A065314 -id:A065314 - cp's OEIS frontend

A065315 Smallest prime divisor of n-th primorial + (n+1)-st prime.

5, 11, 37, 13, 23, 30047, 510529, 9699713, 127, 107, 433, 1093, 375569, 13082761331670077, 941879, 32589158477190044789, 1922760350154212639131, 4129, 92388407, 5879, 40729680599249024150621323549, 1783, 4903, 10279098043, 191, 131, 109, 163, 337, 20261, 673327, 6599, 181

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			For n=3, 3rd primorial=30, prime(4)=7, sum=37, so a(3)=37.

Tyler Busby, Table of n, a(n) for n = 1..84 (terms 1..79 from Sean A. Irvine)
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012-2023. - From N. J. A. Sloane, Jun 13 2012

Cf. A002110, A000040, A006530, A057713, A002584, A002585, A051342, A060881.

A065316 Largest prime divisor of n-th primorial - (n+1)-st prime.

Original entry on oeis.org

23, 199, 2297, 30013, 12451, 9699667, 79139, 122069683, 9241993, 77184383, 211941187, 72280449346243, 73629553, 142226610221, 131076443530861861, 382046844818915214929, 1348764323657, 1822793973448088839487, 217379667530071, 3217644767340672907899084554047, 267064515689275851355624017992701

Offset: 3

Labos Elemer, Oct 29 2001

nonn

			For n=3, 3rd primorial=30, 4th prime=7, difference=23, so a(3)=23.

Tyler Busby, Table of n, a(n) for n = 3..89 (terms 3..80 from Sean A. Irvine)
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A002110, A000040, A006530, A006530, A057713, A002584, A002585, A051342.

A065317 Largest prime divisor of the sum of the n-th primorial and the (n+1)-st prime.

Original entry on oeis.org

5, 11, 37, 17, 101, 30047, 510529, 9699713, 1427, 76829, 789077, 659863, 810104837, 13082761331670077, 652833094897, 32589158477190044789, 1922760350154212639131, 28406001782370300553, 770555057, 94904036422299534098897, 40729680599249024150621323549

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			For n = 4, 4th primorial = 210, prime(5) = 11, sum = 210 + 11 = 13 * 17, a(4) = 17.

Tyler Busby, Table of n, a(n) for n = 1..84 (terms 1..68 from Daniel Suteu, terms 69..79 from Sean A. Irvine)
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A002110, A000040, A060881, A006530, A057713, A002584, A002585, A051342.

A065610 Smallest number m so that n^2 + A000330(m) is also a square, i.e., n^2 + (1 + 4 + 9 + 16 + ... + m^2) = w^2 for some w.

Original entry on oeis.org

1, 47, 2, 5, 767, 16, 1727, 22, 17, 13, 18, 112, 10, 70, 8, 10799, 12287, 21, 82, 17327, 31, 15, 255, 16, 10, 13, 9, 5, 49, 40367, 43199, 117, 17, 1630, 7, 58799, 10, 65711, 34, 73007, 49, 13, 64, 29, 17, 6, 9, 30, 42, 309, 8, 124847, 17, 31, 139967, 13, 150527, 15

Offset: 0

Labos Elemer, Nov 07 2001

nonn

I.e., a(n) is the least solution to n^2 + (x(x+1)(2x+1)/6) = w^2; a(n) is the length of shortest sum of consecutive squares from 1 to a(n) which when added to n^2 gives a new square.

			n = 3: a(3) = 5 because n^2 + 1 + 4 + 9 + 16 + 25 = 9 + (1 + 4 + 9 + 16 + 25) = 64 = 8*8; n = 4: a(4) = 767 because n^2 + (1 + 4 + ... + 767^2) = 150700176 = 12276*12726, where 767 is the length of the shortest such consecutive-square sequence which provides (when summed) a new square, namely 12276^2. Often the least solution is rather large. E.g., at n = 93, a(n) = 415151, which means that 93^2 + A000330(415151) = 8649 + (long square sum) = 154436265^2 = 23850559947150225 is the smallest such square number, sum odd distinct consecutive squares except one repetition(8649).

Cf. A000330, A065311, A065312, A065313, A065314, A065315.

s=n^2 Do[s=s+m^2; If[IntegerQ[Sqrt[s]], Print[m]], {m, 1, 500000}] (* gives solutions of which the smallest is entered into the sequence *)

n^2 + (1 + 4 + 9 + ... + a(n)^2) = w^2, where w depends also on n; i.e., sum of consecutive squares from 1, 4, ... to a(n)^2 + n^2 is also a square.

A309671 Primes prime(m) such that G = prime(m-1)# - prime(m) is prime.

Original entry on oeis.org

7, 11, 13, 17, 23, 83, 89, 97, 151, 373, 433, 857, 4013, 8821, 12959

Offset: 1

Mohamed Sami Gattoufi, Aug 11 2019

G = prime(n-1)# - prime(n) where G is a prime is a special case of A090188 where (k=1).

			7 is a term because 23 = 2*3*5 - 7 is prime.

Cf. A002110, A065314, A060882, A096649, A090188, A065316.

primo(p) = my(ip=primepi(p)); factorback(primes(ip)); \\ A002110
isok(p) = isprime(p) && isprime(primo(precprime(p-1)) - p);

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A065315 Smallest prime divisor of n-th primorial + (n+1)-st prime.

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A065316 Largest prime divisor of n-th primorial - (n+1)-st prime.

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A065317 Largest prime divisor of the sum of the n-th primorial and the (n+1)-st prime.

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A065610 Smallest number m so that n^2 + A000330(m) is also a square, i.e., n^2 + (1 + 4 + 9 + 16 + ... + m^2) = w^2 for some w.

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A309671 Primes prime(m) such that G = prime(m-1)# - prime(m) is prime.

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PARI