A079953 -id:A079953 - cp's OEIS frontend

A039734 a(n)=the prime q

2, 3, 5, 7, 7, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 31, 31, 37, 37, 37, 41, 43, 47, 53, 53, 53, 59, 59, 59, 67, 67, 71, 71, 79, 79, 79, 83, 89, 89, 97, 97, 97, 97, 101, 101, 107, 113, 127, 127, 127, 127, 127, 127, 131, 137, 137, 137

Offset: 2

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A039733, A079953.

Table[Sort[Prime@Range@n, Mod[Prime[n], #1] > Mod[Prime[n], #2] &][[1]], {n, 2, 58}] (* Ivan Neretin, Feb 23 2018 *)

a(n) = {theq = 0; modmax = 0; pr = prime(n); forprime(q = 2, pr-1, mq = pr % q; if (mq > modmax, modmax = mq; theq = q);); theq;} \\ Michel Marcus, Oct 02 2013

A079952 Number of primes less than prime(n)/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 13, 14, 15, 15, 15, 16, 16, 16, 18, 18, 19, 19, 21, 21, 21, 22, 23, 23, 24, 24, 24, 24, 25, 25, 27, 29, 30, 30, 30, 30, 30, 30, 31, 32, 32, 32, 33, 34, 34, 34, 36, 36, 36, 37, 38, 39, 40

Offset: 1

Reinhard Zumkeller, Jan 19 2003

nonn

Previous name: Number of primes p such that prime(n) mod 2*p < prime(n).

Same as A055930, except for a(2). [Noticed by R. J. Mathar, Dec 15 2008, proved by Andrey Zabolotskiy, Oct 26 2017]

			n = 6: prime(6) = 13 and 2, 3, 5 are less than 13/2, therefore a(6) = 3.

Cf. A000040, A000720, A055377, A055930, A079951, A079953, A217564.

Table[PrimePi[Prime[n]/2], {n, 75}] (* Michael De Vlieger, Sep 20 2017 *)

a(n) = primepi(prime(n)/2); \\ Michel Marcus, Sep 20 2017

A079950(n, a(n) + 1) = prime(n).
Where defined, that is for n > 2, prime(a(n)) = A055377(prime(n)). - Peter Munn, Sep 18 2017
0 with partial sums of A217564. - David A. Corneth, Oct 26 2017 (found earlier by Peter Munn).

New name from Peter Munn, Sep 18 2017

A079950 Triangle of n-th prime modulo twice primes less n-th prime.

Original entry on oeis.org

2, 3, 3, 1, 5, 5, 3, 1, 7, 7, 3, 5, 1, 11, 11, 1, 1, 3, 13, 13, 13, 1, 5, 7, 3, 17, 17, 17, 3, 1, 9, 5, 19, 19, 19, 19, 3, 5, 3, 9, 1, 23, 23, 23, 23, 1, 5, 9, 1, 7, 3, 29, 29, 29, 29, 3, 1, 1, 3, 9, 5, 31, 31, 31, 31, 31, 1, 1, 7, 9, 15, 11, 3, 37, 37, 37, 37, 37, 1, 5, 1, 13, 19, 15, 7, 3, 41

Offset: 1

Reinhard Zumkeller, Jan 19 2003

The right border of the triangle are the primes: T(n,n)=A000040(n); T(n,1)=A039702(n), T(n,2)=A039704(n) for n>1, T(n,3)=A007652(n) for n>2, T(n,4)=A039712(n) for n>3;

			Triangle begins:
  2;
  3, 3;
  1, 5, 5;
  3, 1, 7,  7;
  3, 5, 1, 11, 11;
  1, 1, 3, 13, 13, 13;
  1, 5, 7,  3, 17, 17, 17;
  ...

Cf. A001747, A079951, A079952, A079953.

A079950 := proc(n,k)
    modp(ithprime(n),2*ithprime(k)) ;
end proc:
seq(seq(A079950(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 28 2017

T(n,k) = prime(n) % (2*prime(k));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Sep 21 2017

T(n, k) = prime(n) mod 2*prime(k), 1<=k<=n.

A055377 a(n) = largest prime <= n/2.

