A258036 -id:A258036 - cp's OEIS frontend

A358618 First differences of A258036.

2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Clark Kimberling and Robert G. Wilson v, Oct 31 2022

Conjecture: All terms belong to {1, 2, 3}. See third comment in A258036.

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

Cf. A258036, A358619.

P:= :
S:= NULL: count:= 0:
for i from 2 while count < 101 do
  P:= P[2..-1] - P[1..-2];
  if P[1] < 0 then S:= S,i; count:= count+1; fi;
od:
S:= [S]:
S[2..-1]-S[1..-2]; # Robert Israel, Dec 21 2022

nn = 210; p = Prime@ Range@ nn; t = Table[ Differences[p, n][[1]], {n, 0, nn - 1}]; s = Select[ Range@ nn, t[[#]] < 0 &]; d = Differences@ s

A258037 Numbers k such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 27, 29, 31, 33, 35, 37, 39, 40, 42, 44, 46, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121

Offset: 1

Clark Kimberling, Jun 05 2015

Partition of the positive integers: A258036, A258037;
Corresponding partition of the primes: A258038, A258039.
Conjecture: all the terms of the difference sequence of A258037 belong to {1,2,3}.

			D(prime(2), 1) = 3 - 2 > 0, so a(1) = 1;
D(prime(3), 2) = 5 - 2*3 + 2 > 0, so a(2) = 2;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0;

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A258036, A258038, A258039.

u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
p1 = Prime[w1]  (* A258038 *)
p2 = Prime[w2]  (* A258039 *)

is(k) = {my(p=primes(k));sum(i=0,k-1,(-1)^i*p[k-i]*binomial(k-1,i))>0} \\ Jason Yuen, Nov 13 2024

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]

A258038 Numbers prime(k) such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

Original entry on oeis.org

7, 13, 19, 29, 37, 43, 59, 67, 73, 83, 97, 107, 113, 131, 139, 151, 163, 179, 191, 197, 211, 223, 229, 239, 251, 263, 271, 281, 293, 311, 317, 337, 349, 359, 373, 383, 397, 409, 421, 433, 443, 457, 463, 479, 491, 503, 521, 523, 547, 563, 571, 587, 599, 607

Offset: 1

Clark Kimberling, Jun 05 2015

Partition of the positive integers: A258036, A258037;

Corresponding partition of the primes: A258038, A258039.

			D(prime(2), 1) = 3 - 2 > 0;
D(prime(3), 2) = 5 - 2*3 + 2 > 0;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = prime(4) = 7;

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A258036, A258037, A258039.

u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
p1 = Prime[w1]  (* A258038 *)
p2 = Prime[w2]  (* A258039 *)

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]

a(n) = prime(A258036(n)). - Jason Yuen, Nov 13 2024

A258039 Numbers prime(k) such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 31, 41, 47, 53, 61, 71, 79, 89, 101, 103, 109, 127, 137, 149, 157, 167, 173, 181, 193, 199, 227, 233, 241, 257, 269, 277, 283, 307, 313, 331, 347, 353, 367, 379, 389, 401, 419, 431, 439, 449, 461, 467, 487, 499, 509, 541, 557, 569, 577

Offset: 1

Clark Kimberling, Jun 05 2015

Partition of the positive integers: A258036, A258037;

Corresponding partition of the primes: A258038, A258039.

			D(prime(2), 1) = 3 - 2 > 0, so a(1) = prime(1) = 2;
D(prime(3), 2) = 5 - 2*3 + 2 > 0, so a(2) = prime(2) = 3;
D(prime(4), 3) = 7 - 3*5 + 3*3 - 2 < 0.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A258036, A258037, A258038.

u = Table[Prime[Range[k]], {k, 1, 1000}];
v = Flatten[Table[Sign[Differences[u[[k]], k - 1]], {k, 1, 100}]];
w1 = Flatten[Position[v, -1]] (* A258036 *)
w2 = Flatten[Position[v, 1]]  (* A258037 *)
p1 = Prime[w1]  (* A258038 *)
p2 = Prime[w2]  (* A258039 *)

D(prime(k), k-1) = Sum_{i=0..k-1} (-1)^i*prime(k-i)*binomial(k-1,i). [corrected by Jason Yuen, Nov 13 2024]

a(n) = prime(A258037(n)). - Jason Yuen, Nov 13 2024

cp's OEIS Frontend

A358618 First differences of A258036.

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A258037 Numbers k such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

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A258038 Numbers prime(k) such that D(prime(k), k-1) < 0, where D( * , k-1) = (k-1)-st difference.

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A258039 Numbers prime(k) such that D(prime(k), k-1) > 0, where D( * , k-1) = (k-1)-st difference.

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