A083551 -id:A083551 - cp's OEIS frontend

A083552 Quotient when LCM of 2 consecutive prime differences is divided by GCD of the same two differences.

2, 1, 2, 2, 2, 2, 2, 6, 3, 3, 6, 2, 2, 6, 1, 3, 3, 6, 2, 3, 6, 6, 12, 2, 2, 2, 2, 2, 14, 14, 6, 3, 5, 5, 3, 1, 6, 6, 1, 3, 5, 5, 2, 2, 6, 1, 3, 2, 2, 6, 3, 5, 15, 1, 1, 3, 3, 6, 2, 5, 35, 14, 2, 2, 14, 21, 15, 5, 2, 6, 12, 12, 1, 6, 6, 12, 2, 2, 20, 5, 5, 5, 3, 6, 6, 12, 2, 2, 2, 3, 6, 2, 2, 2, 6, 2, 6, 9

Offset: 1

Labos Elemer, May 22 2003

nonn

Conjecture: Every positive integer appears infinitely many times in this sequence. Example: a(834) = a(909) = ... = a(9901) = ... = 4. - Jerzy R Borysowicz, Dec 22 2018

All terms of this sequence are integers because gcd(r,s) divides lcm(r,s) for any r and s. - Jerzy R Borysowicz, Jan 05 2019

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

Cf. A001223, A083538-A083555, A057467.

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

a(n) = my(da=prime(n+2)-prime(n+1), db=prime(n+1)-prime(n)); lcm(da, db)/gcd(da, db) \\ Felix Fröhlich, Jan 05 2019

a(n) = lcm(A001223(n), A001223(n+1))/gcd(A001223(n), A001223(n+1));

a(n) = A083551(n)/A057467(n).

A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 2, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 12, 12, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 12, 40, 24, 30, 30, 24, 56, 56, 24, 32, 224, 210, 570, 532, 28, 36, 60, 480, 672, 70, 30, 44, 44, 32, 864, 864, 544, 782, 46, 36, 900

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=33: cototient(33) = 33-20 = 13, cototient(34) = 34-16 = 18;
lcm(13,18) = 234, gcd(13,18) = 1, so a(34) = 234.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051953, A083538, A083539, A083540, A083541, A083542, A083543, A083544, A083545, A083546, A083547, A083548, A083549, A083550, A083551, A083552, A083553, A083554, A083555, A049586.

f[x_] := x-EulerPhi[x]; Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 69}]
(* Second program: *)
Map[Apply[LCM, #]/Apply[GCD, #] &@ Map[# - EulerPhi@ # &, #] &, Partition[Range[69], 2, 1]] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = lcm(A051953(n), A051952(n+1))/gcd(A051953(n), A051952(n+1)) = lcm(cototient(n+1), cototient(n))/A049586(n).

cp's OEIS Frontend

A083552 Quotient when LCM of 2 consecutive prime differences is divided by GCD of the same two differences.

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