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

2, 3, 4, 5, 10, 11, 23, 29, 41, 53, 57, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 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

Offset

1,1

Comments

Numbers k for which A246277(2k+1) = A246277(k). This in turn implies a looser condition A046523(2k+1) = A046523(k).

Conjecture: Apart from 4 all other terms in this sequence are squarefree. No counterexamples <= 2^22.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

Example

4 is present, because by applying prime shift once to it, we get A003961(4) = 9 = 2*4 + 1.
5 is present, because by applying prime shift twice to it, we get 5 -> 7 -> 11 = 2*5 + 1.

Maple

filter:= proc(n) local F, F1, F2, G, G1, G2;
  F:= sort(ifactors(n)[2], (a, b) -> a[1]<b[1]);
  G:= sort(ifactors(2*n+1)[2], (a, b) -> a[1]<b[1]);
  if nops(F) <> nops(G) then return false fi;
  F2:= map(t -> t[2], F);
  G2:= map(t -> t[2], G);
  if F2 <> G2 then return false fi;
  if nops(F) = 1 then return true fi;
  F1:= map(t -> numtheory:-pi(t[1]), F);
  G1:= map(t -> numtheory:-pi(t[1]), G);
  nops(convert(G1-F1, set))=1;
end proc:
select(filter, [$2..10000]); # Robert Israel, Feb 18 2022

Mathematica

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 *)

Prog

(PARI)
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);
isA348511(n) = (A246277(n)==A246277(1+n+n));

Crossrefs

Cf. A003961, A046523, A246277.

Subsequences: A005384 (primes present), A348514 (terms requiring only one iteration to reach 2k+1).

Cf. also A285706.

Sequence in context: A348255 A117360 A074821 * A338897 A296160 A076707

Adjacent sequences: A348508 A348509 A348510 * A348512 A348513 A348514

Keyword

nonn

Author

Antti Karttunen, Oct 30 2021

Status

approved