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%I #24 Nov 07 2022 21:47:13
%S 1,2,3,10,35,77,286,2431,4199,37145
%N The index of the first occurrence of A002110(n) in A356309.
%C A subsequence of A356314, and probably also of A356316 (from a(2)=3 onward of A356318 as well).
%C Also, from a(2)=3 onward conjectured to be the positions of records in A356302 (after its initial zero), while A358214 gives the conjectured record values.
%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A356309(a(n)) = a(n) + A358214(n).
%F a(n) = A002110(n) - A358214(n).
%t f[nn_] := Block[{m = 1, i = 1, n = nn, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; q = P = 1; Reap[Do[k = j; While[! CoprimeQ[j, f[k]], k++]; If[k == P, Sow[j]; P *= Prime[q]; q++], {j, 0, 2500}] ][[-1, -1]] (* _Michael De Vlieger_, Nov 06 2022 *)
%o (PARI)
%o \\ Very slow:
%o A002110(n) = prod(i=1,n,prime(i));
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A356309(n) = { my(j=n); while(gcd(A276086(j),n)!=1,j++); (j); };
%o A358213(n) = { my(x=A002110(n)); for(i=0,oo,if(A356309(i)==x,return(i))); };
%Y Cf. A002110, A356302, A356309, A356314, A356316, A356318, A358214.
%K nonn,hard,more
%O 0,2
%A _Antti Karttunen_, Nov 05 2022
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