A062042 -id:A062042 - cp's OEIS frontend

A062044 Primes arising in A062042.

2, 5, 7, 11, 17, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 89, 97, 101, 103, 107, 113, 127, 139, 149, 163, 173, 179, 181, 191, 211, 223, 227, 233, 239, 251, 263, 269, 277, 283, 293, 307, 313, 317, 331, 347, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419

Offset: 1

Amarnath Murthy, Jun 05 2001

nonn

			7 and 10 are two consecutive terms of A062042 and their sum 7+10 = 17 is prime and is a member.

Cf. A062042.

More terms from Larry Reeves (larryr(AT)acm.org), Jun 07 2001

A073627 a(1)=a(2)=1; for n > 2, a(n) is the smallest integer such that a(n) > a(n-1) and a(n)+a(n-1) is prime.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 13, 16, 21, 22, 25, 28, 31, 36, 37, 42, 47, 50, 51, 52, 55, 58, 69, 70, 79, 84, 89, 90, 91, 100, 111, 112, 115, 118, 121, 130, 133, 136, 141, 142, 151, 156, 157, 160, 171, 176, 177, 182, 185, 188, 191, 192, 197, 200, 201, 208, 211, 220, 223

Offset: 1

Amarnath Murthy, Aug 08 2002

nonn

Essentially the same as A062042. [John W. Layman, Oct 11 2013]

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

Cf. A073628.

s=1; Join[{1,1}, Table[k=s+1; While[ !PrimeQ[k+s], k++ ]; s=k, {100}]] (* T. D. Noe, Nov 02 2009 *)

FirstTerms(n)={my(x=1,y,a=vector(n),j=2);a[1]=1;a[2]=1;while(j++<=n,y=x+1;while(!isprime(x+y),y++);x=y;a[j]=y);return(a)} \\ R. J. Cano, Jan 18 2017

Edited by Matthew Conroy, Oct 21 2002

Definition corrected by T. D. Noe, Nov 02 2009

A214362 Arithmetic mean of next a(n) successive positive integers is prime.

Original entry on oeis.org

3, 3, 1, 7, 5, 7, 5, 11, 1, 7, 5, 7, 9, 3, 9, 11, 5, 3, 1, 7, 5, 23, 1, 19, 9, 11, 1, 3, 17, 23, 1, 7, 5, 7, 17, 7, 5, 11, 1, 19, 9, 3, 5, 23, 9, 3, 9, 7, 5, 7, 1, 11, 5, 3, 13, 7, 17, 7, 5, 11, 1, 7, 17, 7, 9, 11, 13, 27, 5, 7, 5, 11, 9, 3, 9, 7, 5, 3, 1, 23, 1

Offset: 1

Alex Ratushnyak, Jul 18 2012

Corresponding primes: A062044(n).

All terms are odd. - Robert Israel, Jan 17 2017

			(1+2+3)/3 = 2 is prime, so a(1)=3,
then (4+5+6)/3 = 5 is prime, so a(2)=3,
then 7/1 = 7 is prime, so a(3)=1,
then (8+9+10+11+12+13+14)/7 = 11 is prime, so a(4)=7.

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

Cf. A000040, A062042, A062044, A213815.

t:= 1:
for n from 1 to 100 do
  for s from 1 by 2 do
    if isprime((2*t + s - 1)/2) then
      A[n]:= s; t:= t+s; break
    fi
od od:
seq(A[n],n=1..100); # Robert Israel, Jan 17 2017

firstTerms(n)={
my(k=1,j=1,x,y,a=vector(n));
while(j<=n,x=0;while(!isprime(k+x),x++);y=2*x+1;k+=y;a[j]=y;j++);
return(a)} \\ R. J. Cano, Jan 17 2017

a(n) = 2*(A062042(n)-A062042(n-1))+(-1)^n for n >= 2. - Robert Israel, Jan 17 2017

A215099 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 18, 24, 25, 29, 34, 38, 39, 41, 44, 48, 53, 55, 56, 58, 71, 73, 78, 84, 85, 89, 94, 102, 103, 109, 120, 124, 131, 133, 138, 144, 145, 149, 162, 164, 169, 173, 178, 180, 181, 187, 192, 196, 197, 201

Offset: 0

Alex Ratushnyak, Aug 03 2012

nonn

For n>0 and (n mod 4)<2, a(n) is odd.
Same definition, but k+a(n-2) is a
Fibonacci number: A006498 except first two terms,
Lucas number: A000045 except first two terms,
Pell number: A089928(n-1),
Jacobsthal number: A215095,
factorial: A215096,
square: A194274,
cube: A215097,
triangular number: A011848(n+2),
oblong number: A215098.
Example of a related sequence definition: a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a cube.

Iain Fox, Table of n, a(n) for n = 0..10000

Cf. A062042: a(1) = 2, a(n) = least k>a(n-1) such that k+a(n-1) is a prime.

Cf. A073627, A073628.

first(n) = my(res = vector(n, i, i-1), k); for(x=3, n, k=res[x-1]+1; while(!isprime(k+res[x-2]), k++); res[x]=k); res \\ Iain Fox, Apr 22 2019 (corrected by Iain Fox, Apr 25 2019)

from sympy import prime
prpr = 0
prev = 1
for n in range(77):
    print(prpr, end=', ')
    b = c = 0
    while c<=prev:
        c = prime(b+1) - prpr
        b+=1
    prpr = prev
    prev = c

A231507 a(n) is smallest number greater than a(n-1) such that a(n)+a(n-1) is composite.

Original entry on oeis.org

4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 46, 47, 48, 50, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 65, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 80, 81, 83, 85, 86, 88, 89, 91, 92, 93, 94, 95, 97

Offset: 1

Neil Fernandez, Nov 09 2013

nonn

			a(1) = 4, the first composite. So the smallest a(2) could possibly be 5. 4+5=9, which is composite, so a(2) = 5. a(3) cannot be 6, because 5+6=11, which is prime. But 5+7=12 is composite, so a(3) = 7.

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

Cf. A062042, A073627.

nxt[n_]:=Module[{k=n+1},While[PrimeQ[n+k],k++];k]; NestList[nxt,4,70] (* Harvey P. Dale, Jul 18 2014 *)

A107817 Slowest increasing sequence where 2 consecutive integers sum up to a prime.

Original entry on oeis.org

0, 2, 3, 4, 7, 10, 13, 16, 21, 22, 25, 28, 31, 36, 37, 42, 47, 50, 51, 52, 55, 58, 69, 70, 79, 84, 89, 90, 91, 100, 111, 112, 115, 118, 121, 130, 133, 136, 141, 142, 151, 156, 157, 160, 171, 176, 177, 182, 185, 188, 191, 192, 197, 200, 201, 208, 211, 220, 223, 226

Offset: 0

Eric Angelini, Jun 11 2005

Essentially the same as A073627. [R. J. Mathar, Aug 24 2008]

Essentially the same as A062042. [Zak Seidov, Nov 04 2009]

			0+2=2, which is a prime; 2+3=5=prime; 3+4=7=prime; 4+7=11=prime, etc.

k = 0; Print[k]; Do[p = k + 1; While[ !PrimeQ[k + p], p++ ]; k = p; Print[k], {n, 1, 100}] (* Ryan Propper, Sep 04 2005 *)

More terms from Ryan Propper, Sep 04 2005

A330248 a(1) = 1; for n > 1, a(n) is the least nonnegative number such that a(n) + a(n-1) + n is a prime number.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 3, 0, 2, 1, 1, 0, 0, 3, 5, 2, 4, 1, 3, 0, 2, 1, 1, 0, 0, 5, 3, 0, 2, 3, 1, 2, 0, 1, 1, 0, 0, 3, 5, 2, 4, 1, 3, 0, 2, 5, 1, 4, 0, 3, 1, 0, 0, 1, 5, 0, 4, 3, 3, 2, 2, 1, 1, 0, 0, 1, 5, 0, 4, 3, 3, 2, 2, 1, 1, 0, 0, 5, 7, 4, 6

Offset: 1

Ali Sada, Dec 06 2019

nonn

The primes that result from this sequence are 3, 3, 5, 7, 7, 7, 11, 13, 11, 11, 13, 17, 17, 17, 19, 19, 19, 19, 23, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 37, 41, 37, 37, 41, ...

			When n=5, a(4)=1; we want a(5)+a(4)+5 to be a prime. 1 is the least nonnegative number that satisfies this condition (1+5+1=7). So, a(5)=1.

Cf. A062042.

Nest[Append[#1, Block[{k = 0}, While[! PrimeQ[#1[[-1]] + k + #2], k++]; k]] & @@ {#, Length@ # + 1} &, {1}, 105] (* Michael De Vlieger, Dec 14 2019 *)

for (n=1, 87, print1 (v=if (n==1, 1, nextprime(n+v)-n-v)", ")) \\ Rémy Sigrist, Dec 06 2019

cp's OEIS Frontend

A062044 Primes arising in A062042.

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A073627 a(1)=a(2)=1; for n > 2, a(n) is the smallest integer such that a(n) > a(n-1) and a(n)+a(n-1) is prime.

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A214362 Arithmetic mean of next a(n) successive positive integers is prime.

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A215099 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is prime.

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A231507 a(n) is smallest number greater than a(n-1) such that a(n)+a(n-1) is composite.

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A107817 Slowest increasing sequence where 2 consecutive integers sum up to a prime.

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A330248 a(1) = 1; for n > 1, a(n) is the least nonnegative number such that a(n) + a(n-1) + n is a prime number.

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