A051699 -id:A051699 - cp's OEIS frontend

A132861 Smallest number at distance 3n from nearest prime (variant 2).

2, 26, 53, 532, 211, 1342, 2179, 15704, 16033, 31424, 24281, 31430, 31433, 155960, 58831, 360698, 206699, 370312, 370315, 492170, 1357261, 1357264, 1357267, 2010802, 2010805, 4652428, 12485141, 17051788, 17051791, 17051794, 11117213, 20831416, 10938023, 20831422

Offset: 0

R. J. Mathar, Nov 18 2007

nonn

Let f(m) be the distance to the nearest prime as defined in A051700(m). Then a(n) = min {m: f(m) = 3n} for n > 0. A132470 uses A051699(m) to define the distance. a(n) <= A132470(n) because here primes at the start or end of a prime gap of size 3n may be picked, which would be discarded in A132470 for n>0; this gives a chance to minimize m here further than in A132470.

Michael S. Branicky, Table of n, a(n) for n = 0..76
Michael S. Branicky, Python program

Cf. A132470, A051700.

A051700 := proc(m) if m <= 2 then op(m+1,[2,1,1]) ; else min(nextprime(m)-m,m-prevprime(m)) ; fi ; end: a := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051700(m) = 3 * n then RETURN(m) ; fi ; od: fi ; end: seq(a(n),n=0..18);

# see link for faster program
from sympy import prevprime, nextprime
def A051700(n):
  return [2, 1, 1][n] if n < 3 else min(n-prevprime(n), nextprime(n)-n)
def a(n):
  if n == 0: return 2
  m = 0
  while A051700(m) != 3*n: m += 1
  return m
print([a(n) for n in range(13)]) # Michael S. Branicky, Feb 26 2021

a(n) = min {m : A051700(m) = 3n} for n > 0.

a(n) = A051652(3*n). [From R. J. Mathar, Jul 22 2009]

7 more terms from R. J. Mathar, Jul 22 2009
4 more terms from R. J. Mathar, Aug 21 2018
a(30) and beyond and edits from Michael S. Branicky, Feb 26 2021

A160666 Numbers whose distance to the closest prime number is a prime number.

Original entry on oeis.org

0, 9, 15, 21, 25, 26, 27, 33, 34, 35, 39, 45, 49, 50, 51, 55, 56, 57, 63, 64, 65, 69, 75, 76, 77, 81, 85, 86, 87, 91, 92, 94, 95, 99, 105, 111, 115, 116, 118, 120, 122, 124, 125, 129, 133, 134, 135, 141, 142, 144, 146, 147, 153, 154, 155, 159, 160, 161, 165, 169, 170

Offset: 1

Kyle Stern, May 22 2009

nonn

Terms n=2..31 are identical to terms n=1..30 of A079364.

K. Stern, Table of n, a(n) for n=1..10000

Cf. A000040, A051699.

isA160666 := proc(n) local ppl,pmi ; if isprime(n) then RETURN(false): elif n =0 then RETURN(true): elif n =1 then RETURN(false): fi; ppl := nextprime(n)-n ; pmi := n-prevprime(n) ; RETURN (isprime(min(ppl,pmi)) ) ; end: for n from 0 to 200 do if isA160666(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, May 25 2009

fQ[n_] := PrimeQ[ Min[ NextPrime[n] - n, n - NextPrime[n, -1]]]; Select[ Range[0, 174], !PrimeQ@ # && fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

A193598 Even numbers k such that r(k) < r(k/2), where r(n) is the distance from n to the nearest prime.

Original entry on oeis.org

2, 18, 30, 42, 52, 54, 66, 68, 70, 78, 90, 98, 100, 102, 110, 112, 114, 126, 128, 130, 138, 150, 152, 162, 172, 174, 182, 190, 198, 210, 222, 230, 232, 234, 236, 238, 240, 242, 244, 250, 258, 268, 270, 282, 284, 286, 290, 292, 294, 306, 308

Offset: 1

Vladimir Shevelev, Jul 31 2011

nonn

			18 is in the sequence, since r(18) = 1 < 2 = r(9); 22 is not in the sequence, since r(22) = 1 >= 0 = r(11).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

r(n) is A051699.

r(n)=min(nextprime(n)-n,n-precprime(n))
forstep(k=2,1e3,2,if(r(k)Charles R Greathouse IV, Jul 31 2011

A334208 Number of partitions of 2n into two composite parts, (r,s), such that r and s have the same number of primes less than or equal to them.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2

Offset: 1

Wesley Ivan Hurt, Apr 18 2020

Apparently a(n) = A051699(n) for n>=2. - R. J. Mathar, Apr 22 2020

			a(9) = 2; 2*9 = 18 has two partitions into composite parts, (10,8) and (9,9), such that pi(10) = 4 = pi(8) and pi(9) = 4 = pi(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to partitions.

Cf. A000720, A010051, A051699.

