2012

			From _Gus Wiseman_, Apr 12 2019: (Start)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts of size 2 or larger, or non-hooks, are the following. The Heinz numbers of these partitions are given by A105441.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(5) = 1 through a(9) = 14 partitions of n-1 whose maximum part minus minimum part is at least 2 are the following. The Heinz numbers of these partitions are given by A307516.
  (31)  (41)   (42)    (52)     (53)
        (311)  (51)    (61)     (62)
               (321)   (331)    (71)
               (411)   (421)    (422)
               (3111)  (511)    (431)
                       (3211)   (521)
                       (4111)   (611)
                       (31111)  (3221)
                                (3311)
                                (4211)
                                (5111)
                                (32111)
                                (41111)
                                (311111)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts that are smaller than the largest part are the following. The Heinz numbers of these partitions are given by A307517.
  (211)  (311)   (321)    (322)     (422)
         (2111)  (411)    (421)     (431)
                 (2211)   (511)     (521)
                 (3111)   (3211)    (611)
                 (21111)  (4111)    (3221)
                          (22111)   (3311)
                          (31111)   (4211)
                          (211111)  (5111)
                                    (22211)
                                    (32111)
                                    (41111)
                                    (221111)
                                    (311111)
                                    (2111111)
(End)

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).

Christian G. Bower, Table of n, a(n) for n = 1..500
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Miloslav Znojil, Perturbation theory near degenerate exceptional points, arXiv:2008.00479 [math-ph], 2020.
Index entries for sequences related to trees

Cf. A000041, A206437, A034853, A000147 (diameter 5).

Cf. A006918, A083751, A084835, A105441, A115720, A257990, A307516, A307517, A325164.

g:=x/product(1-x^j,j=1..70)-x-x^2/(1-x)^2: gser:=series(g,x=0,48): seq(coeff(gser,x,n),n=1..46); # Emeric Deutsch, May 01 2006
A000094 := proc(n)
    combinat[numbpart](n-1)-n+1 ;
end proc: # R. J. Mathar, May 17 2016

t=Table[PartitionsP[n]-n,{n,0,45}];
ReplacePart[t,0,1]
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[1/QPochhammer[x]-x/(1-x)^2-1+O[x]^50, x] (* Jean-François Alcover, Feb 04 2016 *)

a(n+1) = A000041(n)-n for n>0. - John W. Layman
G.f.: x/product(1-x^j,j=1..infinity)-x-x^2/(1-x)^2. - Emeric Deutsch, May 01 2006
G.f.: sum(sum(x^(i+j+1)/product(1-x^k, k=i..j), i=1..j-2), j=3..infinity). - Emeric Deutsch, May 01 2006
a(n+1) = Sum_{m=1..n} A083751(m). - Gregory Gerard Wojnar, Oct 13 2020

More terms from Franklin T. Adams-Watters, Jan 13 2006

A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 28 2022

nonn

First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.

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.

			The a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
  1   2   4   6  12  16  24  30  36  48  72  90
      1   2   3   6   8  12  15  18  24  36  45
          1   2   4   4   8   6  12  16  24  30
              1   3   2   6   5   9  12  18  18
                  2   1   4   3   6   8  12  15
                  1       3   2   4   6   9   9
                          2   1   3   4   8   6
                          1       2   3   6   5
                                  1   2   4   3
                                      1   3   2
                                          2   1
                                          1

These divisors belong to A073491, a superset of A055932, complement A073492.
The initial case is A356224.
The complement in the initial case is counted by A356225.
A000005 counts divisors.
A001223 lists the prime gaps.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Divisors[n],nogapQ[primeMS[#]]&]],{n,100}]

A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

Original entry on oeis.org

12, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 52, 56, 60, 63, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 153, 154, 156, 160, 164, 165, 168

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with at least two not necessarily distinct parts less than the largest part. The enumeration of these partitions by sum is given by A000094.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
   76: {1,1,8}

Cf. A000094, A000245, A000961, A001222, A052126, A056239, A061395, A064989, A105441, A112798, A243055, A256617, A307516, A325196.

q:= n-> (l-> add(l[i][2], i=1..nops(l)-1)>1)(sort(ifactors(n)[2])):
select(q, [$1..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimeOmega[#/Power@@FactorInteger[#][[-1]]]>1&]

cp's OEIS Frontend

A000094 Number of trees of diameter 4.

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A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

Views

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Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica