A006987 -id:A006987 - cp's OEIS frontend

A319500 a(n) is the least m for which A006987(n) = binomial(m,k) for some k, 2 <= k <= m/2.

4, 5, 6, 6, 7, 8, 7, 9, 10, 11, 8, 12, 8, 13, 9, 14, 15, 10, 9, 17, 18, 11, 19, 20, 10, 12, 22, 10, 23, 24, 13, 25, 26, 11, 27, 14, 28, 29, 30, 15, 11, 31, 12, 32, 33, 16, 34, 35, 36, 37, 17, 38, 13, 39, 40, 12, 18, 41, 42, 43, 12, 44, 19, 45, 14, 46, 47, 48, 20, 49, 50, 51, 13, 52, 21, 15, 53

Offset: 1

Robert Israel, Sep 20 2018

nonn

First differs from A022911 at n=18.

			a(3) = 6 because A006987(3) = 15 = binomial(6,2).

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

Cf. A006987, A022911, A318955.

N:= 10000: # for binomial(n, k) values <= N
S:= {}:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n, k);
    if v > N then break fi;
    if not member(v,S) then
      S:= S union {v};
      M[v]:= n;
    fi
od od:
A006987:= sort(convert(S,list)):
seq(M[A006987[i]],i=1..nops(A006987));

A006987(n) = binomial(a(n),A318955(n)).

A124000 Semiprimes in A006987(n), or semiprime binomial coefficients: C(n,k), 2 <= k <= n-2.

Original entry on oeis.org

6, 10, 15, 21, 35, 55, 91, 253, 703, 1081, 1711, 1891, 2701, 3403, 5671, 12403, 13861, 15931, 18721, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801

Offset: 1

Alexander Adamchuk, Oct 31 2006

nonn

Conjecture: all a(n) except a(1) = 6 and a(2) = 10 are odd. Conjecture: all a(n) except a(5) = 35 are triangular numbers of the form p*(2p +/- 1) that belong to A068443(n) = {6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, ...} Triangular numbers with two distinct prime factors.
Besides 35 & 371953, all members were found by C(n, 2). - Robert G. Wilson v, Sep 16 2016
Of C(n,k), n: 4, 5, 6, 7, 11, 14, 23, 38, 47, 59, 62, 74, 83, 107, 158, 167, 179, 194, ..., . - Robert G. Wilson v, Sep 16 2016

			C(5,2) = 5!/(3!*2!) = 120/(6*2) = 10 is a semiprime (A001358), so 10 is in the sequence. - _Michael B. Porter_, Sep 17 2016

Robert G. Wilson v, Table of n, a(n) for n = 1..11521

Cf. A006987, A095147, A068443, A000217, A005382, A005384, A001358.

s = {}; Do[b = Binomial[n, k]; If[PrimeOmega@ b == 2, AppendTo[s, b]; Print@ b], {n, 3, 10000}, {k, 2, n/2}]; s (* Robert G. Wilson v, Nov 03 2006; updated Sep 16 2016 *)

Intersection of A001358 and A006987. - Michael B. Porter, Sep 17 2016

More terms from Robert G. Wilson v, Nov 03 2006

A009561 Index of central binomial coefficient C(2n,n) within A006987.

Original entry on oeis.org

1, 4, 13, 28, 61, 119, 226, 423, 790, 1472, 2764, 5224, 9956, 19112, 36923, 71703, 139835, 273625, 536860, 1055588, 2079066, 4100573, 8096789, 16002500, 31652194, 62648225, 124068499, 245826760

Offset: 2

David W. Wilson

nonn

A173017 A006987(a(n))=A000984(n).

Original entry on oeis.org

1, 4, 7, 19, 25, 56, 73, 144, 180, 346, 430, 826

Offset: 2

Dylan Hamilton, Nov 06 2010

nonn

A068443 Triangular numbers which are the product of two primes.

Original entry on oeis.org

6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, 3403, 5671, 12403, 13861, 15931, 18721, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801, 258121, 269011

Offset: 1

Stephan Wagler (stephanwagler(AT)aol.com), Mar 09 2002

These triangular numbers are equal to p * (2p +- 1).
All terms belong to A006987. For n>2 all terms are odd and belong to A095147. - Alexander Adamchuk, Oct 31 2006
A156592 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Triangular numbers with exactly 4 divisors. - Jon E. Schoenfield, Sep 05 2018

			Triangular numbers begin 0, 1, 3, 6, 10, ...; 6=2*3, and 2 and 3 are two distinct primes; 10=2*5, and 2 and 5 are two distinct primes, etc. - _Vladimir Joseph Stephan Orlovsky_, Feb 27 2009
a(11) = 1891 and 1891 = 31 * 61.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.

Cf. A075875.

q:= n-> is(numtheory[bigomega](n)=2):
select(q, [i*(i+1)/2$i=0..1000])[];  # Alois P. Heinz, Mar 27 2024

Select[ Table[ n(n + 1)/2, {n, 1000}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &]
Select[Accumulate[Range[1000]],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 03 2016 *)

list(lim)=my(v=List());forprime(p=2,(sqrtint(lim\1*8+1)+1)\4, if(isprime(2*p-1),listput(v,2*p^2-p)); if(isprime(2*p+1), listput(v,2*p^2+p))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2013

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Edited by Robert G. Wilson v, Jul 08 2002

Definition corrected by Zak Seidov, Mar 09 2008

A001695 a(n) = H_n(2,n) where H_n is the n-th hyperoperator.

Original entry on oeis.org

1, 3, 4, 8, 65536

Offset: 0

N. J. A. Sloane, following a suggestion from Robert G. Wilson v, Aug 31 1994

Originally named: An Ackermann function.

For hyperoperator definitions and links, see A054871.

			a(0) = H_0(2,0) = 0+1 = 1;
a(1) = H_1(2,1) = 2+1 = 3;
a(2) = H_2(2,2) = 2*2 = 4;
a(3) = H_3(2,3) = 2^3 = 8;
a(4) = H_4(2,4) = 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65536;
a(5) = H_5(2,5) = 2^^^5 = 2^^2^^2^^2^^2 = 2^^2^^2^^4 = 2^^2^^65536 = ....

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).

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133.
R. C. Buck, Mathematical induction and recursive definitions, Amer. Math. Monthly, 70 (1963), 128-135.
Y. Sundblad, The Ackermann function. A theoretical, computational and formula manipulative study, Nordisk Tidskr. Informationsbehandling (BIT) 11 (1971), 107-119.
Eric Weisstein's World of Mathematics, Ackermann Function.
R. G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989
R. G. Wilson v, Letters to BYTE Magazine (1988) and N. J. A. Sloane (1994)
Index entries for sequences related to Ackermann function

Cf. A014221, A046859, A054871.

Alternative formula:
With f(x,y)=
{y+1 if x=0
{0 if x=2, y=0
{1 if x>2, y=0
{2 if x=1, y=0
{f(x-1,f(x,y-1)) otherwise
a(n)= f(n,n);

Example, formula and Hyperoperator notation by Natan Arie Consigli with Danny Rorabaugh's help, Oct 25 2015

A052515 Number of ordered pairs of complementary subsets of an n-set with both subsets of cardinality at least 2.

Original entry on oeis.org

0, 0, 0, 0, 6, 20, 50, 112, 238, 492, 1002, 2024, 4070, 8164, 16354, 32736, 65502, 131036, 262106, 524248, 1048534, 2097108, 4194258, 8388560, 16777166, 33554380, 67108810, 134217672, 268435398, 536870852, 1073741762

