A078515 -id:A078515 - cp's OEIS frontend

A079889 Primes indexed by A078515; i.e., primes which start record runs of consecutive primes with distinct first differences.

2, 17, 83, 113, 491, 1367, 1801, 5869, 15919, 34883, 70639, 70657, 214867, 2515871, 3952733, 13010143, 30220163, 60155567, 69931991, 203674907, 1092101119, 1363592621, 1363592677, 2124140323, 23024158649, 30282104173, 196948778371

Offset: 1

N. J. A. Sloane, Jan 07 2003

nonn

			2, 17 and 83 are in the sequence because the 3 consecutive primes 2,3,5 have distinct first differences 1,2, the 4 consecutive primes 17,19,23,29 have distinct differences 2,4,6, and the 5 consecutive primes 83,89,97,101,103 have distinct differences 6,8,4,2.

Cf. A078515, A078516. Same as A079007 with duplicates removed.

a(n) = A078515(n)-th prime.

More terms from Don Reble, Jan 15 2003

a(25)-a(27) from Donovan Johnson, Oct 23 2012

A079007 a(n) = smallest prime p_k such that the n successive differences between the primes p_k through p_(k+n) are all distinct.

Original entry on oeis.org

2, 2, 2, 17, 83, 113, 491, 1367, 1801, 5869, 15919, 34883, 70639, 70657, 214867, 214867, 2515871, 3952733, 13010143, 30220163, 60155567, 69931991, 203674907, 1092101119, 1363592621, 1363592677, 2124140323, 23024158649, 30282104173, 30282104173, 196948778371

Offset: 0

Labos Elemer, Jan 03 2002

nonn

			a(0) = 2; a(1) = 2 from {2,3} with a single difference 1; a(2) = 2 from {2,3,5}, with two distinct differences 1, 2.
a(5) = p_30 = 113 because 113 is followed by 127, 131, 137, 139, 149, with 5 different differences: 14, 4, 6, 2, 10; and no smaller prime has this property.

Cf. A001223, A068843, A053597, A078515. Different from A079889.

f[k_, n_] := Block[{p = Table[ Prime[i], {i, k, k + n - 1}]}, Length[ Union[Drop[p, 1] - Drop[p, -1]]]]; k = 1; Do[ While[ f[k, n] != n - 1, k++ ]; Print[ Prime[k]], {n, 1, 22}]

Edited by Robert G. Wilson v and N. J. A. Sloane, Jan 05 2002
More terms from Don Reble, Jan 15 2003
a(27)-a(30) from Donovan Johnson, Oct 23 2012

A053597 Let prime(i) = i-th prime (A000040), let d(i) = prime(i+1)-prime(i) (A001223); a(n) = number of distinct numbers among d(n), d(n+1), d(n+2), ... before first duplicate is encountered.

Original entry on oeis.org

2, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 1, 2, 3, 3, 3, 3, 2, 3, 4, 3, 2, 2, 2, 3, 2, 5, 4, 3, 2, 3, 2, 1, 2, 2, 1, 3, 2, 3, 2, 3, 2, 1, 3, 2, 3, 4, 3, 3, 2, 1, 1, 2, 3, 5, 4, 4, 4, 3, 2, 5, 5, 5, 4, 5, 4, 3, 2, 2, 1, 2, 3, 3, 2, 4, 3, 2, 2, 4, 3, 2, 3, 4, 3, 2, 4, 3, 3, 2, 2, 6, 5, 4, 5, 4, 3, 2, 2, 1, 2, 3, 2

Offset: 1

N. J. A. Sloane, Jan 07 2003

			The d sequence starting at prime(7) = 17 is d(7) = 2, d(8) = 4, d(9) = 6, d(10) = 2, with three numbers before the first duplication, so a(7) = 3.

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

A078515 gives RECORDS transform of this sequence. See also A079007.

P:= [seq(ithprime(i),i=1..1000)]:
G:= P[2..-1]-P[1..-2]:
R:= Vector(990):
for i from 1 to 990 do
  for k from 1 while nops(convert(G[i..i+k-1],set))=k do od:
  R[i]:= k-1;
od:
convert(R,list);

f[n_] := Block[{k = 1}, While[p = Table[ Prime[i], {i, n, n + k}]; Length[ Union[ Drop[p, 1] - Drop[p, -1]]] == k, k++ ]; k - 1]; Table[ f[n], {n, 1, 105}]

More terms from Robert G. Wilson v, Jan 07 2002

cp's OEIS Frontend

A079889 Primes indexed by A078515; i.e., primes which start record runs of consecutive primes with distinct first differences.

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A079007 a(n) = smallest prime p_k such that the n successive differences between the primes p_k through p_(k+n) are all distinct.

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A053597 Let prime(i) = i-th prime (A000040), let d(i) = prime(i+1)-prime(i) (A001223); a(n) = number of distinct numbers among d(n), d(n+1), d(n+2), ... before first duplicate is encountered.

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