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 A204892 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=prime(k). 249
 2, 3, 3, 4, 4, 5, 7, 5, 5, 6, 6, 7, 10, 7, 7, 8, 8, 9, 13, 9, 9, 10, 16, 10, 16, 10, 10, 11, 11, 12, 19, 12, 20, 12, 12, 13, 22, 13, 13, 14, 14, 15, 24, 15, 15, 16, 25, 16, 26, 16, 16, 17, 29, 17, 30, 17, 17, 18, 18, 19, 31, 19, 32, 19, 19, 20, 33, 20, 20, 21 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Suppose that (s(i)) is a strictly increasing sequence in the set N of positive integers. For i in N, let r(h) be the residue of s(i+h)-s(i) mod n, for h=1,2,...,n+1. There are at most n distinct residues r(h), so that there must exist numbers h and h' such that r(h)=r(h'), where 0<=h=5 ... k(n), j(n): A204908, A204909 ... s(k(n)),s(j(n)): A204910, A204911 ... s(k(n))-s(j(n)): A109043(?), A000034(?) s(n)=prime(n)*prime(n+1) product of consecutive primes ... k(n), j(n): A205146, A205147 ... s(k(n)),s(j(n)): A205148, A205149 ... s(k(n))-s(j(n)): A205150, A205151 s(n)=(prime(n+1)+prime(n+2)/2: averages of odd primes ... k(n), j(n): A205153, A205154 ... s(k(n)),s(j(n)): A205372, A205373 ... s(k(n))-s(j(n)): A205374, A205375 s(n)=2^(n-1), powers of 2 ... k(n), j(n): A204979, A001511(?) ... s(k(n)),s(j(n)): A204981, A006519(?) ... s(k(n))-s(j(n)): A204983(?), A204984 s(n)=2^n, powers of 2 ... k(n), j(n): A204987, A204988 ... s(k(n)),s(j(n)): A204989, A140670(?) ... s(k(n))-s(j(n)): A204991, A204992 s(n)=C(n+1,2), triangular numbers ... k(n), j(n): A205002, A205003 ... s(k(n)),s(j(n)): A205004, A205005 ... s(k(n))-s(j(n)): A205006, A205007 s(n)=n^2, squares ... k(n), j(n): A204905, A204995 ... s(k(n)),s(j(n)): A204996, A204997 ... s(k(n))-s(j(n)): A204998, A204999 s(n)=(2n-1)^2, odd squares ... k(n), j(n): A205378, A205379 ... s(k(n)),s(j(n)): A205380, A205381 ... s(k(n))-s(j(n)): A205382, A205383 s(n)=n(3n-1), pentagonal numbers ... k(n), j(n): A205138, A205139 ... s(k(n)),s(j(n)): A205140, A205141 ... s(k(n))-s(j(n)): A205142, A205143 s(n)=n(2n-1), hexagonal numbers ... k(n), j(n): A205130, A205131 ... s(k(n)),s(j(n)): A205132, A205133 ... s(k(n))-s(j(n)): A205134, A205135 s(n)=C(2n-2,n-1), central binomial coefficients ... k(n), j(n): A205010, A205011 ... s(k(n)),s(j(n)): A205012, A205013 ... s(k(n))-s(j(n)): A205014, A205015 s(n)=(1/2)C(2n,n), (1/2)*(central binomial coefficients) ... k(n), j(n): A205386, A205387 ... s(k(n)),s(j(n)): A205388, A205389 ... s(k(n))-s(j(n)): A205390, A205391 s(n)=n(n+1), oblong numbers ... k(n), j(n): A205018, A205028 ... s(k(n)),s(j(n)): A205029, A205030 ... s(k(n))-s(j(n)): A205031, A205032 s(n)=n!, factorials ... k(n), j(n): A204932, A204933 ... s(k(n)),s(j(n)): A204934, A204935 ... s(k(n))-s(j(n)): A204936, A204937 s(n)=n!!, double factorials ... k(n), j(n): A204982, A205100 ... s(k(n)),s(j(n)): A205101, A205102 ... s(k(n))-s(j(n)): A205103, A205104 s(n)=3^n-2^n ... k(n), j(n): A205000, A205107 ... s(k(n)),s(j(n)): A205108, A205109 ... s(k(n))-s(j(n)): A205110, A205111 s(n)=Fibonacci(n+1) ... k(n), j(n): A204924, A204925 ... s(k(n)),s(j(n)): A204926, A204927 ... s(k(n))-s(j(n)): A204928, A204929 s(n)=Fibonacci(2n-1) ... k(n), j(n): A205442, A205443 ... s(k(n)),s(j(n)): A205444, A205445 ... s(k(n))-s(j(n)): A205446, A205447 s(n)=Fibonacci(2n) ... k(n), j(n): A205450, A205451 ... s(k(n)),s(j(n)): A205452, A205453 ... s(k(n))-s(j(n)): A205454, A205455 s(n)=Lucas(n) ... k(n), j(n): A205114, A205115 ... s(k(n)),s(j(n)): A205116, A205117 ... s(k(n))-s(j(n)): A205118, A205119 s(n)=n*(2^(n-1)) ... k(n), j(n): A205122, A205123 ... s(k(n)),s(j(n)): A205124, A205125 ... s(k(n))-s(j(n)): A205126, A205127 s(n)=ceiling[n^2/2] ... k(n), j(n): A205394, A205395 ... s(k(n)),s(j(n)): A205396, A205397 ... s(k(n))-s(j(n)): A205398, A205399 s(n)=floor[(n+1)^2/2] ... k(n), j(n): A205402, A205403 ... s(k(n)),s(j(n)): A205404, A205405 ... s(k(n))-s(j(n)): A205406, A205407 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 EXAMPLE Let s(k)=prime(k). As in A204890, the ordering of differences s(k)-s(j), follows from the arrangement shown here: k...........1..2..3..4..5...6...7...8...9 s(k)........2..3..5..7..11..13..17..19..23 ... s(k)-s(1)......1..3..5..9..11..15..17..21..27 s(k)-s(2).........2..4..8..10..14..16..20..26 s(k)-s(3)............2..6..8...12..14..18..24 s(k)-s(4)...............4..6...10..12..16..22 ... least (k,j) such that 1 divides s(k)-s(j) for some j is (2,1), so a(1)=2. least (k,j) s.t. 2 divides s(k)-s(j): (3,2), so a(2)=3. least (k,j) s.t. 3 divides s(k)-s(j): (3,1), so a(3)=3. MATHEMATICA s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 50; Table[s[n], {n, 1, 30}] (* A000040 *) u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]] Table[u[m], {m, 1, z1}] (* A204890 *) v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0] w[n_] := w[n] = Table[v[n, h], {h, 1, z1}] d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]] Table[d[n], {n, 1, z2}] (* A204891 *) k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2] m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2] j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2 Table[k[n], {n, 1, z2}] (* A204892 *) Table[j[n], {n, 1, z2}] (* A204893 *) Table[s[k[n]], {n, 1, z2}] (* A204894 *) Table[s[j[n]], {n, 1, z2}] (* A204895 *) Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A204896 *) Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204897 *) (* Program 2: generates A204892 and A204893 rapidly *) s = Array[Prime[#] &, 120]; lk = Table[NestWhile[# + 1 &, 1, Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1, Length[s]}] Table[NestWhile[# + 1 &, 1, Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &], {j, 1, Length[lk]}] (* Peter J. C. Moses, Jan 27 2012 *) PROG (PARI) a(n)=forprime(p=n+2, , forstep(k=p%n, p-1, n, if(isprime(k), return(primepi(p))))) \\ Charles R Greathouse IV, Mar 20 2013 CROSSREFS Cf. A000040, A204890. Sequence in context: A029086 A070046 A130120 * A164512 A127434 A266475 Adjacent sequences: A204889 A204890 A204891 * A204893 A204894 A204895 KEYWORD nonn AUTHOR Clark Kimberling, Jan 20 2012 STATUS approved

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Last modified August 12 04:50 EDT 2024. Contains 375085 sequences. (Running on oeis4.)