A084309 -id:A084309 - cp's OEIS frontend

A084316 a(n) is the smallest number x such that gcd(prime(x)+1,x+1) = n.

1, 3, 20, 11, 24, 5, 6, 39, 98, 29, 120, 23, 64, 13, 104, 15, 1716, 323, 284, 499, 62, 1099, 1264, 215, 1274, 51, 512, 447, 1768, 209, 1332, 31, 32, 373, 34, 1475, 258, 835, 2300, 519, 5780, 419, 5374, 1275, 6974, 1655, 6626, 479, 10240, 10549, 3008, 883, 13938

Offset: 1

Labos Elemer, Jun 13 2003

nonn

a(n) == n+1 (mod 2). - Robert Israel, May 04 2017

			In A066752, n=5 arises first at the 24th position, so a(5)=24.

Ivan Neretin, Table of n, a(n) for n = 1..4000 (first 2880 terms from Robert Israel)

Cf. A000040, A066752, A084309-A084315.

f:= proc(n) local x;
  for x from n-1 by ilcm(n,2) do
    if igcd(x+1, ithprime(x)+1) = n then return x fi
  od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, May 04 2017

f[x_]:=GCD[Prime[x]+1,x+1]; t=Table[0,{256}];Do[s=f[n];If[s<257&&t[[s]] == 0,t[[s]] = n],{n,1,100000}];t (* edited by Harvey P. Dale, Jan 28 2023 *)
Module[{nn=20000,t},t=Table[{x,GCD[Prime[x]+1,x+1]},{x,nn}];Table[SelectFirst[t,#[[2]]==n&],{n,60}]][[All,1]] (* Harvey P. Dale, Jan 28 2023 *)

a(n) = Min{x; A066752(x)=n}.

A084310 a(n) = gcd(prime(n)+1, n).

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 1, 4, 3, 10, 1, 2, 1, 2, 3, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 1, 6, 1, 4, 3, 2, 5, 4, 1, 2, 3, 2, 1, 14, 1, 2, 9, 2, 1, 16, 1, 10, 3, 4, 1, 18, 1, 8, 3, 2, 1, 6, 1, 2, 7, 8, 1, 6, 1, 2, 3, 70, 1, 72, 1, 2, 5, 4, 1, 2, 1, 10, 3, 2, 1, 14, 5, 2, 3, 2, 1, 2, 13, 4, 1, 2, 5, 24, 1, 2, 1, 2

Offset: 1

Labos Elemer, Jun 13 2003

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A084309, A084311, A084312, A084313, A084314, A084315, A084316.

A084310:= n-> igcd(ithprime(n)+1,n): seq(A084310(n), n=1..150); # Wesley Ivan Hurt, Jan 27 2017

Mathematica
```
Table[GCD[Prime[w]+1, w], {w, 128}]
```

a(n) = gcd(prime(n)+1, n); \\ Michel Marcus, Jan 27 2017

A084311 a(n) = gcd(prime(n)-1,n-1).

Original entry on oeis.org

1, 1, 2, 3, 2, 1, 2, 1, 2, 1, 10, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 24, 25, 2, 1, 4, 1, 6, 1, 8, 3, 2, 5, 12, 1, 2, 1, 2, 1, 2, 1, 4, 9, 2, 1, 2, 1, 2, 17, 4, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 8, 1, 66, 1, 2, 3, 2, 1, 6, 1, 2, 1, 4, 11, 2, 1, 2, 3, 2, 1, 6, 17, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 9, 2, 1

Offset: 1

Labos Elemer, Jun 13 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000040, A084309-A084316.

Table[GCD[Prime[w]-1, w-1], {w, 1, 128}]

a(n) = gcd(prime(n)-1, n-1); \\ Michel Marcus, May 04 2017

A084315 a(n) = x is the smallest number such that gcd(prime(x)-1,x-1) = n.

Original entry on oeis.org

1, 3, 4, 13, 36, 19, 120, 33, 46, 11, 78, 37, 560, 239, 496, 1905, 52, 397, 3250, 221, 778, 573, 2738, 25, 26, 287, 352, 477, 552, 271, 8588, 449, 5710, 205, 456, 145, 1000, 4067, 2536, 761, 14638, 295, 4344, 2685, 406, 1151, 612, 3121, 6616, 551, 868, 469, 20088

Offset: 1

Labos Elemer, Jun 13 2003

nonn

			In A084311, n=4 arises first at the 13th position, so a(4)=13.

Ivan Neretin, Table of n, a(n) for n = 1..4000

Cf. A000040, A084309-A084316.

f[x_] := GCD[Prime[x]-1, x-1] t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t

a(n) = Min{x; A084311(x)=n}.

A066752 a(n) = gcd(prime(n)+1, n+1).

