A063444 -id:A063444 - 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)).

A084307 a(n) is the least number x such that gcd(sigma(x), sigma(x+1)) = n.

Original entry on oeis.org

1, 13, 17, 6, 199, 5, 242, 27, 391, 57, 1296, 22, 882, 12, 648, 93, 175232, 89, 3872, 236, 195, 1032, 4875263, 14, 5739271, 467, 35377, 83, 1882384, 58, 3024, 308, 29240, 201, 1627208, 118, 79524, 147, 1682, 56, 46854024, 82, 229441, 1204, 2047, 6301, 83386957823

Offset: 1

Labos Elemer, Jun 13 2003

nonn

a(47) > 10^9 if it exists. - Michel Marcus, Sep 01 2019

			n=9: GCD[sigma[x+1], sigma[x]]=5 holds for {391,799,800,881,...} of which the first is a(9)=391.

Cf. A000203, A060780, A063444.

f[x_] := GCD[DivisorSigma[1, x], DivisorSigma[1, x+1]] t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}]; t

a(n) = my(x=1, sx=sigma(x), y=2, sy=sigma(2)); while(gcd(sx, sy) != n, x++; y++; sx=sy; sy=sigma(y)); x; \\ Michel Marcus, Aug 28 2019

a(41) from Rémy Sigrist, Aug 29 2019
a(42)-a(46) from Michel Marcus, Aug 30 2019
a(47) from Giovanni Resta, Oct 29 2019

cp's OEIS Frontend

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

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