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A047980 a(n) is smallest difference d of an arithmetic progression dk+1 whose first prime occurs at the n-th position. 3

%I #29 Mar 19 2019 00:14:41

%S 1,3,24,7,38,17,184,71,368,19,668,59,634,167,512,757,1028,197,1468,

%T 159,3382,799,4106,227,10012,317,7628,415,11282,361,38032,521,53630,

%U 3289,37274,2633,63334,1637,34108,1861,102296,1691,119074,1997,109474,2053

%N a(n) is smallest difference d of an arithmetic progression dk+1 whose first prime occurs at the n-th position.

%C Definition involves two minimal conditions: (1) the first prime (as in A034693) and (2) dk+1 sequences were searched with minimal d. Present terms are the first ones in sequences analogous to A034780, A034782-A034784, A006093 (called there K(n,m)).

%C Index of the first occurrence of n in A034693. - _Amarnath Murthy_, May 08 2003

%H Jon E. Schoenfield, <a href="/A047980/b047980.txt">Table of n, a(n) for n = 1..150</a> (terms 1..72 from Robert Israel)

%H Jon E. Schoenfield, <a href="/A047980/a047980.txt">Terms <= 5*10^8: Table of n, a(n) for n = 1..406, with -1 for each term > 5*10^8</a>

%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>

%F a(n) = min{k | A034693(k) = n}.

%e For n=2, the sequence with d=1 is 2,3,4,5,... with the prime 2 for k=1. The sequence with d=2 is 3,5,7,9,... with the prime 3 for k=1. The sequence with d=3 is 4,7,10,13,... with the prime 7 for k=2. So a(n)=3. - _Michael B. Porter_, Mar 18 2019

%p N:= 40: # to get a(n) for n <= N

%p count:= 0:

%p p:= 0:

%p Ds:= {1}:

%p while count < N do

%p p:= nextprime(p);

%p ds:= select(d -> (p-1)/d <= N, numtheory:-divisors(p-1) minus Ds);

%p for d in ds do

%p n:= (p-1)/d;

%p if not assigned(A[n]) then

%p A[n]:= d;

%p count:= count+1;

%p fi

%p od:

%p Ds:= Ds union ds;

%p od:

%p seq(A[i],i=1..N); # _Robert Israel_, Jan 25 2016

%t With[{s = Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 10^6}]}, TakeWhile[#, # > 0 &] &@ Flatten@ Array[FirstPosition[s, #] /. k_ /; MissingQ@ k -> {0} &, Max@ s]] (* _Michael De Vlieger_, Aug 01 2017 *)

%o (MATLAB)

%o function [ A ] = A047980( P, N )

%o % Get values a(i) for i <= N with a(i) <= P/i

%o % using primes <= P.

%o % Returned entries A(n) = 0 correspond to unknown a(n) > P/n

%o Primes = primes(P);

%o A = zeros(1,N);

%o Ds = zeros(1,P);

%o for p = Primes

%o ns = [1:N];

%o ns = ns(mod((p-1) * ones(1,N), ns) == 0);

%o newds = (p-1) ./ns;

%o ns = ns(A(ns) == 0);

%o ds = (p-1) ./ ns;

%o q = (Ds(ds) == 0);

%o A(ns(q)) = ds(q);

%o Ds(newds) = 1;

%o end

%o end % _Robert Israel_, Jan 25 2016

%Y Cf. A034693, A034694, A034780, A034782, A034783, A034784, A006093, A047981, A047982.

%K nonn

%O 1,2

%A _Labos Elemer_

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)