A364561 -id:A364561 - cp's OEIS frontend

A005941 Inverse of the Doudna sequence A005940.

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 17, 12, 33, 18, 11, 16, 65, 14, 129, 20, 19, 34, 257, 24, 13, 66, 15, 36, 513, 22, 1025, 32, 35, 130, 21, 28, 2049, 258, 67, 40, 4097, 38, 8193, 68, 23, 514, 16385, 48, 25, 26, 131, 132, 32769, 30, 37, 72, 259, 1026, 65537, 44, 131073, 2050, 39, 64

Offset: 1

N. J. A. Sloane

nonn

a(2^k) = 2^k. - Robert G. Wilson v, Feb 22 2005
Fixed points: A029747. - Reinhard Zumkeller, Aug 23 2006
Question: Is there a simple proof that a(c) = c would never allow an odd composite c as a solution? See also A364551. - Antti Karttunen, Jul 30 2023

J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A103969. Inverse of A005940. One more than A156552.
Cf. A364559 [= a(n)-n], A364557 (Möbius transform), A364558.
Cf. A029747 [known positions where a(n) = n], A364560 [where a(n) <= n], A364561 [where a(n) <= n and n is odd], A364562 [where a(n) > n], A364548 [where n divides a(n)], A364549 [where odd n divides a(n)], A364550 [where a(n) divides n], A364551 [where a(n) divides n and n is odd].

A005941 := proc(n)
    local k ;
    for k from 1 do
    if A005940(k) = n then # code reuse
        return k;
    end if;
    end do ;
end proc: # R. J. Mathar, Mar 06 2010

f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; t = Table[ f[n], {n, 10^5}]; Flatten[ Table[ Position[t, n, 1, 1], {n, 64}]] (* Robert G. Wilson v, Feb 22 2005 *)

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552) - Antti Karttunen, Jul 30 2023

from sympy import primepi, factorint
def A005941(n): return sum((1<Chai Wah Wu, Mar 11 2023

(define (A005941 n) (+ 1 (A156552 n))) ;; Antti Karttunen, Jun 26 2014

a(n) = h(g(n,1,1), 0) / 2 + 1 with h(n, m) = if n=0 then m else h(floor(n/2), 2*m + n mod 2) and g(n, i, x) = if n=1 then x else (if n mod prime(i) = 0 then g(n/prime(i), i, 2*x+1) else g(n, i+1, 2*x)). - Reinhard Zumkeller, Aug 23 2006

a(n) = 1 + A156552(n). - Antti Karttunen, Jun 26 2014

More terms from Robert G. Wilson v, Feb 22 2005

a(61) inserted by R. J. Mathar, Mar 06 2010

A364551 Odd numbers k such that k is a multiple of A005941(k).

Original entry on oeis.org

1, 3, 5, 3125, 7875, 12005, 13365, 22869, 23595, 46475, 703395, 985439, 2084775, 2675673, 13619125, 19144125

Offset: 1

Antti Karttunen, Jul 28 2023

Odd numbers k such that k is a multiple of 1+A156552(k).
Sequence A005940(A364545(n)) sorted into ascending order.
This is a subsequence of A364561, so the comments given in A364564 apply also here (see also the example section).

			In all these cases, the right hand side is a divisor of the left hand side:
      Term   (and its factorization)             A005941(term)
         1   (unity)                         ->    1
         3   (prime)                         ->    3
         5   (prime)                         ->    5
      3125 = 5^5                             ->    125 = 5^3
      7875 = 3^2 * 5^3 * 7                   ->    375 = 3 * 5^3
     12005 = 5 * 7^4                         ->    245 = 5 * 7^2
     13365 = 3^5 * 5 * 11                    ->    1215 = 3^5 * 5
     22869 = 3^3 * 7 * 11^2                  ->    847 = 7 * 11^2
     23595 = 3 * 5 * 11^2 * 13               ->    715 = 5 * 11 * 13
     46475 = 5^2 * 11 * 13^2                 ->    845 = 5 * 13^2
    703395 = 3^2 * 5 * 7^2 * 11 * 29         ->    33495 = 3 * 5 * 7 * 11 * 29
    985439 = 7^3 * 13^2 * 17                 ->    2873 = 13^2 * 17
   2084775 = 3 * 5^2 * 7 * 11 * 19^2         ->    12635 = 5 * 7 * 19^2
   2675673 = 3^5 * 7 * 11^2 * 13             ->    11583 = 3^4 * 11 * 13
  13619125 = 5^3 * 13 * 17^2 * 29            ->    36125 = 5^3 * 17^2
  19144125 = 3^2 * 5^3 * 7 * 11 * 13 * 17    ->    21879 = 3^2 * 11 * 13 * 17.

