A175857 -id:A175857 - cp's OEIS frontend

A175856 a(n) = a(n-1) - 1 if n is composite, a(n) = n otherwise.

1, 2, 3, 2, 5, 4, 7, 6, 5, 4, 11, 10, 13, 12, 11, 10, 17, 16, 19, 18, 17, 16, 23, 22, 21, 20, 19, 18, 29, 28, 31, 30, 29, 28, 27, 26, 37, 36, 35, 34, 41, 40, 43, 42, 41, 40, 47, 46, 45, 44, 43, 42, 53, 52, 51, 50, 49, 48, 59, 58, 61, 60, 59, 58, 57, 56, 67, 66

Offset: 1

Jaroslav Krizek, Sep 29 2010

a(n) = n for noncomposite n, a(n) = 2 * (previous prime (n)) - n for composite n.
See A175859 - absent positive integers (pairs of consecutive numbers) in sequence: 8, 9, 14, 15, 24, 25, 32, 33, ...
a(n) = A075365(n) except at n = 1 and n = 10. - Alexandre Herrera, Nov 11 2023

			a(7) = 7 (7 is a noncomposite number), a(8) = 2*7 - 8 = 6 (8 is composite).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002808, A008578, A175857, A175858, A175859, A175860, A175861, A175862, A075365.

a:= proc(n) option remember;
      `if`(n=1 or isprime(n), n, a(n-1)-1)
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 22 2021

a[1] := 1; a[n_] := a[n] = If[PrimeQ[n], n, a[n - 1] - 1] (* Ben Branman, Jan 02 2011 *)
nxt[{n_,a_}]:={n+1,If[CompositeQ[n+1],a-1,n+1]}; NestList[nxt,{1,1},70][[All,2]] (* Harvey P. Dale, Jan 01 2023 *)

a(n)=if(n==1, 1, 2*precprime(n)-n) \\ Charles R Greathouse IV, Apr 22 2022~

Corrected by Jaroslav Krizek, Oct 01 2010

A175859 a(n) = numbers k in increasing order such that A175856(m) = k has no solution for any m, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

Original entry on oeis.org

8, 9, 14, 15, 24, 25, 32, 33, 38, 39, 54, 55, 62, 63, 74, 75, 90, 91, 92, 93, 98, 99, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 140, 141, 142, 143, 144, 145, 158, 159, 182, 183, 184, 185, 186, 187, 212, 213

Offset: 1

Jaroslav Krizek, Sep 29 2010

nonn

Absent positive integers (pairs of consecutive numbers) in sequence A175856(n).
Complement of A175857(n).
A175860(a(n)) = 0, A175861(a(n)) = 1, A175862(a(n)) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..22312

Cf. A175856, A175857, A175858, A175860, A175861, A175862.

a(2k) = a(2k-1) + 1 for k >= 1.

A175860 a(n) = characteristic function of numbers k such that A175856(m) = k has solution for any m, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Sep 29 2010

nonn

a(n) = characteristic function of numbers from A175857(n). a(n) = 1 if A175856(m) = n for any m, else 0. a(n) = 1 for such n that A175862(n) >= 1. a(n) = 0 for such n that A175862(n) = 0. a(n) + A175861(n) = A000012(n).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Wikipedia, Bertrand's postulate (Generalizations)
Index entries for characteristic functions

Cf. A175856, A175857, A175858, A175859, A175861, A175862.

up_to = 100000;
A175856list(up_to) = { my(v=vector(up_to)); for(n=1,up_to,if((1==n)||isprime(n),v[n] = n,v[n] = v[n-1] - 1)); (v); };
\\ This implementation depends on M. El Bachraoui's proof that there exists a prime between 2n and 3n for n > 1 (see Wikipedia-article).
A175860list(up_to) = { my(v=vector(up_to), A175857 = Set(A175856list(prime(2+primepi(2*up_to))))); for(n=1,up_to,v[n] = (0!=vecsearch(A175857,n))); (v); };
v175860 = A175860list(up_to);
A175860(n) = v175860[n]; \\ Antti Karttunen, Nov 08 2018

a(n) = 1 - A175861(n).

A175861 a(n) = characteristic function of numbers k such that A175856(m) = k has no solution for any m, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Sep 29 2010

nonn

a(n) = characteristic function of numbers from A175859(n). a(n) = 1 if A175856(m) not equal to n for any m, else 0. a(n) = 1 for such n that A175862(n) = 0. a(n) = 0 for such n that A175862(n) >= 1. a(n) + A175860(n) = A000012(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A175856, A175857, A175858, A175859, A175860, A175862.

A175861(n) = (1-A175860(n)); \\ Uses program from A175860. - Antti Karttunen, Nov 08 2018

a(n) = 1 - A175860(n).

A175858 a(n) = all values of A175856(m) in increasing order, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 10, 10, 11, 11, 12, 13, 16, 16, 17, 17, 18, 18, 19, 19, 20, 21, 22, 23, 26, 27, 28, 28, 29, 29, 30, 31, 34, 35, 36, 37, 40, 40, 41, 41, 42, 42, 43, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 58, 59, 59, 60

Offset: 1

Jaroslav Krizek, Sep 29 2010

nonn

A175860(a(n)) = 1, A175862(a(n)) >= 1.

Cf. A175856, A175857, A175859, A175860, A175861, A175862.

Corrected by Jaroslav Krizek, Oct 01 2010

A175862 a(n) = frequency of values n in A175856(m), where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 2, 1

Offset: 1

Jaroslav Krizek, Sep 29 2010

nonn

a(n) >= 1 for such n that A175860(n) = 1, a(n) >= 1 if A175856(m) = n for any m. a(n) = 0 for such n that A175860(n) = 0, a(n) = 0 if A175856(m) = n has no solution for any m.

Cf. A175856, A175857, A175858, A175859, A175860, A175861.

Corrected by Jaroslav Krizek, Oct 01 2010

cp's OEIS Frontend

A175856 a(n) = a(n-1) - 1 if n is composite, a(n) = n otherwise.

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A175859 a(n) = numbers k in increasing order such that A175856(m) = k has no solution for any m, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

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A175860 a(n) = characteristic function of numbers k such that A175856(m) = k has solution for any m, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

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PARI

Formula

A175861 a(n) = characteristic function of numbers k such that A175856(m) = k has no solution for any m, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

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Programs

PARI

Formula

A175858 a(n) = all values of A175856(m) in increasing order, where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

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Extensions

A175862 a(n) = frequency of values n in A175856(m), where A175856(m): a(m) = m for m = noncomposites, a(m) = previous term - 1 for m = composites.

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