A068361
Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.
Original entry on oeis.org
1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054
Offset: 1
A subset of
A029707 (lesser index of twin primes).
Prime index of each (prime) term of
A061351.
For perfect power instead of squarefree we have
A377436, zeros of
A377432.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
Cf.
A000720,
A007674,
A013928,
A057627,
A070321,
A071403,
A072284,
A073247,
A112925,
A122535,
A155752,
A224363,
A240473,
A251092.
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Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)
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isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016
A378032
a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).
Original entry on oeis.org
1, 1, 4, 4, 9, 12, 16, 18, 20, 28, 28, 36, 40, 40, 45, 52, 56, 60, 64, 68, 72, 76, 81, 88, 96, 100, 100, 104, 108, 112, 126, 128, 136, 136, 148, 150, 156, 162, 164, 172, 176, 180, 189, 192, 196, 198, 208, 220, 225, 228, 232, 236, 240, 250, 256, 261, 268, 270
Offset: 1
The terms together with their prime indices begin:
1: {}
1: {}
4: {1,1}
4: {1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
28: {1,1,4}
28: {1,1,4}
36: {1,1,2,2}
40: {1,1,1,3}
40: {1,1,1,3}
45: {2,2,3}
52: {1,1,6}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
68: {1,1,7}
72: {1,1,1,2,2}
Terms appearing twice are
A061351 + 1.
A005117 lists the squarefree numbers.
A070321 gives the greatest squarefree number up to n.
A377046 encodes k-differences of nonsquarefree numbers, zeros
A377050.
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Table[NestWhile[#-1&,Prime[n],#>1&&SquareFreeQ[#]&],{n,100}]
A109945
Primes p such that [p,p+2] is a pair of twin primes and (p*(p+2)-1)/2 is prime.
Original entry on oeis.org
3, 5, 11, 29, 41, 71, 137, 281, 461, 599, 641, 827, 881, 1091, 1301, 1607, 2129, 2267, 2381, 2687, 3527, 3557, 3581, 4127, 4229, 4337, 4547, 5009, 5741, 6131, 6791, 6959, 7211, 7487, 7547, 8009, 8597, 8861, 9041, 9281, 10007, 10037, 10427, 10889, 11117
Offset: 1
3 is in the sequence because [3,5] is a pair of twin primes and (3*5 - 1)/2=7 is prime.
Cf.
A086870 [corresponding primes],
A093706 [primes p such that (p*nextprime(p)-1)/2 is prime],
A061351 [number separating twin pair is squarefree].
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lst={}; d=2; Do[p1=Prime[n]; p2=Prime[n+1]; If[p2-p1==2&&PrimeQ[(p1*p2-1)/2], AppendTo[lst, p1]], {n, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 08 2008 *)
Showing 1-3 of 3 results.
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