A084307 -id:A084307 - cp's OEIS frontend

A067605 Least k such that gcd(prime(k+1)-1, prime(k)-1) = 2n.

2, 6, 11, 24, 42, 121, 30, 319, 99, 1592, 344, 574, 3786, 4196, 650, 4619, 217, 1532, 11244, 5349, 8081, 3861, 12751, 18281, 9221, 5995, 22467, 16222, 43969, 35975, 192603, 108146, 52313, 218234, 15927, 132997, 42673, 78858, 103865, 84483, 111172, 175288, 110734

Offset: 1

Robert G. Wilson v, Jan 31 2002

nonn

Since all consecutive primes, p < q and p greater than 2, are odd, therefore gcd(p-1, q-1) must be even.

			For n = 4: a(4) = 24 = gcd(89-1, 97-1) = gcd(p(24)-1, p(25)-1) = 8 = 2*4.

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A000720, A063444, A084307, A058263, A058264.

N:= 50: # for a(1)..a(N)
V:= Vector(N): count:= 0:
p:= 3:
for k from 2 while count < N do
  q:= p;
  p:= nextprime(p);
  v:= igcd(p-1,q-1)/2;
  if v <= N and V[v] = 0 then
    count:= count+1; V[v]:= k;
  fi
od:
convert(V,list); # Robert Israel, Mar 05 2025

a = Table[0, {100}]; p = 3; q = 5; Do[q = Prime[n + 1]; d = GCD[p - 1, q - 1]/2; If[d < 101 && a[[d]] == 0, a[[d]] = n]; b = c, {n, 2, 10^7}]; a

list(len) = {my(v = vector(len), c = 0, p = 3, k = 2, i); forprime(q = 5, , i = gcd(p-1, q-1)/2; if(i <= len && v[i] == 0, v[i] = k; c++; if(c == len, break)); p = q; k++); v;} \\ Amiram Eldar, Mar 05 2025

a(n) = PrimePi(A058264(n)).

A322569 a(n)=x is the least integer such that gcd(sigma(x), sigma(x+1)) = 2*n.

Original entry on oeis.org

13, 6, 5, 27, 57, 22, 12, 93, 89, 236, 1032, 14, 467, 83, 58, 308, 201, 118, 147, 56, 82, 1204, 6301, 69, 596, 1142, 106, 91, 4167, 87, 432, 381, 393, 1407, 348, 70, 5912, 453, 233, 417, 13692, 166, 56493, 1118, 88, 6987, 54048, 154, 1843, 4490, 6833, 2574, 633, 689, 1538

Offset: 1

Michel Marcus, Aug 29 2019

nonn

Bisection of A084307.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A000203, A060780, A084307.

sol:=[]; for n in [1..55] do k:=1; while Gcd(DivisorSigma(1,k),DivisorSigma(1,k+1)) ne 2*n do k:=k+1; end while; Append(~sol,k); end for; sol; // Marius A. Burtea, Aug 29 2019

Module[{nn=60000,g},g=GCD@@@Partition[DivisorSigma[1,Range[nn]],2,1];Table[ Position[ g,2n,1,1],{n,55}]]//Flatten (* Harvey P. Dale, Jan 28 2023 *)

a(n) = my(x=1); while(gcd(sigma(x), sigma(x+1)) != 2*n, x++); x;

A364890 Least number k such that A060778(k) = n.

Original entry on oeis.org

1, 2, 49, 14, 80, 44, 529983, 104, 16640, 2511, 8212890624, 735, 1019423412224, 29888, 600624, 2295, 54020648488730624, 6075, 3018417549254328320, 5264, 123200, 24151040, 3264402128528250685620224, 5984, 1753599375, 689278976, 2310399, 156735, 27965083137654166225393025024, 180224, 11404289746101879774056466612224, 21735, 170853262335, 2035980763136, 207593229375, 223244

Offset: 1

Seiichi Manyama, Aug 12 2023

nonn

Cf. A000005, A058074, A060778, A084307, A364903.

a(n) = my(x=1, nx=1, ny=2); while(gcd(nx, ny) != n, x++; nx=ny; ny=numdiv(x+1)); x;

a(11),a(13),a(17),a(19),a(23)-a(36) from Max Alekseyev, Feb 18 2024

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A067605 Least k such that gcd(prime(k+1)-1, prime(k)-1) = 2n.

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A322569 a(n)=x is the least integer such that gcd(sigma(x), sigma(x+1)) = 2*n.

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A364890 Least number k such that A060778(k) = n.

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