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%I #21 Oct 15 2020 19:26:45
%S 1,3,2,3,10,3,13,4,5,10,22,3,26,13,10,4,34,5,37,10,13,22,46,4,15,26,6,
%T 13,58,10,61,5,22,34,13,5,73,37,26,10,82,13,86,22,10,46,94,4,14,15,34,
%U 26,106,6,22,13,37,58,118,10,122,61,13,6,26,22,134,34,46,13,142,5,146,73,15,37,22,26,157
%N a(n) is the smallest k such that n divides psi(k!) (k > 0).
%C From _Robert Israel_, Sep 14 2017: (Start)
%C If m and n are coprime then a(m*n) = max(a(m),a(n)).
%C a(n) <= 2n.
%C Suppose p is a prime >= 5. Then
%C a(p) = 2p-1 if p is in A005382, otherwise 2p.
%C a(p^2) = 2p if p is in A005382, otherwise 3p.
%C a(p^3) = 3p if p is in A005382, 4p-1 if p is in A062737, otherwise 4p.
%C (End)
%H Robert Israel, <a href="/A292024/b292024.txt">Table of n, a(n) for n = 1..10000</a>
%e a(4) = 3 because 4 divides psi(3!) = 12 and 3 is the least number with this property.
%p A:= proc(n) option remember;
%p local F, p, e, t, k;
%p F:= ifactors(n)[2];
%p if nops(F)=1 then
%p p:= F[1][1];
%p e:= F[1][2];
%p if p = 3 then
%p t:= 1; if e =1 then return 2 fi
%p else t:= 0:
%p fi;
%p for k from 2*p by p do
%p if isprime(k-1) then
%p t:= t+padic:-ordp(k, p);
%p if t >= e then return(k-1) fi;
%p fi;
%p t:= t + padic:-ordp(k, p);
%p if t >= e then return k fi;
%p od
%p else
%p max(seq(procname(t[1]^t[2]), t=F))
%p fi
%p end proc:
%p A(1):= 1:
%p map(A, [$1..100]); # _Robert Israel_, Sep 14 2017
%t psi[n_] := Module[{p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, FactorInteger[n]}]];
%t a[n_] := Module[{k = 1}, While[!Divisible[psi[k!], n], k++]; k]; a[2] = 3;
%t Array[a, 100] (* _Jean-François Alcover_, Oct 15 2020, after PARI *)
%o (PARI) a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
%o a(n) = {my(k=1); while(a001615(k!) % n, k++); k; } \\ after _Charles R Greathouse IV_ at A001615
%Y Cf. A001615, A005382, A062737, A275985.
%K nonn,look
%O 1,2
%A _Altug Alkan_, Sep 07 2017
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