A067774 -id:A067774 - cp's OEIS frontend

A373826 Sorted positions of first appearances in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

1, 4, 38, 6781, 23238, 26100

Offset: 1

Gus Wiseman, Jun 22 2024

Sorted positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with sorted positions of first appearances a(n).

Sorted positions of first appearances in A373820, cf. A027833.
For runs we have A373824 (unsorted A373825), sorted firsts of A373819.
The unsorted version is A373827.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, 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.
Cf. A001359, A006512, A025584, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

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

A373827 Position of first appearance of n in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

Original entry on oeis.org

4, 1, 38, 6781, 26100, 23238

Offset: 1

Gus Wiseman, Jun 22 2024

Positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with positions of first appearances a(n).

Positions of first appearances in A373820.
For runs instead of antiruns we have A373825, sorted A373824.
The sorted version is A373826.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, 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.
Cf. A001359, A006512, A025584, A027833, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

t=Length/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ],#1+2!=#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A155874 Smallest positive composite number such that a(n)+n is also composite.

Original entry on oeis.org

4, 8, 4, 6, 4, 4, 4, 8, 4, 6, 4, 4, 4, 8, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 8, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 8, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 8, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 8, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 8, 4, 6, 4

Offset: 0

Eric Angelini, Jan 29 2009

a(0)=4. Adding a(0) to n=0 gives 4+0=4, which is a composite number; adding a(1) to n=1 gives 8+1=9 which is composite; adding a(2) to n=2 gives 4+2=6 which is composite; adding a(3) to n=3 gives 6+3=9 which is composite; etc.

At least one of {n+4, n+6, n+8} is divisible by 3, so a(n) is in {4,6,8} for all n. - Charlie Neder, Dec 28 2018

Charlie Neder, Table of n, a(n) for n = 0..1000

Cf. A122984.

a(n) = {forcomposite(k=4, 10, if (!isprime(k+n), return(k)););} \\ Michel Marcus, Dec 28 2018

If n+4 is in A001359, a(n) = 8. If n+4 is in A067774, a(n) = 6. Otherwise, a(n) = 4. - Charlie Neder, Dec 28 2018

More terms from Charlie Neder, Dec 28 2018

A211238 Prime numbers p such that x^2 + x + p produces primes for x = 0..9 but not x = 10.

Original entry on oeis.org

11, 844427, 51448361, 86966771, 122983031, 960959381, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711, 5258591537, 6430890287, 7156316591, 8518049207, 8828280941, 9467776751, 10687380227, 10783636931, 11856793337, 12128287007, 14431067237, 14772642497

Offset: 1

T. D. Noe, Apr 08 2012

nonn

The first term is A164926(10).

Zak Seidov, Table of n, a(n) for n = 1..133

Cf. A067774, A164926.

lookfor = 10; t = {}; n = 0; While[Length[t] < 25, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A373823 Half the sum of the n-th maximal run of first differences of odd primes.

Original entry on oeis.org

2, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 6, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 6, 2, 6, 1, 5, 1, 2, 1, 12, 2, 1, 2, 3, 1, 5, 9, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 6, 2, 3, 4, 2, 4, 5, 1, 5, 1, 3, 2, 3, 4, 2, 1, 2, 6, 4, 2, 4, 2, 3

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Halved run-sums of A001223.

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs:
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with halved sums a(n).

Halved run-sums of A001223.
For run-lengths we have A333254, run-lengths of run-lengths A373821.
Multiplying by two gives A373822.
A000040 lists the primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
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 of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Total/@Split[Differences[Select[Range[3,1000],PrimeQ]]]/2

A107987 Odd composite numbers of the form p+2 where p is prime.

Original entry on oeis.org

9, 15, 21, 25, 33, 39, 45, 49, 55, 63, 69, 75, 81, 85, 91, 99, 105, 111, 115, 129, 133, 141, 153, 159, 165, 169, 175, 183, 195, 201, 213, 225, 231, 235, 243, 253, 259, 265, 273, 279, 285, 295, 309, 315, 319, 333, 339, 351, 355, 361, 369, 375, 381, 385, 391, 399

Offset: 1

Cino Hilliard, Jun 13 2005

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A049591, A067774.

DeleteCases[Prime[Range[2,80]]+2,?PrimeQ] (* _Harvey P. Dale, Aug 15 2013 *)

sum2pr(n) = \\ Composite numbers of form p+2.
{ c=0; cp=0; forprime(x=3, n, cp++; y=x+2; if(isprime(y)==0, c++; print1(y",") ) ); print(); print(c/cp+.) }

a(n) = A049591(n) + 2 = A067774(n+1) + 2. - Amiram Eldar, Jul 05 2024

A210968 Smallest prime product pqr such that p + q + r = 2*n + 1.

