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Search: id:A087985
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| A087985 |
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a(n) = prime(x) is the smallest prime such that 1+n*prime(x) is divisible by prime(x+1), or 0 if no such prime exists. |
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3
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| 2, 0, 3, 2, 23, 19, 2, 3, 47, 2, 5, 43, 2, 7, 83, 2, 0, 3, 2, 11, 79, 2, 3, 0, 2, 0, 103, 2, 17, 13, 2, 5, 3, 2, 0, 7, 2, 3, 5, 2, 0, 163, 2, 257, 263, 2, 7, 3, 2, 0, 0, 2, 3, 0, 2, 61, 223, 2, 11, 5, 2, 47, 3, 2, 41, 73, 2, 3, 7, 2, 317, 11, 2, 5, 19, 2, 0, 3, 2, 7, 5, 2, 3, 829, 2, 17, 0, 2, 67, 0
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Two unsolved problems and supported conjectures: [1] Given arbitrary n, then 1+np[x]=mp[x+1] is solvable at all or not with some m ? For several n (like n=17,24,26,35,41,50,51,54,87, etc) search up to x=10000000 did not provide p[x] solution. [2] If solution exists, then the number of solution is believed to be finite. E.g.: at n=2122, 8 solutions were found after extensive search {p[x]}={2,19,23,37,89,433,4241,7621}
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FORMULA
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a(n)=Min{p[x]; Mod[1+np[x], p[x+1]]=0}; a(n)=p; 1+np=mq where {p, q} are smallest consecutive primes satisfying relation belonging to n.
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EXAMPLE
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n=1: 1+1.p=q is satisfied by {2,3} consecutive prime pair; a[1]=2.
n=2: 1+2p=mq, no solution.
n=6: 1+6p=m.q is satisfied first by {p,q}={19,23} because 6.19+1=115=5.23
It is provable that there are arbitrarily large n to which (a finite number of) solutions exist.
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MATHEMATICA
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{k=0, nu=0; sq={}}; Table[Print[{n-1, Min[Prime[sq]]}]; nu=0; sq={}; Do[s=Mod[n*Prime[x]+1, Prime[x+1]]; If[Equal[s, 0], nu=nu+1; sq=Append[sq, n]], {x, 1, 10000000}], {n, 1, 257}]
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PROGRAM
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(PARI) a(n) = local(i, c, p, q); i = 1; c = 0; q = 2; while (1, p = q; q = prime(i + 1); if (!((1 + n*p)%q), return(p)); if ((i + n*p)/q > n - 1/10, c++; if (c == 8, return(0)), c = 0); i++); (Wasserman)
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CROSSREFS
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Cf. A087986.
Sequence in context: A135351 A079451 A049799 this_sequence A004557 A088276 A099838
Adjacent sequences: A087982 A087983 A087984 this_sequence A087986 A087987 A087988
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KEYWORD
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nonn
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Oct 06 2003
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EXTENSIONS
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More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jun 17 2005
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