A051634 -id:A051634 - cp's OEIS frontend

A175102 1, followed by list of numbers n such that the number of strong primes and the number of weak primes are equal at the n-th prime.

1, 60, 64, 41192, 41194, 41247, 41250, 41252, 41257, 41259, 41261, 41263, 41265, 41267, 41273, 41275, 41277, 41279, 41287, 41317, 41319, 41321, 41323, 41325, 41327, 41328, 41329, 41335, 41336, 41338, 41339, 41341, 41389, 41393, 41397, 41399, 41401, 41404, 41406, 41408, 41412, 41444, 41448, 42112

Offset: 1

G. L. Honaker, Jr., Dec 02 2010

nonn

Also, indices of zeros in A092243. - N. J. A. Sloane, Mar 13 2016

N. J. A. Sloane and Chai Wah Wu, Table of n, a(n) for n = 1..20000, terms for n = 1..446 from N. J. A. Sloane.
Chris Caldwell and G. L. Honaker, Jr., Prime Curio for 281

Cf. A051634, A051635, A092243.

my(c=1, q=3, r=2, s=0); print1(c, ", "); forprime(p=5, default(primelimit), c++;(s+=sign(r+0-2*(r=q)+q=p))||print1(c, ", ")) \\ M. F. Hasler, Dec 03 2010

More terms from Chris K. Caldwell

A363782 Products of three distinct strong primes.

Original entry on oeis.org

5423, 6919, 7667, 11033, 11803, 12529, 13079, 13277, 14773, 16687, 18139, 18241, 18821, 18887, 20009, 20213, 21373, 22649, 23749, 24013, 25201, 25619, 25789, 26609, 27269, 27863, 28897, 29087, 30217, 30481, 30943, 32021, 32153, 32219, 33031, 33473, 34133, 35003, 35629, 35717, 36839

Offset: 1

Massimo Kofler, Jun 21 2023

nonn

Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2.

			5423 = 11*17*29 and 11 > (7+13)/2, 17 > (13+19)/2, 29 > (23+31)/2.
6919 = 11*17*37 and 11 > (7+13)/2, 17 > (13+19)/2, 37 > (31+41)/2.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A007304, A051634, A363167, A364738.

strongQ[p_] := p > 2 && 2*p > Total[NextPrime[p, {-1, 1}]]; Select[Range[1, 37000, 2], (f = FactorInteger[#])[[;; , 2]] == {1, 1, 1} && AllTrue[f[[;; , 1]], strongQ] &] (* Amiram Eldar, Jun 21 2023 *)
Module[{nn=50,strgpr},strgpr=Select[Partition[Prime[Range[nn]],3,1],#[[2]]>(#[[1]]+#[[3]])/2&][[;;,2]];Take[Union[Times@@@Subsets[strgpr,{3}]],nn]] (* Harvey P. Dale, Aug 21 2024 *)

Definition clarified by N. J. A. Sloane, Oct 08 2023

A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

Original entry on oeis.org

1667, 1787, 1867, 1871, 1997, 2381, 2473, 2531, 2539, 3457, 3461, 4217, 4517, 5279, 5417, 5441, 6043, 6659, 7243, 7307, 7757, 7877, 7933, 8167, 8521, 9613, 9619, 11057, 11393, 11593, 11831, 12409, 13877, 14827, 15137, 15551, 16061, 16333

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

First member of pairs of consecutive primes in A054807 (4th of strong prime quartets). - M. F. Hasler, Oct 27 2018

M. F. Hasler, Table of n, a(n) for n = 1..2000, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

a(n) = nextprime(A054810(n)) = prevprime(A054812(n)), nextprime = A151800, prevprime = A151799; A054811 = {m = A054807(n) | nextprime(m) = A054807(n+1)}. - M. F. Hasler, Oct 27 2018

A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

Original entry on oeis.org

1669, 1789, 1871, 1873, 1999, 2383, 2477, 2539, 2543, 3461, 3463, 4219, 4519, 5281, 5419, 5443, 6047, 6661, 7247, 7309, 7759, 7879, 7937, 8171, 8527, 9619, 9623, 11059, 11399, 11597, 11833, 12413, 13879, 14831, 15139, 15559, 16063, 16339

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Second member of pairs of consecutive primes in A054807 (4th term of strong prime quartets). - M. F. Hasler, Oct 27 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

spqQ[c_]:=Module[{d=Differences[c]},d[[1]]>d[[2]]>d[[3]]>d[[4]]]; Transpose[ Select[Partition[Prime[Range[2000]],5,1],spqQ]][[5]] (* Harvey P. Dale, Jan 01 2013 *)

a(n) = nextprime(A054811(n)); A054811 = {m = A054807(n) | prevprime(m) = A054807(n-1)}; nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

A054813 First term of strong prime sextets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3) > p(m+5)-p(m+4).

