Hagen von Eitzen - cp's OEIS frontend

A353010 a(n) = maximal d such that Product_{k=0..m} binomial(m,k) is divisible by m^(m+d), where m = A276710(n).

0, 0, 3, 0, 1, 9, 49, 0, 21, 19, 31, 73, 0, 61, 57, 16, 4, 46, 13, 43, 25, 0, 20, 106, 1, 57, 172, 81, 43, 66, 25, 29, 51, 41, 38, 140, 80, 1, 71, 0, 0, 34, 117, 59, 199, 134, 208, 181, 9, 55, 259, 202, 114, 28, 263, 100, 145, 32, 157, 217, 60, 121, 36, 73, 86, 94, 19, 67, 154, 21, 40, 73, 57, 167, 392, 135, 256

Offset: 1

Hagen von Eitzen, Apr 15 2022

nonn

By definition of A276710, a(n) >= -1.
It is conjectured that a(n) >= 0, computationally verified up to n = 10^7.
Empirically from terms up to n=10^7, a(n) seems to become quite large, small values are rare, and yet a(n)=0 also seems to occur for large n.

			The 7th term of A276710 is 105 because Product_{k=1..105} binomial(36,k) is divisible by 105^(105-1). Actually, it is divisible by 105^(105+49), but not by 105^(105+50). Therefore, a(7) = 49.

Cf. A276710.

from math import prod, comb
from itertools import islice
from sympy import nextprime
def A353010_gen(): # generator of terms
    p, q = 3, 5
    while True:
        for m in range(p+1,q):
            r = m**(m-1)
            c = 1
            for k in range(m+1):
                c = c*comb(m,k) % r
            if c == 0:
                d, (e, f) = -m, divmod(prod(comb(m,k) for k in range(m+1)),m)
                while f == 0:
                    d += 1
                    e, f = divmod(e,m)
                yield d
        p, q = q, nextprime(q)
A353010_list = list(islice(A353010_gen(),40)) # Chai Wah Wu, Jun 09 2022

A261811 Numbers m such that Sum_{k>=0} k^m/3^k is an integer.

Original entry on oeis.org

4, 8, 12, 13, 16, 20, 24, 32, 36, 40, 44, 48, 52, 56, 64, 68, 72, 76, 77, 80, 84, 88, 96, 100, 104, 108, 112, 116, 120, 122, 128, 132, 136, 140, 141, 144, 148, 152, 160, 164, 168, 172, 176, 180, 184, 192, 196, 200, 204, 205, 208, 212, 216, 224, 228, 232, 236, 240, 244, 248, 256, 260

Offset: 1

Hagen von Eitzen, Sep 01 2015

nonn

For a denominator with an exponent base of 2, this series is A000027; for bases more than 3, no integer values are known. All powers of 2 greater than 2 are believed to be in this sequence. - Drake Thomas, May 13 2016

			4 is in the list because 0^4/3^0 + 1^4/3^1 + 2^4/3^2 + ... = 15 is an integer.

RavenclawPrefect, When is Sum_{k>=0} k^n/3^k an integer?

Select[Range@ 260, IntegerQ[Sum[k^#/3^k, {k, 0, Infinity}]] &] (* Michael De Vlieger, Sep 02 2015 *)
Select[Range @ 260, IntegerQ@ HurwitzLerchPhi[1/3, -#, 0] &] (* Giovanni Resta, Sep 10 2015, fixed by Vaclav Kotesovec, Mar 23 2018 *)

A243077 Numbers missing from A025429 (number of partitions of n into 5 nonzero squares).

