A387411 -id:A387411 - 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

A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 639, 1023, 2047, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 126975, 131071, 204799, 229375, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A004767, A048250.

For similar sequences, see A336700, A387411, A387415, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
isA387410(n) = !((1+A000265(A048250(n)))%A000265(1+n));

A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 3, 4, 7, 15, 31, 40, 63, 127, 255, 511, 639, 1023, 2047, 2175, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. Terms 1, 4 and 40 are probably the only terms that are not in A004767.

Cf. A000225 (subsequence), A000265, A003959, A004767.

For similar sequences, see A336700, A387410, A387411, A387415, A387418.

A000265(n) = (n>>valuation(n,2));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
isA387419(n) = !((1+A000265(A003959(n)))%A000265(1+n));

A387414 Numbers k such that the binary expansion of k is a prefix of the binary expansion of A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 4, 10, 18, 57, 348, 1054, 2626, 60625, 68727, 129260, 192276, 675348, 960320, 5368464, 12371554, 30078308, 356311953, 1158654378, 1673018314

Offset: 1

Antti Karttunen, Sep 01 2025

Numbers k such that A003961(k) = 2^e * k + r, for some k >= 1, e >= 0, 0 <= r < 2^e.

			A007088(4) = 100, and A007088(A003961(4)) = A007088(9) = 1001 begins with the same binary string, therefore 4 is included.
A007088(18) = 10010, and A007088(A003961(18)) = A007088(75) = 1001011 begins with the same binary string, therefore 18 is included as a term. Also, 75 = 2^2 * 18 + 3.

Positions of 0's in A387413.
Cf. A000523, A003961, A007088, A387412.
Subsequences: A348514 (which is also a subsequence of A387411).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A387414(n) = { my(s=A003961(n)); while(s>n, s >>= 1); (s==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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A387414 Numbers k such that the binary expansion of k is a prefix of the binary expansion of A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI