A364560 -id:A364560 - cp's OEIS frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608

Offset: 1

N. J. A. Sloane

Fixed points of the Doudna sequence: A005940(a(n)) = A005941(a(n)) = a(n). - Reinhard Zumkeller, Aug 23 2006
Subsequence of A103969. - R. J. Mathar, Mar 06 2010
Question: Is there a simple proof that A005940(c) = c would never allow an odd composite c as a solution? See also my comments in A163511 and in A335431 concerning similar problems, also A364551 and A364576. - Antti Karttunen, Jul 28 & Aug 11 2023

			128 = 2^7 * 1 is in the sequence as well as 160 = 2^5 * 5. - _David A. Corneth_, Sep 18 2020

David A. Corneth, Table of n, a(n) for n = 1..9963 (terms <= 10^1000)
Index entries for linear recurrences with constant coefficients, signature (0,0,2).

Cf. A122132, A000079, A007283, A020714, A130793.
Cf. A005940, A005941, A163511, A335431, A364499, A364551, A364576.
Subsequence of the following sequences: A103969, A253789, A364541, A364542, A364544, A364546, A364548, A364550, A364560, A364565.
Even terms form a subsequence of A320674.

m = 200000; Select[Union @ Flatten @ Outer[Times, {1, 3, 5}, 2^Range[0, Floor[Log2[m]]]], # < m &] (* Amiram Eldar, Oct 15 2020 *)

is(n) = n>>valuation(n, 2) <= 5 \\ David A. Corneth, Sep 18 2020

def A029747(n):
    if n<3: return n
    a, b = divmod(n,3)
    return 1<Chai Wah Wu, Apr 02 2025

a(n) = if n < 6 then n else 2*a(n-3). - Reinhard Zumkeller, Aug 23 2006
G.f.: (1+x+x^2)^2/(1-2*x^3). - R. J. Mathar, Mar 06 2010
Sum_{n>=1} 1/a(n) = 46/15. - Amiram Eldar, Oct 15 2020

Edited by David A. Corneth and Peter Munn, Sep 18 2020

A005941 Inverse of the Doudna sequence A005940.

Original entry on oeis.org

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)));

A364542 Numbers k for which A005940(k) >= k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Sequence A005941(A364560(.)) sorted into ascending order.
A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).
Differs from A343107 for the first time at a(22) = 25, which term is not present in A343107. On the other hand, 35 is the first term of A343107 that is not present in this sequence.

Positions of nonnegative terms in A364499.
Complement of A364540.
Cf. A005940, A005941, A029747 (subsequence), A343107 (not a subsequence), A364560.

nn = 95; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[nn], a[#] >= # &] (* 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 };
isA364542(n) = (A005940(n)>=n);

A364559 a(n) = A005941(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 6, 0, 20, 4, -4, 0, 48, -4, 110, 0, -2, 12, 234, 0, -12, 40, -12, 8, 484, -8, 994, 0, 2, 96, -14, -8, 2012, 220, 28, 0, 4056, -4, 8150, 24, -22, 468, 16338, 0, -24, -24, 80, 80, 32716, -24, -18, 16, 202, 968, 65478, -16, 131012, 1988, -24, 0, 4, 4, 262078, 192, 446, -28, 524218, -16

Offset: 1

Antti Karttunen, Jul 28 2023

sign

			a(528581) = -4 as A005941(528581) = 528577 = 528581-4. Notably, 528581 = 17^2 * 31 * 59, with divisors [1, 17, 31, 59, 289, 527, 1003, 1829, 8959, 17051, 31093, 528581]. Applying A364557 to these divisors gives [1, 64, 1024, 65536, 128, 1024, 65536, 65536, 2048, 131072, 65536, 131072], while applying Euler totient phi (A000010) to them gives [1, 16, 30, 58, 272, 480, 928, 1740, 8160, 15776, 27840, 473280], their differences being [0, 48, 994, 65478, -144, 544, 64608, 63796, -6112, 115296, 37696, -342208], whose sum is -4.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A005941, A364499, A364557, A364558 (Möbius transform).
Cf. A029747 (known positions of 0's), A364560 (of terms <= 0), A364562 (of terms > 0), A364576.
Cf. also A364288.

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)
A364559(n) = (A005941(n)-n);

from sympy import factorint, primepi
def A364559(n): return sum(1<Chai Wah Wu, Jul 29 2023

a(n) = -A364499(A005941(n)).

a(n) = Sum_{d|n} A364558(d).

A364561 Odd numbers k for which A156552(k) < k.

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 25, 27, 35, 45, 49, 55, 63, 75, 77, 81, 91, 99, 105, 121, 125, 135, 143, 147, 165, 169, 175, 187, 189, 195, 221, 225, 231, 243, 245, 273, 275, 289, 297, 315, 323, 325, 343, 351, 357, 363, 375, 385, 405, 425, 429, 441, 455, 495, 507, 525, 539, 561, 567, 585, 595, 605, 625, 627, 637, 663, 665

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Odd numbers k such that A005941(k) <= k.

Antti Karttunen, Table of n, a(n) for n = 1..13739; terms less than 2^21

Odd terms in A364560.
Cf. A005940, A005941, A156552, A364545, A364564 (largest prime factor).
Cf. also A364551, A364576 (subsequences).

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 };
isA364561(n) = ((n%2)&&(A156552(n) < n));

A364550 Numbers k such that k is a multiple of 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, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 3125, 4096, 5120, 6144, 6250, 7875, 8192, 10240, 12005, 12288, 12500, 13365, 15750, 16384, 20480, 22869, 23595, 24010, 24576, 25000, 26730, 31500, 32768, 40960, 45738, 46475

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Numbers k such that k is a multiple of 1+A156552(k).

If k is a term, then also 2*k is present in this sequence, and vice versa.

Cf. A005941, A156552, A364548.
Subsequence of A364560.
Subsequences: A029747, A364551 (odd terms).
Cf. also

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 28 2023
isA364550(n) = !(n%A005941(n));

A364562 Numbers k for which A156552(k) > k.

Original entry on oeis.org

7, 11, 13, 14, 17, 19, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 101, 102, 103, 104, 106, 107, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 127

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Numbers k such that A005941(k) > k.

If k is a term, then also 2*k is present in this sequence, and vice versa.

Positions of strictly positive terms in A364559.

Cf. A005941, A156552, A364560 (complement).

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 };
isA364562(n) = (A156552(n) > n);

cp's OEIS Frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

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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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A364542 Numbers k for which A005940(k) >= k.

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A364559 a(n) = A005941(n) - n.

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A364561 Odd numbers k for which A156552(k) < k.

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A364550 Numbers k such that k is a multiple of A005941(k).

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A364562 Numbers k for which A156552(k) > k.

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