Original entry on oeis.org

1, 1, 2, 1, 6, 7, 2, 1, 2, 1, 4, 1, 14, 1, 16, 1, 6, 1, 4, 3, 2, 1, 12, 5, 2, 3, 4, 1, 10, 1, 32, 33, 2, 35, 6, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 4, 7, 2, 1, 26, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 21, 4, 13, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 76, 1, 78, 1, 2, 1, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 2, 9, 4

Offset: 1

Reinhard Zumkeller, Jan 16 2002

nonn

			a(23) = gcd(prime(23)+1,23+1) = gcd(83+1,24) = gcd(84,24) = 12*gcd(7,2) = 12.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040.

Cf. A084309-A084316.

Table[GCD[Prime[w]+1, w+1], {w, 1, 128}]

{ for (n=1, 1000, write("b066752.txt", n, " ", gcd(prime(n) + 1, n + 1)) ) } \\ Harry J. Smith, Mar 22 2010

More terms from Labos Elemer, Jun 13 2003

A084313 a(n)=x is the smallest number such that gcd(prime(x)-1, x) = n.

Original entry on oeis.org

1, 2, 21, 16, 5, 6, 315, 24, 63, 20, 121, 12, 65, 14, 105, 320, 697, 306, 2185, 60, 399, 286, 299, 216, 575, 156, 513, 644, 1189, 210, 837, 384, 231, 374, 1505, 684, 3515, 1026, 1131, 1480, 2747, 966, 1591, 1012, 1935, 782, 1645, 1776, 30429, 7600, 3009

Offset: 1

Labos Elemer, Jun 13 2003

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 4000 terms from Ivan Neretin)

Cf. A084309-A084316.

f[x_] := GCD[Prime[x]-1, x] t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t
Table[Module[{x=1},While[GCD[Prime[x]-1,x]!=n,x++];x],{n,60}] (* Harvey P. Dale, Aug 23 2025 *)

a(n) = Min{x; A084309(x)=n}.

A084314 a(n) = x is the smallest number such that gcd(prime(x)+1, x) = n.

Original entry on oeis.org

1, 2, 3, 4, 35, 24, 63, 56, 45, 10, 649, 132, 91, 42, 495, 48, 153, 54, 779, 700, 1785, 264, 851, 96, 1125, 286, 1647, 672, 551, 570, 341, 448, 2277, 1462, 385, 144, 1369, 418, 2145, 1000, 4797, 294, 817, 1804, 405, 414, 7003, 2016, 11515, 950, 2193, 2444

Offset: 1

Labos Elemer, Jun 13 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..4000

Cf. A084309-A084316.

f[x_] := GCD[Prime[x]+1, x] t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t
Module[{cc=Table[{x,GCD[Prime[x]+1,x]},{x,20000}]},Transpose[Table[ SelectFirst[ cc,#[[2]]==n&],{n,60}]][[1]]] (* Harvey P. Dale, Nov 28 2014 *)

a(n) = Min{x; A084310(x)=n}.

A180172 a(n) = gcd(prime(n)+2, n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 9, 1, 1, 13, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 11, 1, 1, 1, 1, 71, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Zak Seidov, Aug 15 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003989, A052147, A084309, A084310, A180173.

[GCD(n,NthPrime(n) +2): n in [1..110]]; // G. C. Greubel, Mar 12 2023

Mathematica
```
Table[GCD[n,Prime[n]+2],{n,200}]
```

[gcd(nth_prime(n) + 2, n) for n in range(1,111)] # G. C. Greubel, Mar 12 2023

A180173 a(n) = gcd(prime(n)-2, n).

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 1, 15, 1, 1, 1, 1, 1, 1, 11, 1, 3, 5, 1, 1, 7, 1, 3, 1, 1, 3, 1, 7, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 69, 1, 1, 3, 73, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3

Offset: 1

Zak Seidov, Aug 15 2010

nonn

Cf. A084309, A084310, A180172.

Mathematica
```
Table[GCD[n,Prime[n]-2],{n,200}]
```

cp's OEIS Frontend

A084316 a(n) is the smallest number x such that gcd(prime(x)+1,x+1) = n.

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A084310 a(n) = gcd(prime(n)+1, n).

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A084311 a(n) = gcd(prime(n)-1,n-1).

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A084315 a(n) = x is the smallest number such that gcd(prime(x)-1,x-1) = n.

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A066752 a(n) = gcd(prime(n)+1, n+1).

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A084313 a(n)=x is the smallest number such that gcd(prime(x)-1, x) = n.

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A084314 a(n) = x is the smallest number such that gcd(prime(x)+1, x) = n.

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Keywords

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Crossrefs

Programs

Mathematica

Formula

A180172 a(n) = gcd(prime(n)+2, n).

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