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A348511 Numbers k for which 2k+1 can be obtained with successive prime shifts towards larger primes (by iterating A003961, starting from k).

%I A348511 #20 Feb 18 2022 08:06:22
%S A348511 2,3,4,5,10,11,23,29,41,53,57,83,89,113,131,173,179,191,233,239,251,
%T A348511 281,293,359,419,431,443,491,509,593,641,653,659,683,719,743,761,809,
%U A348511 911,953,1013,1019,1031,1049,1054,1103,1223,1229,1289,1409,1439,1451,1481,1499,1511,1559,1583,1601,1733,1811,1889,1901,1931
%N A348511 Numbers k for which 2k+1 can be obtained with successive prime shifts towards larger primes (by iterating A003961, starting from k).
%C A348511 Numbers k for which A246277(2k+1) = A246277(k). This in turn implies a looser condition A046523(2k+1) = A046523(k).
%C A348511 Conjecture: Apart from 4 all other terms in this sequence are squarefree. No counterexamples <= 2^22.
%H A348511 Robert Israel, <a href="/A348511/b348511.txt">Table of n, a(n) for n = 1..10000</a>
%H A348511 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A348511 4 is present, because by applying prime shift once to it, we get A003961(4) = 9 = 2*4 + 1.
%e A348511 5 is present, because by applying prime shift twice to it, we get 5 -> 7 -> 11 = 2*5 + 1.
%p A348511 filter:= proc(n) local F,F1,F2,G,G1,G2;
%p A348511   F:= sort(ifactors(n)[2],(a,b) -> a[1]<b[1]);
%p A348511   G:= sort(ifactors(2*n+1)[2],(a,b) -> a[1]<b[1]);
%p A348511   if nops(F) <> nops(G) then return false fi;
%p A348511   F2:= map(t -> t[2],F);
%p A348511   G2:= map(t -> t[2],G);
%p A348511   if F2 <> G2 then return false fi;
%p A348511   if nops(F) = 1 then return true fi;
%p A348511   F1:= map(t -> numtheory:-pi(t[1]),F);
%p A348511   G1:= map(t -> numtheory:-pi(t[1]),G);
%p A348511   nops(convert(G1-F1,set))=1;
%p A348511 end proc:
%p A348511 select(filter, [$2..10000]);# _Robert Israel_, Feb 18 2022
%t A348511 f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := NestWhileList[s, n, # < 2*n + 1 &][[-1]] == 2*n + 1; Select[Range[2, 2000], q] (* _Amiram Eldar_, Oct 30 2021 *)
%o A348511 (PARI)
%o A348511 A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
%o A348511 isA348511(n) = (A246277(n)==A246277(1+n+n));
%Y A348511 Cf. A003961, A046523, A246277.
%Y A348511 Subsequences: A005384 (primes present), A348514 (terms requiring only one iteration to reach 2k+1).
%Y A348511 Cf. also A285706.
%K A348511 nonn
%O A348511 1,1
%A A348511 _Antti Karttunen_, Oct 30 2021