Original entry on oeis.org

12, 20, 28, 63, 44, 52, 117, 68, 76, 171, 92, 207, 345, 116, 124, 279, 465, 148, 333, 164, 172, 387, 188, 423, 705, 212, 477, 795, 236, 244, 549, 915, 268, 603, 284, 292, 657, 1095, 316, 711, 332, 747, 1245, 356, 801, 1335, 1869, 388

Offset: 3

Omar E. Pol, Jun 29 2012

From Robert Israel, May 24 2019: (Start)
If p is an odd prime, then a((p+3)/2) = 4*p.
If p > 2 is in A067774, then a((p+5)/2) = 9*p. (End)

Robert Israel, Table of n, a(n) for n = 3..10000

Cf. A067774, A210967.

N:= 100: # for a(3)..a(N)
P:= select(isprime, [2,seq(i,i=3..2*N+1,2)]): nP:= nops(P):
A:= Vector([infinity$(2*N+1)]):
for i from 1 to nP while 2*P[i] <= 2*N+1 do
   p:= P[i];
   for j from i to nP while p+P[j] <= 2*N+1 do
     if p*P[j] < A[p+P[j]] then A[p+P[j]]:= p*P[j] fi
od od:
B:= Vector([infinity$(2*N+1)]):
for i from 1 to nP while 3*P[i] <= 2*N+1 do
  p:= P[i];
  for x from 4 to 2*N+1-p do
    y:= p+x;
    if A[x]*p < B[y] then B[y]:= A[x]*p fi
od od:
[seq(B[2*i+1],i=3..N)]; # Robert Israel, May 24 2019

A211236 Prime numbers p such that x^2 + x + p produces primes for x = 0..7 but not x = 8.

Original entry on oeis.org

21557, 26681, 128981, 2073347, 3992201, 4889237, 6184637, 11900501, 21456047, 24598361, 33771581, 34864211, 50943791, 55793951, 56421347, 61218251, 67787537, 69726647, 76345121, 86145881, 90261707, 92865791, 99624647, 102960281, 108846161

Offset: 1

T. D. Noe, Apr 08 2012

nonn

The first term is A164926(8).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926.

lookfor = 8; t = {}; n = 0; While[Length[t] < 25, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[6250000]],AllTrue[#+{2,6,12,20,30,42,56}, PrimeQ] && !PrimeQ[ #+72]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 04 2016 *)

A211237 Prime numbers p such that x^2 + x + p produces primes for x = 0..8 but not x = 9.

Original entry on oeis.org

51867197, 85776137, 93685301, 97122197, 107599757, 113575727, 118136267, 232728647, 316973621, 483040757, 564537761, 749930717, 840472307, 901288517, 1559839991, 1696818647, 2251028567, 2469604721, 2796607337, 3098938847, 3152692841, 3344410367

Offset: 1

T. D. Noe, Apr 08 2012

nonn

The first term is A164926(9).

T. D. Noe, Table of n, a(n) for n = 1..500

Cf. A067774, A164926.

lookfor = 9; t = {}; n = 0; While[Length[t] < 25, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[31*10^5,65*10^5]],AllTrue[#+{2,6,12,20,30,42,56,72},PrimeQ] && CompositeQ[#+90]&] (* The program generates the first 6 terms of the sequence. To generate more, increase the second Range constant. *) (* Harvey P. Dale, Nov 02 2021 *)

A211239 Prime numbers p such that x^2 + x + p produces primes for x = 0..10 but not x = 11.

Original entry on oeis.org

180078317, 1278189947, 1829187287, 5862143447, 6369321857, 7226006861, 12412643261, 50626299797, 53039299211, 72355485857, 74621287901, 76233413141, 81948881447, 115826556611, 129077263697, 137168442221, 137376420947, 146539105871, 168759510737, 181122284501

Offset: 1

T. D. Noe, Apr 09 2012

nonn

The first term is A164926(11).

Zak Seidov, Table of n, a(n) for n = 1..31

Cf. A067774, A164926.

cp's OEIS Frontend

A373826 Sorted positions of first appearances in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

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A373827 Position of first appearance of n in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

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A155874 Smallest positive composite number such that a(n)+n is also composite.

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A211238 Prime numbers p such that x^2 + x + p produces primes for x = 0..9 but not x = 10.

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A373823 Half the sum of the n-th maximal run of first differences of odd primes.

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Mathematica

A107987 Odd composite numbers of the form p+2 where p is prime.

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A210968 Smallest prime product p*q*r such that p + q + r = 2*n + 1.

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Maple

A211236 Prime numbers p such that x^2 + x + p produces primes for x = 0..7 but not x = 8.

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Mathematica

A211237 Prime numbers p such that x^2 + x + p produces primes for x = 0..8 but not x = 9.

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A210968 Smallest prime product pqr such that p + q + r = 2*n + 1.