Table[Sum[KroneckerDelta[PrimePi[i], PrimePi[2 n - i]] (1 - PrimePi[i] + PrimePi[i - 1]) (1 - PrimePi[2 n - i] + PrimePi[2 n - i - 1]), {i, 2, n}], {n, 100}]

A334208(n) = sum(i=2,n,(!isprime(i) && !isprime(n+n-i) && primepi(i)==primepi(n+n-i))); \\ Antti Karttunen, Jan 29 2025

a(n) = Sum_{i=2..n} [pi(i) = pi(2*n-i)] * (1 - c(i)) * (1 - c(2*n-i)), where [] is the Iverson bracket, pi is the prime counting function (A000720), and c is the prime characteristic (A010051).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A079582 Least positive k such that the distance from k to closest prime = n.

Original entry on oeis.org

2, 1, 9, 26, 93, 118, 119, 120, 531, 532, 897, 1140, 1339, 1340, 1341, 1342, 1343, 1344, 9569, 15702, 15703, 15704, 15705, 19632, 19633, 19634, 19635, 31424, 31425, 31426, 31427, 31428, 31429, 31430, 31431, 31432, 31433, 155958, 155959, 155960, 155961

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

This sequence only differs from A077019 for n = 2: a(2) = 9 whereas A077019(2) = 0. - Rémy Sigrist, Dec 19 2019

Cf. A051699, A077019.

a[n_] := Block[{s = 1}, While[ PrimeQ[s] || Min[s - NextPrime[s, -1], NextPrime[s] - s] != n, s++ ]; s]; a[0] = 2; Table[a[n], {n, 0, 40}]

a(n)=if(n<0,0,s=1; while(abs(n-min(abs(precprime(s)-s),abs(nextprime(s)-s)))>0,s++); s)

More terms from Robert G. Wilson v, Jan 27 2003

Name clarified by Rémy Sigrist, Dec 19 2019

A079669 a(n) = least k such that the distance from Fibonacci(k) to the closest prime is n, or -1 if no such k exists.

Original entry on oeis.org

3, 1, 0, 9, 16, 12, 37, 42, 149, 33, 26, 27, 38, 24, 28, 189, 44, 111, 50, 66, 49, 57, 68, 30, 46, 81, 142, 78, 92, 96, 59, 69, 71, 141, 184, 267, 67, 129, 61, 117, 211, 576, 115, 372, 161, 138, 119, 198

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

Cf. A051699, A000045.

a(n)=if(n<0,0,s=1; while(abs(n-min(abs(precprime(fibonacci(s))-fibonacci(s)),abs(nextprime(fibonacci(s))-fibonacci(s))))>0,s++); s)

Changed "was found" to "exists" in definition. Offset was wrong. Adjusted initial terms. - N. J. A. Sloane, Jan 29 2022

A242561 a(0)=0; thereafter, a(n) is n multiplied by the distance of a(n-1) to the nearest prime.

Original entry on oeis.org

0, 2, 0, 6, 4, 5, 0, 14, 8, 9, 20, 11, 0, 26, 42, 15, 32, 17, 0, 38, 20, 21, 44, 23, 0, 50, 78, 27, 56, 87, 60, 31, 0, 66, 34, 105, 72, 37, 0, 78, 40, 41, 0, 86, 132, 45, 92, 141, 96, 49, 100, 51, 104, 53, 0, 110, 56, 171, 116, 177

Offset: 0

J. M. Bergot, May 17 2014

nonn

It appears that any starting value a(0) will produce a sequence which merges with this one at some point.

Also, if we create a new sequence, call it b(n), from this one by changing one term, say a(k), then it appears that there exists an index m such that a(n)=b(n) for all n>=m. For example, if we replace a(10) by 1341, which is a number within the prime gap 1327-1361, then this new sequence has b(17)=a(17) and so the two sequences agree after that point. - J. M. Bergot, May 21 2014.

			The sequence begins with a(0)=0, so |2-0|=2 and a(1)=1*2=2; find
the next m=|2-2|=0, so a(2)=0*2=0; find the next m=|2-0|=2, so a(3)=3*2=6; find the next m=|7-6|=1, so a(4)=1*4=4.

a(n+1) = n*A051699(a(n)), starting a(0)=0.

Edited by N. J. A. Sloane, May 20 2014

cp's OEIS Frontend

A132861 Smallest number at distance 3n from nearest prime (variant 2).

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A160666 Numbers whose distance to the closest prime number is a prime number.

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A193598 Even numbers k such that r(k) < r(k/2), where r(n) is the distance from n to the nearest prime.

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A334208 Number of partitions of 2n into two composite parts, (r,s), such that r and s have the same number of primes less than or equal to them.

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A079582 Least positive k such that the distance from k to closest prime = n.

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A079669 a(n) = least k such that the distance from Fibonacci(k) to the closest prime is n, or -1 if no such k exists.

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A242561 a(0)=0; thereafter, a(n) is n multiplied by the distance of a(n-1) to the nearest prime.

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