A175856 -id:A175856 - cp's OEIS frontend

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.

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).

A175857 a(n) = possible 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, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Jaroslav Krizek, Sep 29 2010

nonn

Complement of A175859(n). A175860(a(n)) = 1, A175862(a(n)) >= 1.

Cf. A175856, A175858, A175859, A175860, A175861, A175862.

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

A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Mar 29 2004

nonn

Carl Sagan's "Contact" sequence.

Zeros occur at positions given by 1+A110895(k). - Antti Karttunen, Nov 08 2018

W. A. Dembski and J. M. Kushiner, Signs of Intelligence, Baker Book House Co., Grand Rapids, MI, p30-31, 2001,
Carl Sagan, Contact, Simon and Schuster, Chapter 4 "Prime Numbers," pp. 68-82, NY, 1985.

Antti Karttunen, Table of n, a(n) for n = 1..24235 (first 101 runs)
Robin Dougherty, Robert Zemeckis' Contact, Salon.
Alex Kasman, Mathematical Fiction, Contact (1985).
Alex Kasman, How 'Contact' by Carl Sagan Ends.
Index entries for characteristic functions

Cf. A000040, A000042, A005171, A031974, A055976, A056051, A066247, A091247, A110895, A175851, A175856.

a = Table[1, {100}]; Do[ a[[Sum[Prime[i], {i, n}] + n]] = 0, {n, 1, 8}]; a

up_to = 111;
A093521list(up_to) = { my(v=vector(up_to), i=2, j); v[1] = 1; v[2] = 0; forprime(p=2, oo, j=p; while(j, if(i==up_to, return(v), i++; v[i] = 1; j--)); if(i==up_to, return(v), i++; v[i] = 0)); };
v093521 = A093521list(up_to);
A093521(n) = v093521[n];

Data section extended up to n=111 by Antti Karttunen, Nov 08 2018

A345668 Last prime minus distance to last prime.

Original entry on oeis.org

1, 2, 1, 4, 3, 6, 5, 4, 3, 10, 9, 12, 11, 10, 9, 16, 15, 18, 17, 16, 15, 22, 21, 20, 19, 18, 17, 28, 27, 30, 29, 28, 27, 26, 25, 36, 35, 34, 33, 40, 39, 42, 41, 40, 39, 46, 45, 44, 43, 42, 41, 52, 51, 50, 49, 48, 47, 58, 57, 60, 59, 58, 57, 56, 55, 66, 65, 64

Offset: 3

Brian Beard, Jun 21 2021

nonn

Cf. A049711, A151799, A175856.

a[n_] := 2 * NextPrime[n, -1] - n; Array[a, 100, 3] (* Amiram Eldar, Jun 21 2021 *)

a(n) = p - (n-p) = 2*p - n where p is the largest prime < n.
a(n) = A151799(n) - A049711(n).
a(n) = A175856(n-1) - 1.

More terms from Jon E. Schoenfield, Jun 21 2021

cp's OEIS Frontend

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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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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A175857 a(n) = possible 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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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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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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A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

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Programs

Mathematica

PARI

Extensions

A345668 Last prime minus distance to last prime.

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Mathematica

Formula

Extensions