Original entry on oeis.org

188, 259, 304, 308, 372, 394, 483, 497, 594, 634, 685, 705, 722, 729, 740, 750, 756, 766, 780, 782, 834, 850, 864, 884, 885, 888, 922, 925, 946, 955, 956, 1016, 1026, 1048, 1069, 1070, 1071, 1080, 1082, 1108, 1134, 1140, 1154, 1159, 1160, 1161, 1187, 1198, 1266, 1268

Offset: 1

M. F. Hasler, May 30 2014 and Hagen von Eitzen, Jun 05 2014

nonn

For these indices, A080654 and A080673 are ill-defined. Stated another way, a(n) is the n-th index where A080654 and A080673 default to 0.

			a(1)=188 because all the values 0, 1, ..., 187 appear somewhere in A025429, but 188 doesn't.

Hagen von Eitzen, Table of n, a(n) for n = 1..29477
"James47", What is the largest integer with only one representation as a sum of five nonzero squares?, Math.StackExchange, 2014.

Cf. A025429, A080654, A080673.

Edited by N. J. A. Sloane, Jun 06 2014

A181496 Number of degree n polynomials p(x,y) with all coefficients 0 or 1 such that x+y=1 implies p(x,y)=1.

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 140, 474, 1648, 5839, 20980, 76202, 279168, 1029962, 3822336, 14256228, 53400912

Offset: 0

Hagen von Eitzen, Oct 24 2010

The polynomials are of the form P = (x+y-1) * Q + 1.

			For degree 0, the only solution is 1, hence a(0) = 1.
For degree 1, the only solution is x + y, hence a(1) = 1.
For degree 2 there are x^2 + x*y + y and x + x*y + y^2 (and no more), hence a(2) = 2.
For n = 3 the a(3) = 5 polynomials are x^3 + x^2 y + x y + y, x^2 y + x y^2 + x^2 + y, x^2 y + x y^2 + x^2 + x y + y^2, x y^2 + y^3 + x y + x, x^2 y + x y^2 +y^2 + x.

Murray S. Klamkin, Problem #3, Missouri State Math Dept. Challenge Archive, 2005/06 discusses the generalization to m variables.

Cf. A122743.

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {if (n == 0, return (1)); kill(x); kill(y); poln = vector(n+1, i, x^(i-1)*y^(n-i+1)); polm = [1]; for (i = 1, n-1, polm = concat(polm, vector(i+1, j, x^(j-1)*y^(i-j+1)));); nbpol = 0; nbn = #poln; nbm = #polm; for (i = 1, 2^nbn, bi = padbin(i, nbn); poli = sum(ki = 1, nbn, bi[ki]*poln[ki]); for (j = 0, 2^nbm, bj = padbin(j, nbm); pol = poli + sum(kj = 1, nbm, bj[kj]*polm[kj]); if (subst(pol, y, 1-x) == 1, nbpol++;););); nbpol;} \\ Michel Marcus, Aug 25 2013

a(13) from Hagen von Eitzen, Oct 25 2010

a(14)-a(16) from Artem Romanov, Jul 25 2019

A160642 Minimal number k such that n! can be written as product of k (>= 2) consecutive integers.

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 2

Hagen von Eitzen, May 21 2009

nonn

Sequence starts at n=2 because 1! cannot be written as product of 2 (or more) consecutive integers.
For suitable m >= n, we have n! = m!/(m-a(n))!
For n >= 3, we have a(n) <= n-1 because n! = 2*...*n.
For n = m! - 1, we have a(n) <= m!-m because n! = (m+1)*(m+2)*...*(m!-1)*m! = (n+1)!/m!.
For n>=8, it appears that the preceding two inequalities completely describe a(n), i.e. a(n) = m!-m if n=m!-1 and a(n)=n-1 otherwise.

			a(2) = 2 because 2! = 1*2. a(3) = 2 because 3! = 2*3. a(4) = 3 because 4! = 2*3*4. a(5) = 3 because 5! = 4*5*6. a(6) = 3 because 6! = 8*9*10. a(7) = 4 because 7! = 7*8*9*10.

csfac(N, k) = local(d, w=floor(N^(1/k))); while((d=prod(i=1,k,w+i))>N,w=w-1);if(d==N,1,0)
csmin(N) = local(k=2); while(csfac(N,k)==0,k=k+1);k
\p 200; for(n=2,200, print(csmin(n!)))

A160456 Number of triangles that can be built from rods with lengths 1,2,...,n by using and concatenating not necessarily all rods.

Original entry on oeis.org

0, 3, 20, 70, 172, 366, 709, 1274, 2166, 3537, 5573, 8494, 12588, 18227, 25846, 35942, 49124, 66138, 87827, 115132, 149166, 191238, 242800, 305447, 381012, 471602, 579518, 707254, 857627, 1033812, 1239238, 1477589, 1752963

Offset: 3

Hagen von Eitzen, May 14 2009

a(n) is the number of triples (a,b,c) with b+c > a >= b >=c > 0 such that three disjoint subsets A,B,C of {1,2,...,n} with respective element sums a,b,c exist.

			For n = 4, there are 10 triangles with perimeter at most 1+2+3+4 = 10: (1,1,1), (2,2,1), (2,2,2), (3,2,2), (3,3,2), (3,3,3), (4,3,2), (4,3,3), (4,4,1) and (4,4,2). We have a(4)=3 because only 3 of these can be built from rods among 1,2,3,4: (4,3,2), (4,3,3)=(4,3,1+2) and (4,4,2)=(4,1+3,2). For example, it is not possible to build (4,4,1) because the 1-rod must be used for one of the 4-edges.

H. v. Eitzen, Table of n, a(n) for n=3..5262 (i.e. a(n) less than 2^64)
"AI", (Sci.math thread)
H. v. Eitzen, How to Build Triangles from Integers

A002623 is a similar problem where one rod per edge is to be used.
A160455 is a similar problem where all rods must be used.
A160438 is related to this if one drops the triangle inequality condition.

If n<=2, then trivially a(n)=0 because three edges need at least three rods.

If n>=8 then a(n) = A001400(n*(n+1)/2 - 3) - 11 - A133872(n+1).

A160508 Knuth's standard example of an unsorted array.

Original entry on oeis.org

503, 87, 512, 61, 908, 170, 897, 275, 653, 426, 154, 509, 612, 677, 765, 703

Offset: 1

Hagen von Eitzen, May 16 2009

The numbers were randomly generated by Knuth with a set of decimal dice and used throughout his book as sample input for investigated sorting algorithms.

			Apparently, the first roll of three (distinguishable) decimal dice resulted in the first die showing "5", the second showing "0", the third showing "3"; therefore a(1) = 503.

D. E. Knuth, The Art of Computer Programming, Vol. 3 / Sorting and Searching, Addison-Wesley, 1973, page 75 and remainder of chapter 5.

A163188 Numbers of the form n = rs = (r+s)t with gcd(r+s,t) = 1.

Original entry on oeis.org

4, 18, 48, 100, 150, 180, 294, 448, 490, 588, 648, 900, 960, 1134, 1210, 1584, 1620, 2028, 2100, 2178, 2548, 2904, 3150, 3388, 3630, 3718, 3840, 4624, 5040, 5070, 5508, 5850, 6084, 6468, 6498, 6760, 7098, 7600, 8670, 8820, 9900, 9984, 10164, 11638

Offset: 1

Hagen von Eitzen, Jul 22 2009

nonn

Also numbers of the form n = u*v*(u+v)^2 with gcd(u,v) = 1. The connection to the definition is given by r = u*(u+v), s = v*(u+v), t = u*v, resp. u = gcd(r,t), v = gcd(s,t).
Also "primitive" members of A139719: With k as in the definition of A139719, we additionally require that gcd(k+n/k, n/(k+n/k)) = 1.
100 has a non-primitive solution with k=10, resp. (r,s,t) = (10,10,5), resp. (u,v) = (5,5). It is included because there is also the primitive solution k=5, resp. (r,s,t) = (5,20,4), resp. (u,v) = (1,4).
8820 has two primitive solutions: k=21, resp. (r,s,t) = (21,420,20), resp. (u,v) = (1,20) and k=70, resp. (r,s,t) = (70,126,45), resp. (u,v) = (5,9).

			4 is in the sequence because 4 = 2*2 = (2+2)*1, gcd(2+2,1)=1.
18 is in the sequence because 18 = 3*6 = (3+6)*2, gcd(3+6,2)=1.
48 is in the sequence because 48 = 4*12 = (4+12)*3, gcd(4+12,3)=1.
16 = 4*4 = (4+4)*2 is not sufficient to make 16 a member of the sequence because gcd(4+4,2)=2.

Cf. A139719.

L=10000;v=[];for(r=1,L^(1/3),for(s=1,r,if(gcd(r,s)==1, n=r*s*(r+s)^2; if(n>L,break);if(n==8820,print([r,s]));v=concat(v,n))));vecsort(eval(Set(v)))

A165146 Erroneous duplicate version of A181496.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 136, 459, 1594, 5645, 20280, 73659

Offset: 0

Hagen von Eitzen, Sep 05 2009

dead

Previous name was: Number of polynomials P(x,y) of degree n with coefficients only 0 or 1 and such that P(x,1-x) = 1.

A160516 Inverse permutation to A075075.

Original entry on oeis.org

1, 2, 5, 3, 6, 4, 17, 8, 10, 7, 18, 9, 23, 20, 11, 13, 24, 15, 58, 12, 16, 19, 59, 14, 33, 22, 28, 21, 62, 26, 63, 31, 29, 25, 34, 36, 66, 57, 39, 32, 67, 35, 72, 30, 27, 60, 125, 37, 49, 44, 40, 38, 126, 47, 45, 42, 71, 61, 131, 56, 134, 64, 48, 52, 80, 46, 135, 41, 76, 43

Offset: 1

Hagen von Eitzen, May 16 2009

nonn

This is a permutation of the positive integers (provided A075075 really is a permutation).

			A075075(7) = 10, therefore a(10) = 7.
A075055(17) = 7, therefore a(7) = 17.

H. v. Eitzen, Table of n, a(n) for n = 1..50000

Cf. A185635 (fixed points).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a160516 = (+ 1) . fromJust . (`elemIndex` a075075_list)
-- Reinhard Zumkeller, Dec 19 2012

f[s_List] := Block[{m = Numerator[s[[ -1]]/s[[ -2]]]}, k = m; While[MemberQ[s, k], k += m]; Append[s, k]]; s = Nest[f, {1, 2}, 200]; Table[ Position[s, n, 1, 1], {n, 70}] // Flatten (* Robert G. Wilson v, May 20 2009 *)

A075075(a(n)) = n.

cp's OEIS Frontend

User: Hagen von Eitzen

A353010 a(n) = maximal d such that Product_{k=0..m} binomial(m,k) is divisible by m^(m+d), where m = A276710(n).

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A261811 Numbers m such that Sum_{k>=0} k^m/3^k is an integer.

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Mathematica

A243077 Numbers missing from A025429 (number of partitions of n into 5 nonzero squares).

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Extensions

A181496 Number of degree n polynomials p(x,y) with all coefficients 0 or 1 such that x+y=1 implies p(x,y)=1.

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PARI

Extensions

A160642 Minimal number k such that n! can be written as product of k (>= 2) consecutive integers.

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PARI

A160456 Number of triangles that can be built from rods with lengths 1,2,...,n by using and concatenating not necessarily all rods.

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Formula

A160508 Knuth's standard example of an unsorted array.

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A163188 Numbers of the form n = r*s = (r+s)*t with gcd(r+s,t) = 1.

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PARI

A165146 Erroneous duplicate version of A181496.

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A160516 Inverse permutation to A075075.

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A163188 Numbers of the form n = rs = (r+s)t with gcd(r+s,t) = 1.