A376656
Sorted positions of first appearances in the second differences (A036263) of consecutive primes (A000040).
Original entry on oeis.org
1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, 242, 262, 294, 296, 428, 429, 446, 708, 756, 834, 1005, 1182, 1229, 1663, 1830, 1831, 1846, 1879, 2191, 2224, 2343, 2809, 3077, 3086, 3384, 3385, 3427, 3643, 3644, 3793, 3795, 4230
Offset: 1
These are the sorted positions of first appearances in
A036263.
For first differences we had
A373400(n) + 1, except initial terms.
For squarefree instead of prime numbers we have
A376655, sorted firsts of
A376590.
A333254 lists run-lengths of differences between consecutive primes.
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q=Differences[Select[Range[1000],PrimeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
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.
-
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 *)
A379542
Second term of the n-th differences of the prime numbers.
Original entry on oeis.org
3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280
Offset: 0
For all primes (not just odd) we have
A007442.
Including 1 in the primes gives
A030016.
Cf.
A064113,
A065890,
A084758,
A140119,
A173390,
A258025,
A258026,
A293467,
A333214,
A333254,
A377041.
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nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]
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a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 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.
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