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A090318
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a(n) = least positive k such that k, k+1, k+2, ..., k+n-1 is a stapled interval of length n, or 0 if no such sequence exists.
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7
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2184, 27829, 27828, 87890, 87890, 171054, 171054, 323510, 127374, 323510, 151062, 151062, 151062, 151061, 151060, 151059, 151058, 7106718, 7106718, 7567747, 7567746, 7567745, 7567744, 7567743, 7567742, 48595315, 48595314, 48595313
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OFFSET
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1,17
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COMMENTS
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A finite sequence of n consecutive positive integers is called "stapled" if each term in the sequence is not relatively prime to at least one other term in the sequence. Thus a staple joins two terms of the sequence whose gcd is > 1.
It has been proved that stapled intervals of length n >= 17 exist for all n.
An interval is stapled if for every term x there is another term y (different from x) such that gcd(x,y) > 1.
The shortest stapled interval has length 17 and starts with the number 2184.
It is interesting to notice that the intervals [27829,27846] and [27828,27846] are stapled while the interval [27828,27845] is not.
It is clear that a stapled interval [a,b] may not contain a prime number greater than b/2 (as such a prime would be coprime to every other element of the interval).
Together with Bertrand's Postulate this implies a > b/2 or b < 2a. And it follows that
* a stapled interval may not contain prime numbers at all;
* we can determine whether any particular positive integer a is a starting point of some stapled interval. (End)
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REFERENCES
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I. Gassko, Common factor graphs of stapled sequences, Proceedings of the Twenty-eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1997). Congr. Numer. 126 (1997), 163-173.
I. Gassko, Stapling and composite coverings of natural numbers, Proceedings of the Twenty-seventh Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 1996). Congr. Numer. 118 (1996), 109-116.
H. L. Nelson, There is a better sequence, Journal of Recreational Mathematics, Vol. 8(1), 1975, pp. 39-43.
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LINKS
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S. S. Pillai, On m Consecutive Integers I, Proc. Indian Acad. Sci., Sect. A, vol. 11, 1940, pp. 6-12; II ibid., vol. 11, 1940, pp. 73-80, III ibid, vol. 13, 1941, pp. 530-533; IV Bull. Calcutta Math. Soc. 36, 1944, pp. 99-101.
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EXAMPLE
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The shortest possible stapled sequence is [2184, 2185, 2186, 2187, 2188, 2189, 2190, 2191, 2192, 2193, 2194, 2195, 2196, 2197, 2198, 2199, 2200].
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MATHEMATICA
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dd = 41; nn = 10^7; Clear[sp, L]; sp[_] = 0; L[_] = 0; For[ i = 0, i < PrimePi[dd], ++i, p = Prime[i + 1]; For[ n = 0, n < nn + dd, n += p, If[sp[n] == 0, sp[n] = p]]]; Print["init done"]; For[ n = 1, n <= nn, ++n, m = 1; For[ d = 0, d < dd, ++d, s = sp[n + d]; If[s == 0, Break[]]; If[s > d, m = Max[m, d + s]]; If[d >= m && L[d] == 0, L[d] = n]] ]; Reap[For[ i = 1, i <= dd, ++i, Print["a[", i, "] = ", L[i - 1]]; Sow[L[i - 1]]]][[2, 1]] (* Jean-François Alcover, Mar 26 2013, translated and adapted from Max Alekseyev's program *)
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PROG
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(C++)
/* For the current parameters it needs ~ 4GB of RAM to run smoothly.
It first precomputes the smallest prime divisor < 59 of each number below 10^9 and stores them in the array sp. It then uses these divisors to grow up from each particular n < 10^9 a stapled interval of length at most 59. The records found are stored in the array L which is printed out at the end. It uses the following observations:
If a stapled interval contains a number t, then it also contains t-sp(t) or t+sp(t) or both.
If a stapled interval starts with n, n+1, ..., n+k then it must also contain the number m = max{ n + d + sp(n+d) : d=0..k, sp(n+d)>d }.
Moreover, if m <= n+k, then the interval [n, n+k] is stapled. */
#include <iostream>
#include <vector>
using namespace std;
#include <stdint.h>
#define D 59
#define N 1000000000ul
const uint32_t prime[16] =
{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 };
int main() {
vector<uint32_t> sp(N+D);
vector<uint32_t> L(D);
for(int i=0; i<16; ++i) {
uint32_t p = prime[i];
for(uint32_t n=0; n<N+D; n+=p) {
if( sp[n]==0 ) sp[n] = p;
}
}
clog << "Init done\n";
for(uint32_t n=1; n<=N; ++n) {
uint32_t m = 1;
for(int d=0; d<D; ++d) {
uint32_t s = sp[n+d];
if(s==0) break;
if(s>d) m = max(m, d+s);
if(d>=m && L[d]==0) L[d]=n;
}
}
for(int i=1; i<D; ++i) {
cout << i+1 << "\t" << L[i] << endl;
}
return 0;
}
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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