A364549 -id:A364549 - 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)));

A364547 Odd numbers k such that k is a multiple of A005940(k).

Original entry on oeis.org

1, 3, 5, 1035, 524295, 16777217

Offset: 1

Antti Karttunen, Jul 28 2023

Sequence A005941(A364549(.)) sorted into ascending order.
Those terms of A000051 (= 2^k + 1) are included that have A000040(1+k) as one of their prime factors.
a(7) > 402653184.
See also comments in A364963. - Antti Karttunen, Jan 12 2024

			1035 is included because 1034 in binary is "10000001010", which Doudna isomorphism maps to 345 = 3*5*23, which thus divides 1035 (= 3^2 * 5 * 23). Note that there are six 0's in the binary representation between its most significant bit and the trailing "1010", thus we get the prime factors A000040(1+1) = 3, A000040(1+1+1) = 5 and A000040(1+1+1+6) = 23.
524295 is included because 524294 in binary is "10000000000000000110", which Doudna isomorphism maps to 549 = 3^2 * 61, which thus divides 524295 (= 3^2 * 5 * 61 * 191). Note that there are sixteen 0's in the binary representation between its most significant bit and the trailing "110", thus we get the prime factors A000040(2) = 3 and A000040(2+16) = 61.
16777217 = 2^24 + 1 is included because A000040(1+24) = 97, and 16777217 = 97*257*673.

Odd terms in A364546.
Cf. A000040, A000051, A005940, A005941, A364502, A364549.
Cf. also A364545, A364551, A364963.

nn = 2^20 + 2; Array[Set[a[#], #] &, 2]; {1}~Join~Reap[Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], a[n] = k = Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]; If[Divisible[n, a[n]], Sow[n]]], {n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364547(n) = ((n%2)&&!(n%A005940(n)));

A364548 Numbers k such that k divides A005941(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 97, 128, 160, 192, 194, 256, 320, 345, 384, 388, 512, 549, 640, 690, 768, 776, 1024, 1093, 1098, 1280, 1380, 1536, 1552, 2048, 2186, 2196, 2560, 2760, 3072, 3104, 4096, 4372, 4392, 5120, 5520, 6144, 6208, 8192, 8744, 8784, 10240, 11040, 12288, 12416, 16384

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Numbers k such that k divides 1+A156552(k).
Sequence A005940(A364546(.)) sorted into ascending order.
If k is a term, then also 2*k is present in this sequence, and vice versa.

Cf. A005940, A005941, A156552, A364546.
Subsequences: A029747, A364549 (odd terms).
Cf. also A364497.

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)
isA364548(n) = !(A005941(n)%n);

cp's OEIS Frontend

A005941 Inverse of the Doudna sequence A005940.

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

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A364547 Odd numbers k such that k is a multiple of A005940(k).

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A364548 Numbers k such that k divides A005941(k).

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