A073131 -id:A073131 - cp's OEIS frontend

A111410 Erroneous version of A073131.

2, 6, 6, 18, 6, 18, 10, 18, 24, 18, 32, 18, 12, 30, 28, 30, 12

Offset: 1

dead

A073130 a(n) = gcd(p(n+1) - p(n), p(p(n+1)) - p(p(n))), where p(n) is the n-th prime.

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 6, 2, 2, 4, 6, 6, 2, 6, 2, 2, 2, 2, 6, 8, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 4, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 2, 4, 2, 2, 2, 6, 6, 6, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 6, 2, 2, 2, 6, 4, 6, 6, 2, 6, 4, 2, 2, 2, 2, 2, 2, 6, 2, 6, 4, 2, 2, 4, 4, 2, 4, 2, 2, 2, 12, 2, 6, 2, 2, 2, 6, 2

Offset: 1

Labos Elemer, Jul 16 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A073131, A073132.

[GCD(NthPrime(n+1) - NthPrime(n), NthPrime(NthPrime(n+1)) - NthPrime(NthPrime(n))): n in [1..120]]; // G. C. Greubel, Oct 20 2019

seq(gcd(ithprime(n+1) - ithprime(n), ithprime(ithprime(n+1)) - ithprime(ithprime(n))), n=1..120); # G. C. Greubel, Oct 20 2019

Table[GCD[Prime[n+1]-Prime[n], Prime[Prime[n+1]]-Prime[Prime[n]]], {n, 120}]

vector(120, n, gcd(prime(n+1) - prime(n), prime(prime(n+1)) - prime(prime(n))) ) \\ G. C. Greubel, Oct 20 2019

[gcd(nth_prime(n+1) - nth_prime(n), nth_prime(nth_prime(n+1)) - nth_prime(nth_prime(n))) for n in (1..120)] # G. C. Greubel, Oct 20 2019

A073132 Smallest subscript j such that d=p[p(j+1)]-p[p(j)]=2n, or 0 if j does not exist (at d=4).

Original entry on oeis.org

1, 0, 2, 7, 5, 13, 4, 8, 6, 14, 12, 27, 9, 51, 11, 40, 21, 16, 25, 39, 36, 96, 58, 18, 132, 134, 56, 106, 108, 72, 34, 102, 42, 158, 202, 68, 53, 118, 121, 46, 124, 101, 383, 91, 157, 30, 80, 97, 204, 126, 258, 139, 381, 145, 222, 47, 62, 240, 242, 363, 66, 177, 565

Offset: 1

Labos Elemer, Jul 16 2002

nonn

			d=20 appears first at n=14, p(15)=47,p(14)=43, d=p(47)-p(43)=211-191=20, so a(20/2)=a(10)=14.

Cf. A073130, A073131.

t=Table[0, {100}]; Do[s = Prime[ Prime[x+1]] - Prime[ Prime[x]]; If[ s < 202 && t[[s/2]]==0, t[[s/2]]=n], {n, 1, 1000}]; t

a(n)=Min{x : A073131[x]=2n}

Edited by Robert G. Wilson v, Jul 17 2002

A098043 Primes of the form (prime(prime(k+1)) - prime(prime(k)))/2.

Original entry on oeis.org

3, 3, 7, 5, 13, 11, 3, 11, 7, 17, 19, 17, 31, 7, 37, 23, 61, 5, 19, 47, 31, 17, 29, 7, 5, 19, 41, 31, 41, 11, 79, 7, 7, 23, 37, 31, 13, 29, 47, 13, 83, 29, 13, 11, 59, 17, 23, 17, 11, 61, 5, 23, 83, 7, 7, 79, 5, 5, 31, 41, 61, 5, 29, 19, 19, 47, 67, 7, 13, 31, 29, 13, 137, 61, 53, 43

Offset: 1

Cino Hilliard, Sep 10 2004

Primes of the form A073131(k)/2. - Amiram Eldar, Jul 08 2024

			prime(prime(3)) - prime(prime(2)) = 6. 6/2 = 3 = first term.
prime(prime(4)) - prime(prime(3)) = 6. 6/2 = 3 = second term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006450, A098042, A073131.