Original entry on oeis.org

1831, 2477, 3413, 9551, 21433, 22973, 25189, 26053, 32143, 33359, 33893, 39047, 40771, 41203, 44221, 47251, 48787, 55849, 57751, 66977, 70079, 74231, 74653, 74687, 75083, 75109, 82913, 84263, 87811, 88339, 88609, 103723, 103843, 106219, 106921, 108139, 110881, 112979, 118093

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2500, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(n) = A151799(A054814(n)), A054813 = { m = A054808(n) | m = A151799(A054808(n+1)) }, where A151799 = next smaller prime. - M. F. Hasler, Oct 27 2018

More terms and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054814 Second term p(m) of strong prime sextets: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

Original entry on oeis.org

1847, 2503, 3433, 9587, 21467, 22993, 25219, 26083, 32159, 33377, 33911, 39079, 40787, 41213, 44249, 47269, 48799, 55871, 57773, 67003, 70099, 74257, 74687, 74699, 75109, 75133, 82939, 84299, 87833, 88379, 88643, 103769, 103867, 106243, 106937, 108161, 110899, 112997, 118127, 120371

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2500, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Subsequence of A054808.

Select[Partition[Prime[Range[12000]],6,1],Max[Differences[#,2]]<0&][[;;,2]] (* Harvey P. Dale, Jun 17 2023 *)

a(n) = A151800(A054813(n)) = A151799(A054815(n)), A151800 = nextprime, A151799 = prevprime; A054814 = { m = A054809(n) | m = nextprime(A054809(n-1)) }. - M. F. Hasler, Oct 27 2018

Edited and offset changed to 1 by M. F. Hasler, Oct 26 2018

A054815 Third term of strong prime sextets: p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

Original entry on oeis.org

1861, 2521, 3449, 9601, 21481, 23003, 25229, 26099, 32173, 33391, 33923, 39089, 40801, 41221, 44257, 47279, 48809, 55889, 57781, 67021, 70111, 74279, 74699, 74707, 75133, 75149, 82963, 84307, 87853, 88397, 88651, 103787, 103889

Offset: 0

Henry Bottomley, Apr 10 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..2500

Cf. A051634, A054800-A054840.

Select[Partition[Prime[Range[10000]],6,1],Max[Differences[#,2]]<0&] [[All,3]] (* Harvey P. Dale, Apr 30 2017 *)

A155188 Sophie Germain primes that are also strong primes and lesser of twin prime pairs.

Original entry on oeis.org

11, 29, 41, 179, 191, 239, 281, 419, 431, 641, 659, 809, 1019, 1031, 1049, 1229, 1289, 1451, 1481, 1931, 2129, 2141, 2339, 2549, 2969, 3299, 3329, 3359, 3389, 3539, 3821, 3851, 4019, 4271, 4481, 5231, 5279, 5441, 5501, 5639, 5741, 5849, 6131, 6269, 6449

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

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

Cf. A045536, A005384, A051634, A006562, A051635.

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];If[p1>(p0+p2)/2,If[PrimeQ[p1*2+1],If[PrimeQ[p1+2],AppendTo[lst,p1]]]],{n,7!}];lst

A363167 Products of four distinct strong primes.

Original entry on oeis.org

200651, 222343, 283679, 319957, 363341, 385033, 408221, 428417, 452353, 463573, 483923, 491249, 513689, 526031, 544357, 546601, 547723, 580261, 605693, 671143, 688721, 696377, 698819, 739211, 740333, 742951, 743699, 747881, 771661, 774367, 783343, 790801, 808027, 820369

Offset: 1

Massimo Kofler, Sep 07 2023

nonn

Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2.

			200651 = 11*17*29*37 and 11 > (7+13)/2, 17 > (13+19)/2, 29 > (23+31)/2, 37 > (31+41)/2.
222343 = 11*17*29*41 and 11 > (7+13)/2, 17 > (13+19)/2, 29 > (23+31)/2, 41 > (37+43)/2.
283679 = 11*17*37*41 and 11 > (7+13)/2, 17 > (13+19)/2, 37 > (31+41)/2, 41 > (37+43)/2.

Cf. A046386, A051634, A364778, A363782.

strongQ[p_] := p > 2 && 2*p > Total[NextPrime[p, {-1, 1}]]; Select[Range[1, 10^6, 2], (f = FactorInteger[#])[[;; , 2]] == {1, 1, 1, 1} && AllTrue[f[[;; , 1]], strongQ] &] (* Amiram Eldar, Sep 08 2023 *)

A108415 a(n) = 1, 2 or 3 (resp.) if prime(n) is weak, balanced or strong (resp.).

Original entry on oeis.org

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

Offset: 2

Zak Seidov, Jun 02 2005

nonn

n >= 2: a(n) = 1, 2 or 3 (resp.) if n-th prime is in A051635, A006562 or A051634 (resp.).

Robert Israel, Table of n, a(n) for n = 2..10000
Wikipedia, Balanced prime

Cf. A006562, A051634, A051635.

p:= 2: q:= 3: r:= 5:
for i from 2 to 200 do
  t:= q - (p+r)/2;
  A[i]:= piecewise(t<0,1,t=0,2,3);
  p:= q; q:= r; r:= nextprime(r);
od:
seq(A[i],i=2..200); # Robert Israel, Mar 25 2018

A108415[n_]:=2+Sign[Prime[n]-1/2(Prime[n-1]+Prime[n+1])]

cp's OEIS Frontend

A175102 1, followed by list of numbers n such that the number of strong primes and the number of weak primes are equal at the n-th prime.

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A363782 Products of three distinct strong primes.

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A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

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A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

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A054813 First term of strong prime sextets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3) > p(m+5)-p(m+4).

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A054814 Second term p(m) of strong prime sextets: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

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A054815 Third term of strong prime sextets: p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

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A155188 Sophie Germain primes that are also strong primes and lesser of twin prime pairs.

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Mathematica

A363167 Products of four distinct strong primes.

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