A058263 -id:A058263 - cp's OEIS frontend

A083555 Quotient of LCM of prime(n+1)-1 and prime(n)-1 and GCD of the same two numbers.

2, 2, 6, 15, 30, 12, 72, 99, 154, 210, 30, 90, 420, 483, 598, 754, 870, 110, 1155, 1260, 156, 1599, 1804, 132, 600, 2550, 2703, 2862, 756, 72, 4095, 4420, 4692, 5106, 5550, 650, 702, 6723, 7138, 7654, 8010, 342, 9120, 2352, 9702, 1155, 1295, 12543, 12882

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=25: prime(25)=97, prime(26)=101; a(25) = lcm(96,100)/gcd(96,100) = 2400/4 = 600.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006093, A083538..A083554, A051537, A058263.

P:= seq(ithprime(i),i=1..100):
seq(ilcm(P[i+1]-1,P[i]-1)/igcd(P[i+1]-1,P[i]-1),i=1..99); # Robert Israel, Jun 11 2017

f[x_] := Prime[x]-1 Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 1, 128}]
(* Second program: *)
Table[Apply[LCM[#1, #2]/GCD[#1, #2] &, Prime[n + {1, 0}] - 1], {n, 49}] (* Michael De Vlieger, Jun 11 2017 *)

first(n)=my(v=vector(n),p=2,k,g); forprime(q=3,, g=gcd(p-1,q-1); v[k++]=(p-1)*(q-1)/g^2; p=q; if(k==n, break)); v \\ Charles R Greathouse IV, Jun 11 2017

a(n) = lcm(A006093(n+1), A006093(n))/gcd(A006093(n+1), A006093(n));
a(n) = A083554(n)/A058263(n).
a(n) = A051537(A006093(n+1), A006093(n)). - Robert Israel, Jun 11 2017

A063091 Prime(n) such that gcd(1+prime(n+1), 1+prime(n)) = gcd(-1+prime(n+1), -1+prime(n)).

Original entry on oeis.org

2, 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 283, 311, 317, 337, 347, 419, 431, 461, 521, 547, 569, 577, 599, 617, 641, 659, 773, 787, 809, 821, 827, 857, 863, 881, 1019, 1031, 1049, 1061, 1091, 1129, 1151, 1153

Offset: 1

Labos Elemer, Aug 06 2001

nonn

			p=101 is here because gcd(102,104) = 2 = gcd(100,102).

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

Cf. A058263.

lst={};Do[p0=Prime[n];p1=Prime[n+1];If[GCD[p0-1,p1-1]==GCD[p0+1,p1+1],AppendTo[lst,p0]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 11 2010 *)
Transpose[Select[Partition[Prime[Range[200]],2,1],GCD[First[#]+1, Last[#]+1] == GCD[First[#]-1,Last[#]-1]&]][[1]] (* Harvey P. Dale, Jan 22 2012 *)

{ n=0; for (m=1, 10^9, if(gcd(prime(m+1) + 1, prime(m) + 1) == gcd(prime(m+1) - 1, prime(m) - 1), write("b063091.txt", n++, " ", prime(m)); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 17 2009

A063086 a(n) = gcd(1 + prime(n+1), 1 + prime(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 4, 6, 2, 2, 2, 2, 4, 6, 6, 2, 2, 4, 2, 2, 4, 6, 2, 2, 2, 4, 2, 2, 2, 4, 6, 2, 10, 2, 2, 2, 4, 6, 6, 2, 2, 2, 2, 2, 4, 4, 4, 2, 2, 6, 2, 2, 6, 6, 6, 2, 2, 2, 2, 2, 14, 4, 2, 2, 2, 2, 2, 2, 2, 6, 8, 2, 2, 4, 6, 2, 2, 2, 10, 2, 2, 2, 2, 4, 6, 2, 2, 2, 4, 12, 8, 4, 4, 4, 6, 6, 2, 2, 2, 2, 6

Offset: 1

Labos Elemer, Aug 06 2001

			n=34: gcd(1 + 139, 1 + 149) = 10 = a(34).

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

Cf. A058263.

GCD[First[#]+1,Last[#]+1]&/@ Partition[Prime[Range[110]],2,1] (* Harvey P. Dale, May 02 2012 *)

a(n)={gcd(1 + prime(n+1), 1 + prime(n))} \\ Harry J. Smith, Aug 17 2009

A083553 Product of prime(n+1)-1 and prime(n)-1.

Original entry on oeis.org

2, 8, 24, 60, 120, 192, 288, 396, 616, 840, 1080, 1440, 1680, 1932, 2392, 3016, 3480, 3960, 4620, 5040, 5616, 6396, 7216, 8448, 9600, 10200, 10812, 11448, 12096, 14112, 16380, 17680, 18768, 20424, 22200, 23400, 25272, 26892, 28552, 30616, 32040

Offset: 1

Labos Elemer, May 22 2003

nonn

The conductor of x*prime(n) + y*prime(n+1); that is, for all k >= a(n), there exist nonnegative integers x and y such that k = x*prime(n) + y*prime(n+1). - T. D. Noe, Sep 22 2004

			n=25: a(25) = (97-1)*(101-1) = 9600.