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) is the number of binary sequences of length n having at least two 0's and at least two 1's. a(4)=6 because there are six binary sequences of length four that have two or more 0's and two or more 1's: 0011, 0101, 0110, 1100, 1010, 1001. - Geoffrey Critzer, Feb 11 2009
For n>3, a(n) is also the sum of those terms from the n-th row of Pascal's triangle which also occur in A006987: 6, 10+10, 15+20+15, 21+35+35+21,...  - Douglas Latimer, Apr 02 2012
From Dennis P. Walsh, Apr 09 2013: (Start)
Column 2 of triangle A200091.
Number of doubly-surjective functions f:[n]->[2].
Number of ways to distribute n different toys to 2 children so that each child gets at least 2 toys. (End)
a(n) is the number of subsets of an n-set of cardinality k with 2 <= k <= n - 2. - Rick L. Shepherd, Dec 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 81
Dennis Walsh, Notes on doubly-surjective finite functions
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(x)-1-x)^2 )); [0,0,0,0] cat [Factorial(n+3)*b[n]: n in [1..m-5]]; // G. C. Greubel, May 13 2019

Pairs spec := [S,{S=Prod(B,B),B=Set(Z,2 <= card)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0,0,0}, LinearRecurrence[{4,-5,2}, {0,6,20}, 35]] (* G. C. Greubel, May 13 2019 *)
With[{nn=30},CoefficientList[Series[(Exp[x]-x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 29 2023 *)

concat([0,0,0,0],Vec((6-4*x)/(1-x)^2/(1-2*x)+O(x^35))) \\ Charles R Greathouse IV, Apr 03 2012

x='x+O('x^35); concat([0,0,0,0],Vec(serlaplace((exp(x)-x-1)^2))) \\ Joerg Arndt, Apr 10 2013

(2*x^4*(3-2*x)/((1-x)^2*(1-2*x))).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, May 13 2019

E.g.f.: (exp(x) - x - 1)^2. - Joerg Arndt, Apr 10 2013
n*a(n+2) - (1+3*n)*a(n+1) + 2(1+n)*a(n) = 0, with a(0) = .. = a(3) = 0, a(4) = 6.
For n>2, a(n) = 2^n - 2n - 2 = A005803(n) - 2 = A070313(n) - 1 = A071099(n) - A071099(n+1) + 1 = 2*A000247(n-1). - Ralf Stephan, Jan 11 2004
G.f.: 2*x^4*(3-2*x)/((1-x)^2*(1-2*x)). - Colin Barker, Feb 19 2012

More terms from Ralf Stephan, Jan 11 2004
Definition corrected by Rainer Rosenthal, Feb 12 2010
Definition further clarified by Rick L. Shepherd, Dec 05 2014

A058084 Smallest m such that binomial(m,k) = n for some k.

Original entry on oeis.org

0, 2, 3, 4, 5, 4, 7, 8, 9, 5, 11, 12, 13, 14, 6, 16, 17, 18, 19, 6, 7, 22, 23, 24, 25, 26, 27, 8, 29, 30, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 42, 43, 44, 10, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 8, 57, 58, 59, 60, 61, 62, 63, 64, 65, 12, 67, 68, 69, 8, 71, 72, 73, 74, 75, 76

Offset: 1

Fabian Rothelius, Nov 25 2000

Index of first row of Pascal's triangle (A007318) containing n.

			a(28)=8 because 28 is first found in row 8 of Pascal's triangle (where the first row is counted as 0).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, with 3 corrections).

Cf. A007318, A006987, A137905, A357327.

import Data.List (findIndex); import Data.Maybe (fromJust)
a058084 n = fromJust $ findIndex (elem n) a007318_tabl
-- Reinhard Zumkeller, Nov 09 2011

with(combinat): for n from 2 to 150 do flag := 0: for m from 1 to 150 do for k from 1 to m do if binomial(m,k) = n then printf(`%d,`,m); flag := 1; break fi: od: if flag=1 then break fi; od: od:

nmax = 76; t = Table[Binomial[m, k], {m, 0, nmax}, {k, 0, m}]; a[n_] := Position[t, n, 2, 1][[1, 1]]-1; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Nov 30 2011 *)

a(n) = my(k=0); while (!vecsearch(vector((k+2)\2, i, binomial(k, i-1)), n), k++); k; \\ Michel Marcus, Dec 07 2021

def A058084(n):
    if n == 1: return 0
    c = [1]
    for m in range(n):
        d = [1]
        for i in range(m):
            a = c[i]+c[i+1]
            if a == n:
                return m+1
            d.append(a)
        c = d+[1] # Chai Wah Wu, Aug 23 2025

a(A006987(n)) < A006987(n); a(A137905(n)) = A137905(n). - Reinhard Zumkeller, Mar 20 2009

A007318(a(n), A357327(n)) = n. - Pontus von Brömssen, Sep 24 2022

More terms from James Sellers, Nov 27 2000

A022911 Arrange the nontrivial binomial coefficients C(m,k) (2 <= k <= m/2) in increasing order (not removing duplicates); record the sequence of m's.

