A151799 -id:A151799 - cp's OEIS frontend

A249666 Numbers n such that the sum of n and the largest primeA151799(n)) is prime.

3, 4, 6, 10, 12, 16, 22, 24, 30, 36, 42, 46, 50, 54, 56, 66, 70, 76, 78, 84, 90, 92, 100, 114, 116, 120, 126, 130, 132, 142, 144, 156, 160, 170, 174, 176, 180, 186, 192, 196, 202, 210, 220, 222, 226, 232, 234, 240, 246, 250, 252, 276, 280, 282, 286, 288, 294, 300, 306, 310, 324

Offset: 1

Antonio Roldán, Dec 03 2014

nonn

			66 is in the sequence because A151799(66)=61, and 66+61=127 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A151799, A249624, A249667, A249676.

Select[Range[400],PrimeQ[#+NextPrime[#,-1]]&] (* Harvey P. Dale, Aug 04 2019 *)

{for(i=3,10^3,k=i+precprime(i-1);if(isprime(k),print1(i,", ")))}

A249676 Terms k of A249667 such that k-A151799(k) = A151800(k)-k.

Original entry on oeis.org

6, 30, 50, 144, 300, 560, 610, 650, 660, 714, 780, 810, 816, 870, 1120, 1176, 1190, 1806, 2130, 2470, 2490, 2550, 2922, 3030, 3240, 3330, 3390, 3480, 3600, 3620, 3840, 4266, 4368, 5796, 5850, 6270, 6786, 6954, 7074, 7710, 8280, 9400, 9990, 10146, 10350, 10380, 10530, 10660, 11064

Offset: 1

Antonio Roldán, Dec 03 2014

nonn

			610 is in A249667: the least prime>610 is 613, and 610+613=1223 is prime; the largest prime<610 is 607, and 610+607=1217 is prime. Also, 613-610=610-607=3, then 610 is in the current sequence.

Cf. A249624, A249666, A249667.

{for(i=3,2*10^4,m=nextprime(i+1);k=i+m;n=precprime(i-1);q=i+n;if(isprime(k)&&isprime(q)&&m-i==i-n,print1(i,", ")))}

A286208 Greater of Wilson's pseudo-twin primes: primes q such that p! == 1 (mod q), where p=A151799(q) is the previous prime before q, and q-p>2.

Original entry on oeis.org

7853, 594278556271609021, 4259842839142238791410741595983041626644087509

Offset: 1

Max Alekseyev and Thomas Ordowski, May 04 2017

By Wilson's theorem, p! == 1 (mod p+2) whenever (p,p+2) are twin primes. This sequence and A286181 concern consecutive primes (p,q) satisfying p! = 1 (mod q), where d = q-p > 2.
It follows that (d-1)! == 1 (mod q). Listed terms correspond to d = 12, 30, 76 (cf. A286230). Further terms should have d>=140.
Also, primes q=prime(n) such that A275111(n-1)=1, and (prime(n-1),prime(n)) are not twin primes (i.e., q is not a term of A006512).

Subsequence of A166864.

Cf. A275111, A286181, A286230.

A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

Original entry on oeis.org

0, 1, -1, 2, 0, 1, -2, 3, 2, -2, 0, 8, 12, -8, -7, 14, -1, 10, 2, 4, 6, -3, 20, -2, 5, -5, -27, 4, -16, 5, 5, 4, -8, 11, 13, -8, -19, 8, -36, 3, 2, -14, -5, 2, -3, -55, -19, -6, 14, -54, -13, -53, 63, -26, 38, -2, 21, 38, -30, 7, 39, 2, -23, 41, 2, -8, 5, 5, -5, -110

Offset: 2

Hugo Pfoertner, Jan 29 2021

sign

a(n) > 0 if the difference nextprime(2^n) - 2^n = A013597(n) is greater than the difference 2^n - previousprime(2^n) = A013603(n).

			a(4) = -1: 2^4 = 16, (13 + 17 - 32)/2 = -1;
a(5) = 2: 2^5 = 32, (31 + 37 - 64)/2 = 2;
a(6) = 0: 2^6 = 64, (61 + 67 - 128)/2 = 0.

Hugo Pfoertner, Table of n, a(n) for n = 2..5000 (from T. D. Noe's b-files in A013597 and A013603).

Cf. A151799, A151800, A013597, A013603, A058249, A226178.

a:= (p-> (nextprime(p)+prevprime(p))/2-p)(2^n):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 29 2021

Array[(NextPrime[2^#] + NextPrime[2^#, -1] - 2^(# + 1))/2 &, 60, 2] (* Michael De Vlieger, Aug 07 2022 *)

for(k=2,71,my(p2=2^k,pp=precprime(p2),pn=nextprime(p2));if(print1((pp+pn-2*p2)/2", ")))

a(n) = (A013597(n) - A013603(n))/2.

a(A226178(n)) = 0.

Name made more precise by Peter Luschny, Aug 08 2022

A357362 Primes q such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where p = A151799(q).

Original entry on oeis.org

7, 53, 59, 151057, 240733, 911135857

Offset: 1

Felix Fröhlich, Sep 25 2022

Cf. A151799, A151800, A357363, A357364, A357365.

is(n) = my(b=precprime(n-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=3, , if(is(p), print1(p, ", ")))

from sympy import nextprime
from itertools import islice
def agen():
    p, q = 2, 3
    while True:
        if pow(p, q-1, q*q) == 1 or pow(q, p-1, p*p) == 1: yield q
        p, q = q, nextprime(q)
print(list(islice(agen(), 5))) # Michael S. Branicky, Sep 30 2022

a(6) from Michael S. Branicky, Sep 26 2022

A357365 Primes q such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where p = A151799(A151799(A151799(A151799(q)))).

Original entry on oeis.org

19, 67, 349, 2011, 22307, 13699249, 2018905087, 9809844767

Offset: 1

Felix Fröhlich, Sep 25 2022

Cf. A151799, A151800, A357362, A357363, A357364.

is(n) = my(b=precprime(precprime(precprime(precprime(n-1)-1)-1)-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=11, , if(is(p), print1(p, ", ")))

from sympy import nextprime
from itertools import islice
def agen():
    p, m1, m2, m3, q = 2, 3, 5, 7, 11
    while True:
        if pow(p, q-1, q*q) == 1 or pow(q, p-1, p*p) == 1: yield q
        p, m1, m2, m3, q = m1, m2, m3, q, nextprime(q)
print(list(islice(agen(), 5))) # Michael S. Branicky, Sep 30 2022

a(7)-a(8) from Michael S. Branicky, Sep 26 2022

A035031 For n >= 7, max(A151799(n), 2*A151799(floor((n-1)/2))).

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 14, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 26, 29, 29, 31, 31, 31, 31, 31, 34, 37, 37, 37, 38, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53

Offset: 1

N. J. A. Sloane

nonn

S. K. Stein and S. Szabo, Algebra and Tiling, MAA Carus Monograph 25, 1994, page 104.

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

Cf. A035032, A151799.

f:= proc(n) max(prevprime(n), 2*prevprime(floor((n-1)/2))) end proc:
for i from 1 to 6 do f(i):= [1, 2, 2, 3, 3, 5, 5][i] od:
map(f, [$1..100]); # Robert Israel, Dec 15 2023

Join[{1,2},Table[Max[NextPrime[n,-1],2NextPrime[Floor[(n-1)/2],-1]],{n,3,70}]] (* Harvey P. Dale, Mar 12 2022 *)

A335185 a(n) = nextprime(ceiling(n/2)-1) - prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

Original entry on oeis.org

0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 2, 2, 2, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 6, 6, 6

Offset: 4

Wesley Ivan Hurt, May 25 2020

a(n) is the difference of the smallest prime appearing among the largest parts of the partitions of n into two parts and the largest prime appearing among the smallest parts of the partitions of n into two parts.
a(n) = 0 if and only if n = 2p, where p is prime. All terms are even except a(5).
The values in the n-th run of positive integers are all equal to the n-th prime gap (A001223).
Each value specifies the run length of the block (of positive integers) in which it appears. If a(n) = 0, then it appears once. If a(n) > 0, it has a run length of 2k - 1.

			a(5) = 1; n=5 has 2 partitions into two parts: (4,1) and (3,2). Among the largest parts, the smallest prime is 3. Among the smallest parts, 2 is the largest. So a(5) = 3 - 2 = 1.
a(6) = 0; n=6 has 3 partitions into two parts: (5,1), (4,2) and (3,3). Among the largest parts, the smallest prime is 3. Among the smallest parts, the largest prime is 3. So a(6) = 3 - 3 = 0.
a(7) = 2; n=7 has 3 partitions into two parts: (6,1), (5,2) and (4,3). Among the largest parts, 5 is the smallest. Among the smallest parts, 3 is the largest. So a(7) = 5 - 3 = 2.

Index entries for sequences related to partitions

Cf. A001223 (prime gaps), A151799, A151800, A335186.

[NextPrime(Ceiling(n/2)-1) - PreviousPrime(Floor(n/2)+1) : n in [4..100]];

Table[NextPrime[Ceiling[n/2] - 1, 1] - NextPrime[Floor[n/2] + 1, -1], {n, 4, 100}]

a(n) = A151800(ceiling(n/2)-1) - A151799(floor(n/2)+1).

A335186 a(n) = nextprime(ceiling(n/2)-1) + prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

Original entry on oeis.org

4, 5, 6, 8, 8, 8, 10, 12, 12, 12, 14, 18, 18, 18, 18, 18, 18, 18, 22, 24, 24, 24, 26, 30, 30, 30, 30, 30, 30, 30, 34, 36, 36, 36, 38, 42, 42, 42, 42, 42, 42, 42, 46, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 58, 60, 60, 60, 62, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 74

Offset: 4

Wesley Ivan Hurt, May 25 2020

a(n) is the sum of the smallest prime appearing among the largest parts of the partitions of n into two parts and the largest prime appearing among the smallest parts of the partitions of n into two parts.
If n = 2p where p is prime, then a(n) = n. The converse is not true since a(8) = 8, but n = 2*4 and 4 is not prime.
All terms are even except a(5).

			a(5) = 5; n=5 has 2 partitions into two parts: (4,1) and (3,2). Among the largest parts, the smallest prime is 3. Among the smallest parts, 2 is the largest. So a(5) = 3 + 2 = 5.
a(6) = 6; n=6 has 3 partitions into two parts: (5,1), (4,2) and (3,3). Among the largest parts, the smallest prime is 3. Among the smallest parts, the largest prime is 3. So a(6) = 3 + 3 = 6.
a(7) = 8; n=7 has 3 partitions into two parts: (6,1), (5,2) and (4,3). Among the largest parts, 5 is the smallest. Among the smallest parts, 3 is the largest. So a(7) = 5 + 3 = 8.

Index entries for sequences related to partitions

Cf. A001223 (prime gaps), A151799, A151800, A335185.

[NextPrime(Ceiling(n/2)-1) + PreviousPrime(Floor(n/2)+1) : n in [4..100]];

Table[NextPrime[Ceiling[n/2] - 1, 1] + NextPrime[Floor[n/2] + 1, -1], {n, 4, 100}]

a(n) = A151800(ceiling(n/2)-1) + A151799(floor(n/2)+1).

A308022 a(n) = prevprime(2*n-1) - nextprime(n-1), where prevprime = A151799 and nextprime = A151800.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 2, 2, 6, 8, 6, 10, 6, 6, 12, 14, 12, 12, 14, 14, 18, 20, 14, 18, 18, 18, 24, 24, 22, 28, 24, 24, 24, 30, 30, 34, 32, 32, 32, 38, 36, 40, 36, 36, 42, 42, 36, 36, 44, 44, 48, 50, 44, 48, 50, 50, 54, 54, 52, 52, 46, 46, 46, 60, 60, 64, 60, 60

Offset: 2

Wesley Ivan Hurt, May 09 2019

a(n) is the difference of the largest and smallest prime(s) in the closed interval [n, 2n-2].

Also, the maximum distance between all pairs of primes (not necessarily distinct) appearing among the largest parts of the partitions of 2n into two parts < 2n-1.

			a(10) = 6; The primes in the closed interval [10, 18] are 11, 13 and 17. The difference of the largest and smallest primes is 17 - 11 = 6.

Index entries for sequences related to partitions

Cf. A151799, A151800.

Table[NextPrime[2 n - 1, -1] - NextPrime[n - 1, 1], {n, 2, 100}]

a(n) = A151799(2*n-1) - A151800(n-1).

cp's OEIS Frontend

A249666 Numbers n such that the sum of n and the largest primeA151799(n)) is prime.

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A249676 Terms k of A249667 such that k-A151799(k) = A151800(k)-k.

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A286208 Greater of Wilson's pseudo-twin primes: primes q such that p! == 1 (mod q), where p=A151799(q) is the previous prime before q, and q-p>2.

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A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

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A357362 Primes q such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where p = A151799(q).

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A357365 Primes q such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where p = A151799(A151799(A151799(A151799(q)))).

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A035031 For n >= 7, max(A151799(n), 2*A151799(floor((n-1)/2))).

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A335185 a(n) = nextprime(ceiling(n/2)-1) - prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

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A335186 a(n) = nextprime(ceiling(n/2)-1) + prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.