A378084
Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).
Original entry on oeis.org
9, 25, 27, 28, 36, 45, 49, 50, 52, 56, 64, 76, 81, 88, 92, 96, 99, 100, 117, 120, 121, 124, 125, 126, 135, 136, 144, 147, 148, 153, 156, 162, 169, 171, 172, 176, 188, 189, 204, 207, 208, 216, 220, 225, 236, 243, 244, 245, 248, 250, 256, 261, 268, 275, 276, 280
Offset: 1
The terms together with their prime indices begin:
9: {2,2}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
36: {1,1,2,2}
45: {2,2,3}
49: {4,4}
50: {1,3,3}
52: {1,1,6}
56: {1,1,1,4}
64: {1,1,1,1,1,1}
76: {1,1,8}
81: {2,2,2,2}
88: {1,1,1,5}
92: {1,1,9}
96: {1,1,1,1,1,2}
A005117 lists the squarefree numbers.
A070321 gives the greatest squarefree number up to n.
A112925 gives least squarefree number > prime(n), differences
A378038.
A112926 gives greatest squarefree number < prime(n), differences
A378037.
A377046 encodes k-differences of nonsquarefree numbers, zeros
A377050.
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nn=100;
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,nn}];
Complement[Select[Range[Prime[nn]],!SquareFreeQ[#]&],y]
A378083
Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).
Original entry on oeis.org
4, 8, 32, 44, 104, 140, 284, 464, 572, 620, 644, 824, 860, 1232, 1292, 1304, 1484, 1700, 1724, 1880, 2084, 2132, 2240, 2312, 2384, 2660, 2732, 2804, 3392, 3464, 3560, 3920, 3932, 4004, 4220, 4244, 4424, 4640, 4724, 5012, 5444, 5480, 5504, 5660, 6092, 6200
Offset: 1
The terms together with their prime indices begin:
4: {1,1}
8: {1,1,1}
32: {1,1,1,1,1}
44: {1,1,5}
104: {1,1,1,6}
140: {1,1,3,4}
284: {1,1,20}
464: {1,1,1,1,10}
572: {1,1,5,6}
620: {1,1,3,11}
644: {1,1,4,9}
824: {1,1,1,27}
860: {1,1,3,14}
1232: {1,1,1,1,4,5}
Terms not appearing at all are
A378084.
A005117 lists the squarefree numbers.
A378086(n) =
A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf.
A053797,
A053806,
A070321,
A072284,
A112929,
A120992,
A224363,
A337030,
A377430,
A377431,
A377703.
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y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==2&]
A378374
Perfect powers p such that the interval from the previous perfect power to p contains a unique prime.
Original entry on oeis.org
128, 225, 256, 64009, 1295044
Offset: 1
The consecutive perfect powers 125 and 128 have interval (125, 126, 127, 128) with unique prime 127, so 128 is in the sequence.
For prime powers instead of perfect powers we have
A345531, difference
A377281.
If the same interval contains at least one prime we get
A378250.
For next instead of previous perfect power we have
A378355.
Swapping "prime" with "perfect power" gives
A378364.
A069623 counts perfect powers <= n.
A080769 counts primes between perfect powers.
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radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
y=Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==1&]
A380413
Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).
Original entry on oeis.org
0, 1, 11, 14, 39, 53, 109, 179, 222, 240, 251, 319, 337, 481, 505, 508, 578, 664, 674, 738, 818, 835, 877, 905, 933, 1041, 1069, 1098, 1325, 1352, 1392, 1535, 1539, 1567, 1652, 1663, 1732, 1817, 1849, 1960, 2134, 2148, 2158, 2220, 2387, 2428, 2457, 2622, 2625
Offset: 1
A070321 gives the greatest squarefree number up to n.
A112925 gives the greatest squarefree number between primes, least
A112926.
-
y=Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}];
Select[Most[Union[y]],Count[y,#]==2&]
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