With[{t = Table[Prime[Prime[n]], {n, 1, 400}]}, Select[(Rest[t] - Most[t])/2, PrimeQ]] (* Amiram Eldar, Jul 08 2024 *)

lista(n) = for(x=1,n,y=prime(prime(x+1)) - prime(prime(x)); if(y%2==0&isprime(y/2),print1(y\2",")))

Offset corrected by Amiram Eldar, Jul 08 2024

A299644 a(n) = prime(prime(n+1)) + prime(prime(n)).

Original entry on oeis.org

8, 16, 28, 48, 72, 100, 126, 150, 192, 236, 284, 336, 370, 402, 452, 518, 560, 614, 684, 720, 768, 832, 892, 970, 1056, 1110, 1150, 1186, 1216, 1326, 1448, 1512, 1570, 1656, 1736, 1796, 1886, 1958, 2022, 2094, 2150, 2240, 2324, 2372, 2418, 2514, 2706, 2842, 2880, 2918

Offset: 1

Vincenzo Librandi, Mar 20 2018

nonn

All terms are even. - Michel Marcus, Mar 20 2018

Cf. A000040, A006450, A073131.

[NthPrime(NthPrime(n+1))+NthPrime(NthPrime(n)):n in [1..50]];

Table[Prime[Prime[n + 1]] + Prime[Prime[n]], {n, 1, 50}]

a(n) = prime(prime(n+1)) + prime(prime(n)); \\ Michel Marcus, Mar 20 2018

a(n) = A006450(n+1) + A006450(n).

A378677 a(n)=a(n-1) + prime(n) for n prime, and a(n)=-a(n-1) otherwise, with a(0)=0, with duplicates removed afterwards.

Original entry on oeis.org

0, 3, 8, -8, -3, 14, -14, 17, -17, 24, -24, 35, -35, 32, -32, 51, -51, 58, -58, 69, -69, 88, -88, 91, -91, 100, -100, 111, -111, 130, -130, 147, -147, 136, -136, 195, -195, 158, -158, 209, -209, 192, -192, 239, -239, 222, -222, 287, -287, 260, -260, 303, -303

Offset: 0

Bill McEachen, Dec 03 2024

sign

Let b = subset of positive terms for n>4. We have A073131= b(m+2)-b(m) , A006450= b(m+2)+b(m) and A299644= b(m+2)+2*b(m+1)+b(m).

			n=1 is not prime, so a(1)= -a(0)= 0. n=2 is prime, so a(2)=a(1)+prime(2)=0+3=3. n=5 is prime, so a(5)=3, but note that it duplicates a(2). n=6 is not prime, so a(6)= -a(5)=-3. After terms are computed, duplicates are only then removed, which will alter indices accordingly.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A006450, A073131, A299644.

Module[{n = 0}, DeleteDuplicates[NestList[If[PrimeQ[++n], # + Prime[n], -#] &, 0, 200]]] (* Paolo Xausa, Dec 06 2024 *)

a(n) = a(n-1) + a prime for n odd >4.

a(n) = -a(n-1) for a(n-1)>0, n>1.

cp's OEIS Frontend

A111410 Erroneous version of A073131.

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A073130 a(n) = gcd(p(n+1) - p(n), p(p(n+1)) - p(p(n))), where p(n) is the n-th prime.

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A073132 Smallest subscript j such that d=p[p(j+1)]-p[p(j)]=2n, or 0 if j does not exist (at d=4).

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A098043 Primes of the form (prime(prime(k+1)) - prime(prime(k)))/2.

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A299644 a(n) = prime(prime(n+1)) + prime(prime(n)).

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A378677 a(n)=a(n-1) + prime(n) for n prime, and a(n)=-a(n-1) otherwise, with a(0)=0, with duplicates removed afterwards.

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