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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.

%I A047980 #29 Mar 19 2019 00:14:41
%S A047980 1,3,24,7,38,17,184,71,368,19,668,59,634,167,512,757,1028,197,1468,
%T A047980 159,3382,799,4106,227,10012,317,7628,415,11282,361,38032,521,53630,
%U A047980 3289,37274,2633,63334,1637,34108,1861,102296,1691,119074,1997,109474,2053
%N A047980 a(n) is smallest difference d of an arithmetic progression dk+1 whose first prime occurs at the n-th position.
%C A047980 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 A047980 Index of the first occurrence of n in A034693. - _Amarnath Murthy_, May 08 2003
%H A047980 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 A047980 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 A047980 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F A047980 a(n) = min{k | A034693(k) = n}.
%e A047980 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 A047980 N:= 40: # to get a(n) for n <= N
%p A047980 count:= 0:
%p A047980 p:= 0:
%p A047980 Ds:= {1}:
%p A047980 while count < N do
%p A047980     p:= nextprime(p);
%p A047980     ds:= select(d -> (p-1)/d <= N, numtheory:-divisors(p-1) minus Ds);
%p A047980     for d in ds do
%p A047980       n:= (p-1)/d;
%p A047980       if not assigned(A[n]) then
%p A047980         A[n]:= d;
%p A047980         count:= count+1;
%p A047980       fi
%p A047980     od:
%p A047980     Ds:= Ds union ds;
%p A047980 od:
%p A047980 seq(A[i],i=1..N); # _Robert Israel_, Jan 25 2016
%t A047980 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 A047980 (MATLAB)
%o A047980 function [ A ] = A047980( P, N )
%o A047980 %   Get values a(i) for i <= N with a(i) <= P/i
%o A047980 %   using primes <= P.
%o A047980 %   Returned entries A(n) = 0 correspond to unknown a(n) > P/n
%o A047980 Primes = primes(P);
%o A047980 A = zeros(1,N);
%o A047980 Ds = zeros(1,P);
%o A047980 for p = Primes
%o A047980    ns = [1:N];
%o A047980    ns = ns(mod((p-1) * ones(1,N), ns) == 0);
%o A047980    newds = (p-1) ./ns;
%o A047980    ns = ns(A(ns) == 0);
%o A047980    ds = (p-1) ./ ns;
%o A047980    q = (Ds(ds) == 0);
%o A047980    A(ns(q)) = ds(q);
%o A047980    Ds(newds) = 1;
%o A047980 end
%o A047980 end % _Robert Israel_, Jan 25 2016
%Y A047980 Cf. A034693, A034694, A034780, A034782, A034783, A034784, A006093, A047981, A047982.
%K A047980 nonn
%O A047980 1,2
%A A047980 _Labos Elemer_