Cf. A005940, A005941, A156552, A364545, A364549, A364564.

Subsequence of A364561, which is a subsequence of A364560.

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
isA364551(n) = ((n%2)&&!(n%A005941(n)));

A364560 Numbers k for which A156552(k) < k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 55, 60, 63, 64, 70, 72, 75, 77, 80, 81, 84, 90, 91, 96, 98, 99, 100, 105, 108, 110, 120, 121, 125, 126, 128, 135, 140, 143, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 180, 182, 187, 189, 192, 195, 196

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Numbers k such that A005941(k) <= k.
Sequence A005940(A364542(.)) sorted into ascending order.
If k is a term, then also 2*k is present in this sequence, and vice versa.

Positions of nonpositive terms in A364559.
Cf. A005941, A156552, A364542, A364562 (complement).
Subsequences: A029747, A364550, A364561 (odd terms).

A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
isA364560(n) = (A156552(n) < n);

A364576 Starting from k=1, each subsequent term is the next larger odd k such that A156552(k) < k and the ratio A156552(k)/k is nearer to 1.0 than for any previous k in the sequence.

Original entry on oeis.org

1, 3, 5, 21, 323, 66297, 139965, 263375, 264845, 528581

Offset: 1

Antti Karttunen, Aug 06 2023

All the odd fixed points of map n -> A005940(n) [and its inverse, map n -> A005941(n)] are included in this sequence. This includes both the known odd fixed points, 1, 3 and 5 (see A029747), and any additional hypothetical odd composites that would satisfy the condition n == A005940(n).

This is a subsequence of A364561, so the comments given in A364564 apply also here.

			       k  A156552(k)    A156552(k)/k  k-(1+A156552(k)) factorization of k
       1:       0         0                0
       3:       2         0.6666667        0
       5:       4         0.8              0
      21:      18         0.8571429        2           (3 * 7)
     323:     320         0.9907121        2           (17 * 19)
   66297:   65714         0.9912062      582           (3 * 7^2 * 11 * 41)
  139965:  139306         0.9952917      658           (3 * 5 * 7 * 31 * 43)
  263375:  262364         0.9961614     1010           (5^3 * 7^2 * 43)
  264845:  264244         0.9977307      600           (5 * 7^2 * 23 * 47)
  528581:  528576         0.9999905        4           (17^2 * 31 * 59).

Subsequence of A364561.
Cf. A005940, A005941, A029747, A156552.
Cf. also A364551, A364564, A364572.

A364564 Largest prime factor computed for those odd numbers k for which A156552(k) < k.

Original entry on oeis.org

1, 3, 5, 3, 5, 7, 5, 3, 7, 5, 7, 11, 7, 5, 11, 3, 13, 11, 7, 11, 5, 5, 13, 7, 11, 13, 7, 17, 7, 13, 17, 5, 11, 3, 7, 13, 11, 17, 11, 7, 19, 13, 7, 13, 17, 11, 5, 11, 5, 17, 13, 7, 13, 11, 13, 7, 11, 17, 7, 13, 17, 11, 5, 19, 13, 17, 19, 5, 11, 13, 3, 7, 19, 17, 13, 11, 17, 13, 11, 17, 7, 11, 19, 17, 7, 19, 13, 13

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

These primes must stay reasonably small, but how small?

See also the example section of A364551.

Antti Karttunen, Table of n, a(n) for n = 1..13739

Cf. A006530, A156552, A364551, A364561.

a(n) = A006530(A364561(n)).

cp's OEIS Frontend

A005941 Inverse of the Doudna sequence A005940.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Scheme

Formula

Extensions

A364551 Odd numbers k such that k is a multiple of A005941(k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A364560 Numbers k for which A156552(k) < k.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A364576 Starting from k=1, each subsequent term is the next larger odd k such that A156552(k) < k and the ratio A156552(k)/k is nearer to 1.0 than for any previous k in the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

A364564 Largest prime factor computed for those odd numbers k for which A156552(k) < k.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula