A049711 -id:A049711 - cp's OEIS frontend

A175077 Final number reached by iterating r -> A049711(r) starting at r = n.

1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

See A175071 for starting n that reach 1, and A175072 for starting n that reach 2.

			Iteration procedure for n = 6: 6 mod 5 = 1 = a(6).
Iteration procedure for n = 7: 7 mod 5 = 2 = a(7).

Cf. A049711, A121559, A151799, A175071, A175072.

A151799 := proc(n) prevprime(n) ; end proc:
A049711 := proc(n) if n <=2 then n; else n-A151799(n) ; end if; end proc:
A175077 := proc(n) local r ; r := n ; while r > 2 do r := A049711(r) ; end do: r ; end proc:
seq(A175077(n),n=1..100) ; # R. J. Mathar, Feb 19 2010

f[n_] := Switch[n, 1, 1, 2, 2, _, n - NextPrime[n, -1]];
a[n_] := FixedPoint[f, n];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 13 2023 *)

a(A175071(k)) = 1; a(A175072(k)) = 2, any k. - R. J. Mathar, Feb 19 2010

a(n) = A121559(n-1) + 1 for n >= 2. - Pontus von Brömssen, Jul 31 2022

More terms from R. J. Mathar, Feb 19 2010

A013603 Difference between 2^n and the nearest prime less than or equal to 2^n.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 5, 3, 3, 9, 3, 1, 3, 19, 15, 1, 5, 1, 3, 9, 3, 15, 3, 39, 5, 39, 57, 3, 35, 1, 5, 9, 41, 31, 5, 25, 45, 7, 87, 21, 11, 57, 17, 55, 21, 115, 59, 81, 27, 129, 47, 111, 33, 55, 5, 13, 27, 55, 93, 1, 57, 25, 59, 49, 5, 19, 23, 19, 35, 231, 93, 69, 35, 97, 15

Offset: 1

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

If a(n) = 1, then n is prime and 2^n - 1 is a Mersenne prime. - Franz Vrabec, Sep 27 2005
Using the first variant A007917 (rather than A151799) of the prevprime() function, the sequence is well defined for n = 1, with a(1) = 2^1 - prevprime(2^1) = 2 - 2 = 0. - M. F. Hasler, Sep 09 2015
In Mathematica, one can use NextPrime with a second argument of -1 to obtain the next smaller prime. As almost all the powers of 2 are composite, this produces the proper results for most of this sequence. However, NextPrime[2, -1] returns -2 rather than the expected 2, which would consequently mean a(1) = 4 rather than 0. - Alonso del Arte, Dec 10 2016

T. D. Noe, Table of n, a(n) for n = 1..5000 (corrected by Sean A. Irvine, Jan 18 2019)
V. Danilov, Table for large n.
Corbin Simpson, 2^255 - 19 and Elliptic Curve Cryptography (MegaFavNumbers), Youtube video (2020).

Cf. A013597, A014234, A049711, A007917, A151799.

Equivalent sequence for next prime: A092131.

Maple
```
seq(2^i-prevprime(2^i),i=2..100);
```

{0} ~Join~ Array[With[{c = 2^#}, c - NextPrime[c, -1]] &, 80, 2] (* Harvey P. Dale, Jul 23 2013 *)
Table[2^n - Prime[PrimePi[2^n]], {n, 80}] (* Alonso del Arte, Dec 10 2016 *)

a(n) = 2^n - precprime(2^n); \\ Michel Marcus, Apr 04 2020

a(n) = A049711(2^n). - R. J. Mathar, Nov 28 2016

a(n) = 2^n - prevprime(2^n) = 2^n - prime(primepi(2^n)). - Alonso del Arte, Dec 10 2016

Extended to a(1) = 0 by M. F. Hasler, Sep 09 2015

A064722 a(1) = 0; for n >= 2, a(n) = n - (largest prime <= n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3

Offset: 1

Reinhard Zumkeller, Oct 13 2001

			a(26) = 26 - 23 = 3, a(37) = 37 - 37 = 0.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A000040, A000720, A007917, A007920, A010051, A049711.

0, seq(n - prevprime(n+1), n=2..100); # Robert Israel, Aug 25 2014

Join[{0},Table[n-NextPrime[n+1,-1],{n,2,110}]] (* Harvey P. Dale, Aug 23 2011 *)

{ for (n = 1, 1000, if (n>1, a=n - precprime(n), a=0); write("b064722.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 23 2009

a(n) = n - A007917(n).
a(n) = 0 iff n is 1 or a prime.
Computable also as a "commutator": pi(prime(m)) - prime(pi(m)) = A000720(A000040(m))-A000040(A000720(m)). Labels position of composites between 2 consecutive primes. - Labos Elemer, Oct 19 2001
a(n) = a(n-1)*0^A010051(n) + 1 - A010051(n), a(1) = 0. - Reinhard Zumkeller, Mar 23 2006
a(n) = n mod A007917(n). - Michel Marcus, Aug 22 2014
a(n) = A049711(n+1) - 1 for n >= 2. - Pontus von Brömssen, Jul 31 2022

A049653 a(n) = 2n - prevprime(2n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 1, 3, 1, 3, 5, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 1, 3, 5, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 5, 7, 9, 11, 13, 1, 3, 1, 3, 5, 1, 1, 3, 5, 7, 9, 1, 1, 3, 5, 1, 3, 5, 1, 3, 1, 3, 5, 1, 3, 5, 1, 1, 3, 5, 7

Offset: 2

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A049613, A049711, A049716, A049847.

[2*n - NthPrime(#PrimesUpTo(2*n)): n in [2..50]]; // G. C. Greubel, Dec 05 2017

Table[2*n - NextPrime[2*n, -1], {n, 2, 50}] (* G. C. Greubel, Dec 05 2017 *)

for(n=2,50, print1(2*n - precprime(2*n), ", ")) \\ G. C. Greubel, Dec 05 2017

a(n) = A049711(2*n). - R. J. Mathar, Oct 26 2015

A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4

Offset: 1

Jaroslav Krizek, Sep 29 2010

Sequence is cardinal and not fractal. Cardinal sequence is sequence with infinitely many times occurring all natural numbers. Fractal sequence is sequence such that when the first instance of each number in the sequence is erased, the original sequence remains.

Ordinal transform of the nextprime function, A151800(1..) = 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, ..., also ordinal transform of A304106. - Antti Karttunen, Jun 09 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000720, A007917, A008578, A151799, A151800, A305300.
Cf. A065358 for another way of visualizing prime gaps.
Cf. A304106 (ordinal transform of this sequence).
Cf. A049711.

a[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

A175851(n) = if(1==n,n,1 + n - precprime(n)); \\ Antti Karttunen, Mar 04 2018

a(1) = 1, a(n) = n - A007917(n) + 1 for n >= 2. a(1) = 1, a(2) = 1, a(n) = n - A151799(n+1) + 1 for n >= 3.
a(n) = Sum_{i=1..n} floor(pi(i)/pi(n)), for n>1 with pi(n) = A000720(n). - Ridouane Oudra, Jun 24 2024
a(n) = A049711(n+1), for n>1. - Ridouane Oudra, Jul 16 2024

A135528 1, then repeat 1,0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

This is Guy Steele's sequence GS(2, 1) (see A135416).
2-adic expansion of 1/3 (right to left): 1/3 = ...01010101010101011. - Philippe Deléham, Mar 24 2009
Also, with offset 0, parity of A036467(n-1). - Omar E. Pol, Mar 17 2015
Appears to be the Gilbreath transform of 1,2,3,5,7,11,13,... (A008578). (This is essentially the same as the Gilbreath conjecture, see A036262.) - N. J. A. Sloane, May 08 2023

			G.f. = x + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 + x^14 + x^16 + x^18 + x^20 + ...

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A008578, A036467, A049711, A135416.

a135528 n = a135528_list !! (n-1)
a135528_list = concat $ iterate ([1,0] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (0:ps) * qs         = 0 : ps * qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

Maple
```
GS(2,1,200); [see A135416].
```

Prepend[Table[Mod[n + 1, 2], {n, 2, 60}], 1] (* Michael De Vlieger, Mar 17 2015 *)
PadRight[{1},120,{0,1}] (* Harvey P. Dale, Apr 23 2024 *)

G.f.: x*(1+x-x^2)/(1-x^2). - Philippe Deléham, Feb 08 2012
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x)))). - Michael Somos, Apr 02 2012
a(n) = A049711(n+2) mod 2. - Ctibor O. Zizka, Jan 28 2019

A336496 Products of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 34560, 36864, 41472, 49152, 55296

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A317804 in having 34560, which is the first term with more than two distinct prime factors.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}

A001013 is the version for factorials, with complement A093373.
A181818 is the version for superprimorials, with complement A336426.
A336497 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178.
A174605 is the maximum prime multiplicity in A000178.
A303279 counts prime factors of superfactorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[1000],facsusing[Rest[Array[supfac,30]],#]!={}&]

A013604 a(n) = 3^n - prevprime(3^n), where prevprime(x) is the largest prime < x.

Original entry on oeis.org

1, 2, 4, 2, 2, 2, 8, 8, 2, 20, 16, 58, 22, 8, 16, 98, 10, 10, 14, 8, 4, 2, 20, 28, 20, 20, 26, 22, 14, 26, 4, 34, 52, 56, 28, 10, 2, 10, 58, 38, 2, 122, 74, 34, 22, 22, 46, 44, 50, 46, 16, 124, 4, 10, 106, 10, 26, 220, 38, 160, 74, 8, 28, 104, 104, 38, 298, 94, 16

Offset: 1

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000244, A013598, A049711.

[3^n-PreviousPrime(3^n): n in [1..70]]; // Vincenzo Librandi, Sep 16 2016

Maple
```
seq(3^i-prevprime(3^i),i=1..100);
```

Table[3^i - NextPrime[3^i, - 1], {i, 70}] (* Vincenzo Librandi, Sep 16 2016 *)

a(n) = 3^n - precprime(3^n-1); \\ Michel Marcus, Sep 16 2016

a(n) = A049711(3^n) = A049711(A000244(n)). - Michel Marcus, Sep 16 2016

A049613 a(n) = 2n - (largest prime < 2n-2).

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 9, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 9, 11, 13, 15, 3, 5, 3, 5, 7, 3, 3, 5, 7, 9, 11, 3, 3, 5, 7, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5

Offset: 3

David M. Elder (elddm(AT)rhodes.edu)

			a(14)=28 - (largest prime < 26) = 28 - 23 = 5.

T. D. Noe, Table of n, a(n) for n=3..10000

Cf. A049653, A049711, A049716, A049847.

Cf. A002373, A020481, A007917.

a049613 n = 2 * n - a007917 (2 * n - 2)
-- Reinhard Zumkeller, Jan 02 2015

Table[2n-NextPrime[2n-2,-1],{n,3,100}] (* Harvey P. Dale, Aug 16 2011 *)

a(n) <= A002373(n). - R. J. Mathar, Mar 19 2008

a(n) = 2*n - A007917(2*n-2). - Reinhard Zumkeller, Jan 02 2015

A049716 a(n) = 2n + 1 - prevprime(2n + 1).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 2, 4, 2, 4, 6, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 2, 4, 6, 2, 4, 2, 4, 6, 2, 4, 6, 8, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 6, 8, 10, 12, 14, 2, 4, 2, 4, 6, 2, 2, 4, 6, 8, 10, 2, 2, 4, 6, 2, 4, 6, 2, 4, 2, 4, 6, 2, 4, 6, 2, 2, 4

Offset: 1

N. J. A. Sloane

nonn

			n:     1  2  3  4  5  6  7  8 ...
2n+1:  3  5  7  9 11 13 15 17 ...
pp:    2  3  5  7  7 11 13 13 ...
diff:  1  2  2  2  4  2  2  4 ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A049613, A049653, A049711, A049847.

seq(2*n+1-prevprime(2*n+1), n=1..100); # Robert Israel, Jul 05 2018

Table[2n+1-NextPrime[2n+1,-1],{n,100}] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = 2*n+1-precprime(2*n); \\ Michel Marcus, Jul 06 2018

a(n) = A049711(2*n+1). - R. J. Mathar, Oct 26 2015

cp's OEIS Frontend

A175077 Final number reached by iterating r -> A049711(r) starting at r = n.

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A013603 Difference between 2^n and the nearest prime less than or equal to 2^n.

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A064722 a(1) = 0; for n >= 2, a(n) = n - (largest prime <= n).

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A049653 a(n) = 2*n - prevprime(2*n).

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Magma

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A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

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A135528 1, then repeat 1,0.

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Haskell

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A336496 Products of superfactorials (A000178).

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A049653 a(n) = 2n - prevprime(2n).

A049716 a(n) = 2n + 1 - prevprime(2n + 1).