A051729 -id:A051729

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A023186

Lonely (or isolated) primes: increasing distance to nearest prime.

+10
25

2, 5, 23, 53, 211, 1847, 2179, 3967, 16033, 24281, 38501, 58831, 203713, 206699, 413353, 1272749, 2198981, 5102953, 10938023, 12623189, 72546283, 142414669, 162821917, 163710121, 325737821, 1131241763, 1791752797, 3173306951, 4841337887, 6021542119, 6807940367, 7174208683, 8835528511, 11179888193, 15318488291, 26329105043, 31587561361, 45241670743

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Erdős and Suranyi call these reclusive primes and prove that there are an infinite number of them. They define these primes to be between two primes. Hence their first term would be 3 instead of 2. Record values in A120937. - T. D. Noe, Jul 21 2006

REFERENCES

Paul Erdős and Janos Suranyi, Topics in the theory of numbers, Springer, 2003.

LINKS

Dmitry Petukhov, Table of n, a(n) for n = 1..56 (first 40 terms from Ken Takusagawa, terms 41..52 from Giovanni Resta)

EXAMPLE

The nearest prime to 23 is 4 units away, larger than any previous prime, so 23 is in the sequence.

The prime a(4) = A120937(3) = 53 is at distance 2*3 = 6 from its neighbors {47, 59}. The prime a(5) = A120937(4) = A120937(5) = A120937(6) = 211 is at distance 2*6 = 12 from its neighbors {199, 223}. Sequence A120937 requires the terms to have 2 neighbors, therefore its first term is 3 and not 2. - M. F. Hasler, Dec 28 2015

MATHEMATICA

p = 0; q = 2; i = 0; Do[r = NextPrime[q]; m = Min[r - q, q - p]; If[m > i, Print[q]; i = m]; p = q; q = r, {n, 1, 152382000}]

Join[{2}, DeleteDuplicates[{#[[2]], Min[Differences[#]]}&/@Partition[Prime[ Range[ 2, 10^6]], 3, 1], GreaterEqual[ #1[[2]], #2[[2]]]&][[;; , 1]]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Aug 31 2023 *)

CROSSREFS

Related sequences: A023186, A023187, A023188, A046929, A046930, A046931, A051650, A051652, A051697-A051702, A051728, A051729, A051730, A102723.

The distances are in A023187.

KEYWORD

nonn,nice

AUTHOR

David W. Wilson

EXTENSIONS

More terms from Jud McCranie, Jun 16 2000

More terms from T. D. Noe, Jul 21 2006

STATUS

approved

A051701

Closest prime to n-th prime p that is different from p (break ties by taking the smaller prime).

+10
5

3, 2, 3, 5, 13, 11, 19, 17, 19, 31, 29, 41, 43, 41, 43, 47, 61, 59, 71, 73, 71, 83, 79, 83, 101, 103, 101, 109, 107, 109, 131, 127, 139, 137, 151, 149, 151, 167, 163, 167, 181, 179, 193, 191, 199, 197, 199, 227, 229, 227, 229, 241, 239, 257, 251, 257, 271, 269

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A227878 gives the terms occurring twice. - Reinhard Zumkeller, Oct 25 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

EXAMPLE

Closest primes to 2,3,5,7,11 are 3,2,3,5,13.

MATHEMATICA

a[n_] := (p = Prime[n]; np = NextPrime[p]; pp = NextPrime[p, -1]; If[np-p < p-pp, np, pp]); Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Oct 20 2011 *)

cp[{a_, b_, c_}]:=If[c-b<b-a, c, a]; Join[{3}, cp/@Partition[Prime[Range[ 60]], 3, 1]] (* Harvey P. Dale, Oct 08 2012 *)

PROG

(Haskell)

a051701 n = a051701_list !! (n-1)

a051701_list = f 2 $ 1 : a000040_list where

f d (q:ps@(p:p':_)) = (if d <= d' then q else p') : f d' ps

where d' = p' - p

-- Reinhard Zumkeller, Oct 25 2013

(Python)

from sympy import nextprime

def aupton(terms):

prv, cur, nxt, alst = 0, 2, 3, []

while len(alst) < terms:

alst.append(prv if 2*cur - prv <= nxt else nxt)

prv, cur, nxt = cur, nxt, nextprime(nxt)

return alst

print(aupton(58)) # Michael S. Branicky, Jun 04 2021

CROSSREFS

Related sequences: A023186, A023187, A023188, A046929, A046930, A046931, A051650, A051652, A051697, A051698, A051699, A051700, A051701, A051702, A051728, A051729, A051730.

Cf. A116946 (semiprime analog).

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

A062816

a(n) = phi(n)*tau(n) - 2n = A000010(n)*A000005(n) - 2*n.

+10
2

-1, -2, -2, -2, -2, -4, -2, 0, 0, -4, -2, 0, -2, -4, 2, 8, -2, 0, -2, 8, 6, -4, -2, 16, 10, -4, 18, 16, -2, 4, -2, 32, 14, -4, 26, 36, -2, -4, 18, 48, -2, 12, -2, 32, 54, -4, -2, 64, 28, 20, 26, 40, -2, 36, 50, 80, 30, -4, -2, 72, -2, -4, 90, 96, 62, 28, -2, 56, 38, 52, -2, 144, -2, -4, 90, 64, 86, 36, -2, 160, 108, -4, -2, 120, 86, -4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

It can be shown that phi(n)*tau(n) >= n, which means that quotient = n/tau(n) <= phi(n); note: a(n)+5 is positive.

The value is always positive except when a(n) = 0 for {8,9,12}; or a(n) = -2 for primes together with 4 (i.e., for A046022 but without 1); or a(n) = -4 for A001747 (without 2 and 4); or a(n) = -1 for n = 1.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..2000

FORMULA

a(n) = A062355(n) - 2*n. - Amiram Eldar, Jul 10 2024

MATHEMATICA

Table[EulerPhi[n]DivisorSigma[0, n]-2n, {n, 90}] (* Harvey P. Dale, Feb 03 2021 *)

PROG

(PARI) a(n)={eulerphi(n)*numdiv(n) - 2*n} \\ Harry J. Smith, Aug 11 2009

CROSSREFS

Cf. A000010, A000005, A062355, A062516, A046022, A001747, A036762, A036763, A051728, A051729, A051730.

KEYWORD

sign

AUTHOR

Labos Elemer, Jul 20 2001

EXTENSIONS

Offset changed from 0 to 1 by Harry J. Smith, Aug 11 2009

STATUS

approved

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