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 46.

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

Cf. A000040, A006093, A058263, A083538-A083555, A099407 (terms halved), A172042 [= A000010(a(n))], A256617.
One more than A037165.
Column 3 of A379010.

f[x_] := Prime[x]-1; Table[f[w+1]*f[w], {w, 1, 128}]

A083553(n) = ((prime(1+n)-1)*(prime(n)-1)); \\ Antti Karttunen, Dec 14 2024

a(n) = A006093(n+1)*A006093(n) = (prime(n+1)-1)*(prime(n)-1).
a(n) = A037165(n) + 1.
a(n) = 2*A099407(n). - Antti Karttunen, Dec 14 2024

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

Original entry on oeis.org

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)).

A083554 Least common multiple of prime(n+1)-1 and prime(n)-1.

Original entry on oeis.org

2, 4, 12, 30, 60, 48, 144, 198, 308, 420, 180, 360, 840, 966, 1196, 1508, 1740, 660, 2310, 2520, 936, 3198, 3608, 1056, 2400, 5100, 5406, 5724, 3024, 1008, 8190, 8840, 9384, 10212, 11100, 3900, 4212, 13446, 14276, 15308, 16020, 3420, 18240, 9408

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=25: a(25) = lcm(97-1, 101-1) = lcm(96,100) = 2400.

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

Cf. A006093, A083538-A083555, A058263.

f[x_] := Prime[x]-1; Table[LCM[f[w+1], f[w]], {w, 1, 128}]

a(n) = lcm(prime(n+1)-1, prime(n)-1); \\ Michel Marcus, Mar 15 2018

a(n) = lcm(A006093(n+1), A006093(n)) = lcm(prime(n+1)-1, prime(n)-1).

A058264 Smallest prime p of two consecutive primes, p < q, such that gcd( p-1, q-1 ) = 2n.

Original entry on oeis.org

3, 13, 31, 89, 181, 661, 113, 2113, 523, 13421, 2311, 4177, 35543, 39901, 4831, 44417, 1327, 12853, 119321, 52321, 82657, 36389, 136897, 203713, 95651, 59281, 255259, 178697, 531919, 427621, 2640581, 1414849, 643303, 3021173, 175141, 1770337, 514967, 1004797, 1354393

Offset: 1

Labos Elemer, Dec 06 2000

nonn

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

			a(4) = 89 because gcd(89-1, 97-1) = gcd(8*11, 8*16) = 8 = 2*4 and these primes are the smallest with this property.
a(49) = 604073 because gcd(604073-1, 604171-1) = gcd(6164*98, 6165*98) = 98 = 2*49.

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

Cf. A006093, A058263, A067605.

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
With[{tsp={#[[1]],#[[2]],GCD[#[[1]]-1,#[[2]]-1]}&/@Partition[Prime[ Range[ 300000]],2,1]}, Transpose[Flatten[Table[Select[tsp, Last[#]==2n&,1],{n,40}],1]][[1]]] (* Harvey P. Dale, Jul 07 2013 *)

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

a(n) = prime(A067605(n)). - Amiram Eldar, Mar 05 2025

Edited by Robert G. Wilson v, Feb 01 2002

A080373 a(n) is the smallest number k such that GCD of n values of prime(j)-1 for successive j values starting with k is greater than 2, where prime(j) = j-th prime.

Original entry on oeis.org

3, 6, 24, 77, 271, 271, 1395, 1395, 1395, 13717, 34369, 172146, 172146, 804584, 804584, 804584, 12762142, 16138563, 16138563, 56307979, 56307979, 56307979, 56307979, 56307979, 1857276773, 3511121443

Offset: 1

Labos Elemer, Feb 26 2003

			For n = 2: a(2) = 6 = A067605(2).
For n = 3: a(3) = 24 means: firstly occurs that for three consecutive p-1 terms GCD[prime(24)-1, prime(25)-1, prime(26)-1] = GCD[88, 96, 100] = 4 > 2;

Cf. A058263, A067605.

a(n) = {my(k = 0, v = vector(n, i, prime(i)-1)); if(gcd(v) > 2, return(0)); forprime(p = v[n]+1, , k++; v = concat(vecextract(v, "^1"), p-1); if(gcd(v) > 2, return(k)));} \\ Amiram Eldar, Jun 22 2024

a(n) = Min{x; gcd[prime(x)-1, ..., prime(x+n-1)] > 2}, where prime() = A000040().

a(1) corrected and a(17)-a(26) added by Amiram Eldar, Jun 22 2024

cp's OEIS Frontend

A083555 Quotient of LCM of prime(n+1)-1 and prime(n)-1 and GCD of the same two numbers.

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A063091 Prime(n) such that gcd(1+prime(n+1), 1+prime(n)) = gcd(-1+prime(n+1), -1+prime(n)).

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A063086 a(n) = gcd(1 + prime(n+1), 1 + prime(n)).

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A083553 Product of prime(n+1)-1 and prime(n)-1.

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

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A083554 Least common multiple of prime(n+1)-1 and prime(n)-1.

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A058264 Smallest prime p of two consecutive primes, p < q, such that gcd( p-1, q-1 ) = 2n.

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