A387414 -id:A387414 - 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

A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 3, 4, 7, 10, 15, 18, 23, 27, 31, 47, 57, 63, 95, 119, 127, 255, 348, 383, 415, 447, 511, 575, 695, 767, 959, 1023, 1054, 1071, 1535, 1919, 2047, 2626, 3471, 3839, 4095, 4415, 6815, 8191, 8703, 13823, 16383, 31743, 32767, 39895, 42367, 48127, 64607, 65535, 68727, 74495, 81919, 92159, 98303, 113535, 124671, 131071

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to select for numbers with a long tail of trailing 1-bits. Terms that are not in A004767 are: 1, 4, 10, 18, 57, 348, 1054, 2626, 675348, 1869741, 12371554, 14070141, 1158654378, 1673018314, etc.

Antti Karttunen, Table of n, a(n) for n = 1..159
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A004767.
Subsequences: A000225, A348514 (which is also a subsequence of A387414).
For similar sequences, see A336700, A387410, A387415, A387410, A387418, A387419.

a3961[x_] := Apply[Times, Prime[PrimePi[#1] + 1]^#2 & @@@ FactorInteger[x]] - Boole[x == 1];
a265[x_] := x/2^IntegerExponent[x, 2];
Select[Range[2^17], Divisible[1 + a3961[#], a265[# + 1] ] &] (* Michael De Vlieger, Sep 01 2025 *)

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA387411(n) = !((1+A003961(n))%A000265(1+n));

A387413 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 2, 3, 3, 0, 3, 3, 3, 3, 3, 3, 2, 0, 3, 4, 4, 3, 4, 4, 1, 2, 3, 3, 2, 3, 4, 5, 1, 5, 3, 5, 4, 3, 4, 3, 2, 2, 3, 5, 3, 3, 5, 2, 4, 5, 5, 5, 4, 5, 5, 5, 0, 5, 3, 5, 5, 4, 5, 5, 6, 6, 3, 5, 4, 6, 4, 5, 3, 6, 6, 6, 4, 6, 5, 5, 5, 5, 4, 6, 5, 5, 5, 4, 6, 5, 3, 5, 1, 5, 4, 6, 3, 6, 6, 4, 2, 6, 4, 5, 4

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000523, A003961, A387412, A387414 (positions of 0's).

Cf. also A387423.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387413(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };

a(n) = (1+A000523(n)) - A387412(n).

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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A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A387413 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

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