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
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).
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.
A162309
a(n) is the number of isolated primes up to the smaller component of the n-th twin prime pair.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 5, 6, 10, 10, 13, 13, 17, 17, 17, 19, 20, 23, 24, 26, 29, 39, 39, 43, 50, 54, 57, 59, 60, 62, 80, 80, 80, 82, 84, 101, 101, 102, 102, 104, 110, 119, 122, 123, 124, 125, 133, 136, 138, 138, 153, 154, 158, 159, 160, 167, 174, 174, 178, 178, 182, 185, 189, 189
Offset: 1
a(1)=1 counts the isolated prime 2, which smaller than 3;
a(2)=1 counts the isolated prime 2, which is smaller than 5;
a(5)=2 counts the isolated primes 2 and 23, which are smaller than 29;
a(6)=3 counts 2, 23 and 37, which are smaller than 41.
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read("transforms3") ; tp := BFILETOLIST("b001359.txt") ;
A162309 := proc(n) global tp; a := 0 ; for j from 2 to op(n,tp)-1 do if isprime(j) then if ( j in tp ) or (j-2) in tp then ; else a :=a +1; fi; fi; od: a ; end:
seq(A162309(n),n=1..130 ); # R. J. Mathar, Aug 29 2009
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A027833 =
Differences[Flatten[Position[Differences[Prime[Range[500]]], 2]]];
ReplacePart[Accumulate[Join[{2}, A027833 - 2]], 1 -> 1]
(* Jean-François Alcover, Jan 23 2023, after Harvey P. Dale in A027833 *)
53 replaced with 54, 100 removed twice, etc., by
R. J. Mathar, Aug 29 2009
Original entry on oeis.org
1, 2, 3, 4, 6, 12, 20, 24, 27, 29, 42, 54, 72, 75, 103, 128, 131, 151, 153, 162, 164, 181, 204, 208, 209, 211, 237, 265, 285, 286, 326, 335, 340, 356, 368, 392, 409, 432, 439, 441, 444, 446
Offset: 1
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nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[ NextPrime@ q - q > 2, q = NextPrime@ q]; q]; p = 2; q = 3; px = 1; qx = 2; mxd = 0; tpx = 0; lst = {}; While[p <
5090000001, d = qx - px; If[ d > mxd, mxd = d; AppendTo[ lst, d]; Print@ d]; p = q; px = qx; q = nextLesserTwinPrime@ q; qx = PrimePi@ q; tpx++]; lst (* Robert G. Wilson v, May 21 2014 *)
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def A242459_list(n) :
a = [ 1 ]
st = 3
for i in (4..n) :
if (nth_prime(i+1)-nth_prime(i) == 2) :
if i-st > a[len(a)-1] :
a.append(i-st)
st = i
return(a)
A242459_list(10^(5))
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.
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