A364560 -id:A364560

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A029747

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

+10
50

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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).

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

A005941

Inverse of the Doudna sequence A005940.
(Formerly M0510)

+10
34

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

J. H. Conway, personal communication.

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

LINKS

R. J. Mathar, Table of n, a(n) for n=1,..,5000

Index entries for sequences that are permutations of the natural numbers

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(Python)

from sympy import primepi, factorint

def A005941(n): return sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n, multiple=True)))+1 # Chai Wah Wu, Mar 11 2023

(PARI) 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

CROSSREFS

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].

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

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

STATUS

approved

A364551

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..16.

EXAMPLE

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.

PROG

isA364551(n) = ((n%2)&&!(n%A005941(n)));

CROSSREFS

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

Subsequence of A364561, which is a subsequence of A364560.

KEYWORD

nonn,more

AUTHOR

Antti Karttunen, Jul 28 2023

STATUS

approved

A364542

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..81.

MATHEMATICA

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 *)

PROG

(PARI)

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);

CROSSREFS

Positions of nonnegative terms in A364499.

Complement of A364540.

Cf. A005940, A005941, A029747 (subsequence), A343107 (not a subsequence), A364560.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 28 2023

STATUS

approved

A364559

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

LINKS

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

FORMULA

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

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

EXAMPLE

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.

PROG

(PARI)

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);

(Python)

from sympy import factorint, primepi

def A364559(n): return sum(1<<primepi(p)+i for i, p in enumerate(factorint(n, multiple=True), -1))+1-n # Chai Wah Wu, Jul 29 2023

CROSSREFS

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.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jul 28 2023

STATUS

approved

A364561

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

PROG

(PARI)

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));

CROSSREFS

Odd terms in A364560.

Cf. A005940, A005941, A156552, A364545, A364564 (largest prime factor).

Cf. also A364551, A364576 (subsequences).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 28 2023

STATUS

approved

A364550

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..59.

PROG

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

CROSSREFS

Cf. A005941, A156552, A364548.

Subsequence of A364560.

Subsequences: A029747, A364551 (odd terms).

Cf. also

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 28 2023

STATUS

approved

A364562

Numbers k for which A156552(k) > k.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k such that A005941(k) > k.

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

LINKS

Table of n, a(n) for n=1..74.

PROG

(PARI)

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);

CROSSREFS

Positions of strictly positive terms in A364559.

Cf. A005941, A156552, A364560 (complement).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 28 2023

STATUS

approved

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