Original entry on oeis.org

2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37

Offset: 4

Labos Elemer, Jun 22 2000; David W. Wilson, Jun 10 2005

nonn

Also largest prime factor of any composite <= n. E.g., a(15) = 7 since 7 is the largest prime factor of {4,6,8,9,10,12,14,15}, the composites <= 15.
Also largest prime dividing A025527(n) = n!/lcm[1,...,n]. [Comment from Ray Chandler, Apr 26 2007: Primes > n/2 don't appear as factors of A025527(n) since they appear once in n! and again in the denominator lcm[1,...,n]. Primes <= n/2 appear more times in the numerator than the denominator so they appear in the fraction.]
a(n) is the largest prime factor whose exponent in the factorization of n! is greater than 1. - Michel Marcus, Nov 11 2018

			n = 10, n! = 3628800, lcm[1,...,10] = 2520, A025527(10) = 1440 = 32*9*5 so a(7) = 5 (offset = 3).

Michael De Vlieger, Table of n, a(n) for n = 4..10000

Cf. A000040, A000142, A003418, A006530, A007917, A025527, A060265, A079952, A079953.

Table[Prime@ PrimePi[n/2], {n, 4, 78}] (* Michael De Vlieger, Sep 21 2017 *)

a(n) = precprime(n/2); \\ Michel Marcus, Sep 20 2017

a(n) = Max(gpf((n+2) mod k): 1 < k < (n+2) and k not prime), with gpf=A006530 (greatest prime factor). - Reinhard Zumkeller, Mar 27 2004

Where defined, that is for n > 2, a(A000040(n)) = A000040(A079952(n)). - Peter Munn, Sep 18 2017

More terms from James Sellers, Jul 04 2000

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007

A336298 Greatest prime < prime(n)/2.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 29, 29, 31, 31, 31, 37, 41, 43, 47, 47, 47, 53, 53, 53, 61, 61, 67, 67, 73, 73, 73, 79, 83, 83, 89, 89, 89, 89, 97, 97, 103, 109, 113, 113, 113, 113, 113, 113, 127, 131, 131, 131, 137, 139, 139, 139, 151, 151

Offset: 3

Clark Kimberling, Nov 16 2020

nonn

The n-th prime appears A102820(n) times. - Flávio V. Fernandes, Apr 08 2021

A080191 lists the distinct terms of this sequence. - Flávio V. Fernandes, Jun 19 2021

			Prime(3)/2 = 2.5, so a(3) = 2.

Cf. A000040, A039734, A336299.

z = 120; t = Table[NextPrime[Prime[n]/2], {n, 3, z}]; (* A039734, A079953 *)
u = NextPrime[t, -1]  (* A336298 *)
t - u (* A336299 *)
Table[NextPrime[Prime[n]/2, -1], {n, 3, 80}] (* Wesley Ivan Hurt, Nov 26 2020 *)

a(n) = precprime(prime(n)/2); \\ Michel Marcus, Nov 16 2020

from sympy import prime, prevprime
def A336298(n):
    return prevprime(prime(n)//2+1) # Chai Wah Wu, Nov 26 2020

a(n) = A151799(A000040(n)/2) for n >= 3. - Wesley Ivan Hurt, Nov 26 2020

A336299 (Least prime > prime(n)/2) - (greatest prime < prime(n)/2).

Original entry on oeis.org

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

Offset: 3

Clark Kimberling, Nov 16 2020

nonn

			Prime(3)/2 = 2.5, so that a(3) = 3 - 2 = 1.
Prime(4)/2 = 3.5, so that a(4) = 5 - 3 = 2.

Cf. A000040, A039734, A336298.

z = 120; t = Table[NextPrime[Prime[n]/2], {n, 3, z}]; (* cf. A039734, A079953 *)
u = NextPrime[t, -1]  (* A336298 *)
t - u  (* A336299 *)

a(n) = my(x=prime(n)/2); nextprime(x) - precprime(x); \\ Michel Marcus, Nov 18 2020

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A039734 a(n)=the prime q

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A079952 Number of primes less than prime(n)/2.

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A079950 Triangle of n-th prime modulo twice primes less n-th prime.

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A055377 a(n) = largest prime <= n/2.

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A336298 Greatest prime < prime(n)/2.

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A336299 (Least prime > prime(n)/2) - (greatest prime < prime(n)/2).

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