A307517 -id:A307517 - 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

A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 1, 1, 3, 1, 1, 0, 3, 0, 1, 2, 0, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 1, 2, 1, 2, 0, 4, 0, 1, 0, 3, 1, 1, 1, 3, 0, 3, 1, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 2, 0, 3, 2

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The prime factors of 300 are {2,2,3,5,5} of which {2,2,3} are less than the largest, so a(300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 0's are A000961. Positions of 1's are A325230. Positions of terms > 1 are A307517.

Cf. A000094, A000245, A001222, A052126, A053585, A061395, A062977, A064989, A071178, A105441, A108951, A256617.

Table[PrimeOmega[n/Power@@FactorInteger[n][[-1]]],{n,100}]

A071178(n) = if(1==n, 0, factor(n)[omega(n), 2]);
A325226(n) = (bigomega(n) - A071178(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A001222(n/A053585(n)).

a(n) = A001222(n) - A071178(n) = A062977(A108951(n)). - Antti Karttunen, Nov 17 2019

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

Original entry on oeis.org

6, 10, 14, 15, 18, 21, 22, 26, 33, 34, 35, 38, 39, 46, 50, 51, 54, 55, 57, 58, 62, 65, 69, 74, 75, 77, 82, 85, 86, 87, 91, 93, 94, 95, 98, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 147, 155, 158, 159, 161, 162, 166, 177, 178

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   54: {1,2,2,2}
   55: {3,5}
   57: {2,8}
   58: {1,10}

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

Positions of 1's in A325226.

Cf. A056239, A093641, A112798, A174090, A257541, A307517, A325223, A325224, A325225, A325231.

filter:= proc(n) local F;
   F:= sort(ifactors(n)[2],(a,b)-> a[1]Robert Israel, Apr 14 2019

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

from sympy import factorint
A325230_list = [n for n, m in ((n, factorint(n)) for n in range(2,10**6)) if len(m) == 2 and m[min(m)] == 1] # Chai Wah Wu, Apr 16 2019

A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

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 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 whose maximum and minimum parts differ by more than 1. The enumeration of these partitions by sum is given by A000094.

Differs from A069900 first at n = 43.

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   14: {1,4}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   52: {1,1,6}
   55: {3,5}

Positions of numbers > 1 in A243055. Complement of A000961 and A256617.

Cf. A000094, A000245, A001222, A052126, A056239, A061395, A064989, A069900, A105441, A112798, A307517, A325196.

with(numtheory):
q:= n-> (l-> pi(l[-1])-pi(l[1])>1)(sort([factorset(n)[]])):
select(q, [$2..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]>1&]

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 18 2023

nonn

			The prime factorization of 360 is 2*2*2*3*3*5, with factors greater than the least 3*3*5, so a(360) = 3.

Positions of 0's are A000961.
Positions of numbers > 0 are A024619.
Positions of first appearances appear to be A099856.
For "less than greatest" instead of "greater than least" we have A325226.
For multiplicities instead of parts we have A363131.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A047966 counts uniform partitions, ranks A072774.
A363128 counts partitions with more than one non-mode, complement A363129.
Cf. A001221, A001222, A052126, A053585, A061395, A064989, A071178, A105441, A307517, A325230.

Table[PrimeOmega[n]-If[n==1,0,FactorInteger[n][[1,2]]],{n,30}]

a(n) = A001222(n) - A067029(n).

a(n) = A001222(A028234(n)).

cp's OEIS Frontend

A000094 Number of trees of diameter 4.

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A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

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A325230 Numbers of the form p^k * q, p and q prime, p > q, k > 0.

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A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

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A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

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