A378620
Lesser prime index of twin primes with nonsquarefree mean.
Original entry on oeis.org
2, 5, 7, 17, 20, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 230, 236, 253, 256, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 373, 377, 390, 395, 405, 428, 430, 444
Offset: 1
A subset of
A029707 (twin prime lesser indices).
Prime indices of the primes listed by
A061368.
Indices of twin primes with squarefree mean are
A068361.
A038664 finds the first position of a prime gap of 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
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Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)
A211240
Prime numbers p such that x^2 + x + p produces primes for x = 0..11 but not x = 12.
Original entry on oeis.org
1761702947, 11085833111, 177558051107, 473787509537, 557149355507, 715464238661, 2236159108277, 2751203698151, 3247566894821, 3288002848511, 3424305123047, 3490420408691, 3729352769561, 3801308473871, 4296903559301, 4656625081181, 4837112125121, 5125165470701
Offset: 1
A382766
Odd primes p such that p + 4, p + 6 and p + 8 are composite.
Original entry on oeis.org
113, 137, 139, 179, 181, 197, 199, 211, 239, 241, 281, 283, 293, 317, 337, 409, 419, 421, 467, 509, 521, 523, 547, 577, 617, 619, 631, 659, 661, 691, 709, 773, 787, 797, 809, 811, 827, 829, 839, 863, 887, 919, 953, 997, 1019, 1021, 1039, 1049, 1051, 1069
Offset: 1
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P:= select(isprime,{seq(i,i=3..10008,2)}):
R:= P minus (P -~ 4) minus (P -~ 6) minus (P -~ 8):
sort(convert(R,list)); # Robert Israel, Apr 28 2025
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Select[Table[
Module[{p = 2, q},
While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
p = NextPrime[p]]; If[p == 11, q, Nothing]], {n, 2, 1000}], # =!=
Nothing &]
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isok(p) = (p%2) && isprime(p) && !isprime(p+4) && !isprime(p+6) && !isprime(p+8); \\ Michel Marcus, Apr 07 2025
A373828
Run-sums (differing by 0) of run-lengths (differing by 2) of odd primes.
Original entry on oeis.org
3, 4, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 3, 4, 4, 6, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 10, 4, 4, 2, 7, 2, 4, 2, 3, 2, 2, 2, 1, 2, 2, 2, 18, 6, 2, 2, 2, 2, 17, 4, 1, 4, 2, 2, 6, 2, 9, 2, 3, 2, 1, 2, 1, 2, 1, 2, 8, 2, 3, 2, 2, 4, 15, 2, 1, 2, 4, 2, 1, 2, 1, 2, 7, 2
Offset: 1
The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with runs:
{3,5,7}, {11,13}, {17,19}, {23}, {29,31}, {37}, {41,43}, {47}, {53}, ...
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, ...
with runs:
{3}, {2,2}, {1}, {2}, {1}, {2}, {1,1}, {2}, {1}, {2}, {1,1,1,1}, {2,2}, ...
with sums a(n).
A001223 gives first differences of primes.
A027833 gives antirun-lengths of primes > 3 (prepended run-lengths
A373820).
A046933 counts composite numbers between primes.
A071148 gives partial sums of odd primes.
A333254 gives run-lengths of first differences of primes.
A373821 gives run-lengths of run-lengths of first differences of odd primes.
A383485
a(n) = 2*(2*(n - 1)! + n + 2) (mod n*(n + 2)).
Original entry on oeis.org
1, 4, 3, 12, 5, 16, 0, 20, 31, 24, 11, 28, 0, 32, 49, 36, 17, 40, 0, 44, 67, 48, 0, 52, 54, 56, 85, 60, 29, 64, 0, 68, 70, 72, 109, 76, 0, 80, 121, 84, 41, 88, 0, 92, 139, 96, 0, 100, 102, 104, 157, 108, 0, 112, 114, 116, 175, 120, 59, 124, 0, 128, 130, 132, 199, 136, 0, 140
Offset: 1
Comments