A066753 -id:A066753 - cp's OEIS frontend

A066805 a(n) is the least k such that n + Sum_{i=1..k} A018252(i) is prime; or 0 if none exists.

1, 1, 5, 1, 6, 1, 8, 2, 5, 1, 7, 1, 5, 2, 5, 1, 7, 1, 5, 3, 6, 1, 6, 2, 5, 2, 13, 1, 6, 1, 5, 2, 5, 4, 6, 1, 8, 2, 5, 1, 6, 1, 5, 10, 5, 1, 7, 2, 8, 3, 5, 1, 7, 2, 5, 2, 7, 1, 6, 1, 5, 2, 6, 4, 6, 1, 8, 2, 5, 1, 6, 1, 5, 2, 5, 27, 7, 1, 5, 11, 5, 1, 15, 2, 5

Offset: 1

Joseph L. Pe, Jan 19 2002

nonn

Is a(n) nonzero for all n? If so, then every n can be represented as the difference of a prime and a partial sum of the nonprime numbers series. See A066753 for a related possible representation of n as the difference of a prime and a partial sum of the prime numbers series.

			3 + (1 + 4 + 6 + 8 + 9) = 31, a prime and 5 consecutive nonprime numbers starting with 1 are required to achieve this. Hence a(3) = 5.

Cf. A018252, A066753.

a(n) = my(c=0, s=n); for(k=1, oo, until(!isprime(c), c++); if(isprime(s+=c), return(k))); \\ Jinyuan Wang, Jul 30 2020

More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2002

A066818 a(n) is the least k such that n + Sum_{i=1..k} A005384(i) is prime; or 0 if none exists.

Original entry on oeis.org

1, 2, 1, 12, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 4, 5, 2, 7, 2, 1, 6, 1, 6, 3, 2, 3, 6, 1, 2, 3, 2, 1, 4, 1, 2, 3, 8, 1, 4, 11, 2, 3, 4, 1, 4, 5, 2, 13, 2, 1, 4, 1, 8, 3, 2, 3, 6, 1, 2, 7, 2, 1, 10, 1, 8, 3, 2, 15, 4, 1, 2, 3, 4, 1, 4, 5, 2, 7, 4

Offset: 1

Joseph L. Pe, Jan 19 2002

nonn

There is some empirical evidence to suggest a(n) is nonzero for every n. That is, every n can be expressed as the difference between a prime and a partial sum of the Sophie Germain primes series. See A066753 for a similar conjecture.

			7 + (2 + 3 + 5) = 17, a prime and three consecutive Sophie Germain primes starting from 2, the first Sophie Germain prime, are needed to achieve this. So a(7) = 3.

Cf. A005384, A066753.

a(n) = my(p=0, s=n); for(k=1, oo, until(isprime(2*p+1), p=nextprime(p+1)); if(isprime(s+=p), return(k))); \\ Jinyuan Wang, Jul 30 2020

a(53) corrected by and more terms from Jinyuan Wang, Jul 30 2020

A066868 a(n) is the least k such that n + Sum_{i=1..k} A005385(i) is prime; or 0 if none exists.

Original entry on oeis.org

2, 1, 10, 5, 2, 1, 2, 1, 12, 5, 2, 1, 4, 1, 4, 5, 2, 1, 2, 3, 4, 7, 12, 1, 2, 1, 4, 7, 2, 3, 2, 1, 4, 5, 2, 1, 4, 1, 6, 17, 2, 1, 4, 3, 6, 5, 2, 1, 2, 3, 4, 11, 12, 1, 2, 1, 4, 5, 2, 3, 2, 1, 4, 5, 14, 1, 2, 1, 10, 5, 2, 7, 10, 1, 6, 7, 2, 1, 8, 3, 4, 7, 26, 1

Offset: 1

Joseph L. Pe, Jan 21 2002

Conjecture: a(n) is nonzero for all n, so every n can be represented as the difference between a prime and a partial sum of the safe primes series. See A066753 for a similar conjecture.

			4 + (5 + 7 + 11 + 23 + 47) = 97, a prime and 5 consecutive safe primes, starting from the first safe prime 5, are needed to achieve this. Hence a(4) = 5.

Cf. A005385, A066753.

a(n) = my(p=3, s=n); for(k=1, oo, until(isprime((p-1)/2), p=nextprime(p+1)); if(isprime(s+=p), return(k))); \\ Jinyuan Wang, Jul 30 2020

Offset changed to 1 by Jinyuan Wang, Jul 30 2020

cp's OEIS Frontend

A066805 a(n) is the least k such that n + Sum_{i=1..k} A018252(i) is prime; or 0 if none exists.

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A066818 a(n) is the least k such that n + Sum_{i=1..k} A005384(i) is prime; or 0 if none exists.

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A066868 a(n) is the least k such that n + Sum_{i=1..k} A005385(i) is prime; or 0 if none exists.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions