A068780 -id:A068780 - cp's OEIS frontend

A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

1, 2, 11, 13, 29, 33, 45, 51, 57, 59, 69, 75, 105, 129, 211, 227, 301, 313, 321, 341, 407, 413, 447, 459, 537, 679, 709, 767, 1113, 1301, 1405, 1411, 1429, 1439, 1709, 1829, 1923, 2491, 2543, 2791, 2865, 3301, 3471, 3641, 4199, 4611, 5181, 5231, 6345, 6555

Offset: 1

Gus Wiseman, Jun 21 2024

nonn

Sorted positions of first appearances in A373819.

			The runs of odd primes differing by 2 begin:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
  3
  2 2
  1
  2
  1
  2
  1 1
  2
  1
  2
  1 1 1 1
  2 2
  1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3,...
with sorted positions of first appearances a(n).

Sorted firsts of A373819 (run-lengths of A251092).
The unsorted version is A373825.
For antiruns we have A373826, unsorted A373827.
A000040 lists the primes.
A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths, run-lengths of A027833.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A373403, A373404.
Cf. A006560, A006562, A029707, A037201, A038664, A069010, A122535, A176246, A373401, A373402.

t=Length/@Split[Length/@Split[Select[Range[3,10000],PrimeQ],#1+2==#2&]];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A121495 Numbers k such that k and k+1 are composite and squarefree.

Original entry on oeis.org

14, 21, 33, 34, 38, 57, 65, 69, 77, 85, 86, 93, 94, 105, 110, 114, 118, 122, 129, 133, 141, 142, 145, 154, 158, 165, 177, 182, 185, 186, 194, 201, 202, 205, 209, 213, 214, 217, 218, 221, 230, 237, 246, 253, 254, 258, 265, 266, 273, 285, 286, 290, 298, 301, 302

Offset: 1

Klaus Brockhaus, Aug 03 2006

Numbers that are in A068780 and in A007674.

			21 = 3*7 and 22 = 2*11 are squarefree, so 21 is in the sequence.

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

Cf. A002808, A005117, A007674, A068780.

q[n_] := CompositeQ[n] && SquareFreeQ[n]; Select[Range[300], q[#] && q[# + 1] &] (* Amiram Eldar, Feb 22 2021 *)

for(n=1,310,if(!isprime(n)&&!isprime(n+1)&&issquarefree(n)&&issquarefree(n+1),print1(n,",")))

A144291 Triangular numbers n*(n-1)/2 with n and n -1 nonprime.

Original entry on oeis.org

0, 36, 45, 105, 120, 210, 231, 300, 325, 351, 378, 528, 561, 595, 630, 741, 780, 990, 1035, 1176, 1225, 1275, 1326, 1485, 1540, 1596, 1653, 1953, 2016, 2080, 2145, 2346, 2415, 2775, 2850, 2926, 3003, 3240, 3321, 3570, 3655, 3741, 3828, 4095, 4186, 4278

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2008

nonn

			If n=1, then 1*(1-1)/2=0=a(1).
If n=9, then 9*(9-1)/2=36=a(2).
etc.

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

Cf. A000040, A000217, A068780, A141468.

p:= -1: Res:= NULL: count:= 0:
while count < 100 do
  q:= p; p:= nextprime(p);
  if p - q > 2 then
    count:= count + p-q-2;
    Res:= Res, seq(k*(k+1)/2, k=q+1..p-2);
  fi
od:
Res; # Robert Israel, Jul 03 2018

Reap[For[n = 1, n <= 100, n++, If[!PrimeQ[n] && !PrimeQ[n-1], Sow[n(n-1)/2] ] ] ][[2, 1]] (* Jean-François Alcover, Mar 27 2019 *)

a(n) = A000217(A068780(n-1)), n>1. - R. J. Mathar, Dec 10 2008

3570 inserted by R. J. Mathar, Dec 10 2008

A174113 Smallest number k such that k, k+1, and k+2 are all divisible by an n-th power.

Original entry on oeis.org

48, 1375, 33614, 2590623, 26890623, 2372890624, 70925781248, 2889212890624, 61938212890624, 4497636425781248, 8555081787109375, 2665760081787109375, 98325140081787109375, 198816740081787109374, 11776267480163574218750, 872710687480163574218750, 50783354512519836425781248

Offset: 2

Michel Lagneau, Mar 08 2010

nonn

Least of the smallest trio of consecutive numbers divisible by an n-th power.

			a(3) = 1375 because
  1375 =  11 * 5^3;
  1376 = 172 * 2^3;
  1377 =  51 * 3^3.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1375, p. 135, Ellipses, Paris 2008.

Charles R Greathouse IV, Table of n, a(n) for n = 2..677 (next term has 1001 digits)

Cf. A068780, A068781, A068140, A068782, A068783, A068784, A045330, A059737, A063528, A051903, A051903.

with(numtheory):for n from 2 to 6 do: i:=0:for k from 1 to 3000000 while(i=0) do:j:=0:
for a from 0 to 2 do: ii:=0:for m from 1 to 4  while(ii=0) do:p:=ithprime(m)^n:if irem(k+a,p)=0 then j:=j+1:ii:=1:else fi:od:od:if j=3 then i:=1:print(k):else fi:od:od:

a(n)=my(ch,t,best=30^n);forprime(a=2, 29, forprime(b=2, 29, if(a==b,next); ch=chinese(Mod(0,a^n), Mod(-1,b^n)); if(lift(ch)>=best, next); forprime(c=2, 29, if(a==c || b==c, next); t=lift(chinese(ch, Mod(-2, c^n))); if(tCharles R Greathouse IV, Jan 16 2012

5^n < a(n) < 30^n. Can the lower bound be improved? - Charles R Greathouse IV, Jan 16 2012

a(8)-a(18) from Charles R Greathouse IV, Jan 16 2012

A350460 Positive integers k such that if p is the next prime > k then p - k is prime.

Original entry on oeis.org

3, 5, 8, 9, 11, 14, 15, 17, 20, 21, 24, 26, 27, 29, 32, 34, 35, 38, 39, 41, 44, 45, 48, 50, 51, 54, 56, 57, 59, 62, 64, 65, 68, 69, 71, 74, 76, 77, 80, 81, 84, 86, 87, 90, 92, 94, 95, 98, 99, 101, 104, 105, 107, 110, 111, 114, 116, 120, 122, 124, 125, 128, 129

Offset: 1

Ryan Bresler, Jan 01 2022

nonn

a(n) is only prime when n is the lesser of a twin prime pair (A001359). All other terms are composite.

			3 is a term because the next prime > 3 is 5, and 5 - 3 = 2, which is prime.
14 is a term because the next prime > 14 is 17, and 17 - 14 = 3, which is prime.

Cf. A038711, A001359, A068780.

Select[Range[130], PrimeQ[NextPrime[#] - #] &] (* Amiram Eldar, Jan 01 2022 *)

isok(k) = my(p=nextprime(k+1)); isprime(p-k); \\ Michel Marcus, Jan 01 2022

from sympy import isprime, nextprime
def ok(n): return n > 0 and isprime(nextprime(n) - n)
print([k for k in range(130) if ok(k)]) # Michael S. Branicky, Jan 01 2022

A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between kn and k(n + 1), or -1 if no such k exists.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 3, 4, 1, 2, 3, 2, 4, 2, 9, 1, 1, 6, 2, 3, 2, 7, 3, 5, 2, 6, 1, 6, 7, 10, 1, 3, 4, 2, 4, 5, 1, 2, 5, 1, 2, 3, 8, 1, 7, 1, 2, 2, 3, 5, 4, 7, 11, 3, 4, 2, 3, 1, 2, 10, 1, 4, 9, 1, 2, 6, 1, 4, 15, 4, 6, 2, 5, 8, 1, 2, 3, 4, 1, 3, 2, 3, 5, 8, 3, 5, 2, 4, 7, 2, 5, 1, 3, 12, 1, 2, 2, 4, 7

Offset: 1

Juri-Stepan Gerasimov, Nov 25 2023

			Triangle begins:
1;
1;
1;
2;
1;
4;
2;
1, 3, 4;
1;
2;
3;
2, 4;
2, 9;
1;
1, 6;
2, 3;
2, 7;
3, 5;
2, 6;
1, 6, 7, 10;
1, 3, 4;
2;
4, 5;
1, 2, 5;
1, 2, 3, 8.

Cf. A068780, A047845, A110835, A278425.

cp's OEIS Frontend

A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

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Mathematica

A121495 Numbers k such that k and k+1 are composite and squarefree.

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A144291 Triangular numbers n*(n-1)/2 with n and n -1 nonprime.

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A174113 Smallest number k such that k, k+1, and k+2 are all divisible by an n-th power.

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A350460 Positive integers k such that if p is the next prime > k then p - k is prime.

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A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between k*n and k*(n + 1), or -1 if no such k exists.

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A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between kn and k(n + 1), or -1 if no such k exists.