Original entry on oeis.org

4, 5, 6, 6, 7, 8, 7, 9, 10, 11, 8, 12, 8, 13, 9, 14, 15, 16, 10, 9, 17, 18, 11, 19, 20, 21, 10, 12, 22, 10, 23, 24, 13, 25, 26, 11, 27, 14, 28, 29, 30, 15, 11, 31, 12, 32, 33, 16, 34, 35, 36, 37, 17, 38, 13, 39, 40, 12, 18, 41, 42, 43, 12, 44, 19, 45, 14, 46, 47, 48

Offset: 1

Clark Kimberling

nonn

In case of duplicates, the m values are listed in decreasing order. Thus a(18)=16 and a(19)=10 corresponding to binomial(16,2)=binomial(10,3)=120. - Robert Israel, Sep 18 2018

Robert Israel, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)

Cf. A003015, A006987, A022912, A319382.

N:= 10000: # for binomial(n,k) values <= N
Res:= NULL:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n,k);
    if v > N then break fi;
    Res:= Res,[v,n,k]
od od:
Res:= sort([Res],proc(p,q) if p[1]<>q[1] then  p[1]q[2] then p[2]>q[2]
  fi end proc):
map(t -> t[2], Res); # Robert Israel, Sep 18 2018

A319382(n) = binomial(a(n),A022912(n)). - Robert Israel, Sep 18 2018

A022912 Arrange the nontrivial binomial coefficients C(m,k) (2 <= k <= m/2) in increasing order (not removing duplicates); record the sequence of k's.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 3, 4, 2, 2, 3, 2, 2, 2, 4, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 2, 3, 5, 2, 4, 2, 2, 3, 2, 2, 2, 2, 3, 2, 4, 2, 2, 5, 3, 2, 2, 2, 6, 2, 3, 2, 4, 2, 2, 2, 3, 2, 2, 2, 5, 2, 3, 4, 2, 2, 2, 2, 3, 2, 2, 2, 6, 2, 3, 4, 2, 2, 2, 5, 2, 3, 2, 2, 2

Offset: 1

Clark Kimberling

nonn

In case of duplicates, the k values are listed in increasing order. Thus a(18)=2 and a(19)=3 corresponding to binomial(16,2)=binomial(10,3)=120.

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

Cf. A003015, A006987, A022911, A319382.

N:= 10000: # for binomial(n, k) values <= N
Res:= NULL:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n, k);
    if v > N then break fi;
    Res:= Res, [v, n, k]
od od:
Res:= sort([Res], proc(p, q) if p[1]<>q[1] then  p[1]q[2] then p[2]>q[2]
fi end proc): map(t -> t[3], Res); # Robert Israel, Sep 18 2018

A319382(n) = binomial(A022911(n),a(n)). - Robert Israel, Sep 18 2018

Corrected by Robert Israel, Sep 18 2018

cp's OEIS Frontend

A319500 a(n) is the least m for which A006987(n) = binomial(m,k) for some k, 2 <= k <= m/2.

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A124000 Semiprimes in A006987(n), or semiprime binomial coefficients: C(n,k), 2 <= k <= n-2.

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A009561 Index of central binomial coefficient C(2n,n) within A006987.

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A173017 A006987(a(n))=A000984(n).

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A068443 Triangular numbers which are the product of two primes.

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A001695 a(n) = H_n(2,n) where H_n is the n-th hyperoperator.

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A052515 Number of ordered pairs of complementary subsets of an n-set with both subsets of cardinality at least 2.

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A058084 Smallest m such that binomial(m,k) = n for some k.

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