A348511 -id:A348511 - cp's OEIS frontend

A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

4, 10, 57, 1054, 2626, 68727, 12371554, 1673018314, 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878

Offset: 1

Antti Karttunen, Oct 29 2021

Numbers k such that A064216(1+k) = k.
It seems that after 4, all other terms are squarefree. See conjecture in A348511.
a(9)..a(13) <= 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878, which are also terms. - David A. Corneth, Oct 30 2021

Index entries for sequences computed from indices in prime factorization

Fixed points of map A108228. (Compare to A048674).
Positions of ones in A252748.
Subsequence of the following sequences: A246282, A319630, A348511, A378980 (see also A379216), A387411, A387414.
Cf. A000230, A003961, A064216, A326134.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], s[#] == 2*# + 1 &] (* Amiram Eldar, Oct 30 2021 *)

a(9)-a(11) verified by Amiram Eldar, Nov 01 2021

a(12)-a(13) verified by Martin Ehrenstein, Nov 08 2021

A349622 Numbers k for which 2k-1 can be obtained with successive prime shifts towards larger primes (by iterating A003961, starting from k).

Original entry on oeis.org

1, 2, 3, 7, 19, 25, 26, 31, 33, 37, 79, 93, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 841, 877, 937, 967, 979, 997, 1009, 1034, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011, 2029, 2089, 2131, 2137

Offset: 1

Antti Karttunen, Nov 26 2021

nonn

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

Nonsquarefree terms are rare: 25, 841 (= 29^2), 970225 ( = 5^2 * 197^2), ..., also 414690595, which is not a square. Some of these are also terms of A048674. Compare to A348511.

Index entries for sequences computed from indices in prime factorization

Subsequences: A005382 (primes present), A048674 (terms requiring only one iteration to reach 2k-1).
Cf. A003961, A046523, A246277.
Cf. also 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);
isA349622(n) = (A246277(n)==A246277(n+n-1));

cp's OEIS Frontend

A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

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