A373574
Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.
Original entry on oeis.org
1, 2, 4, 6, 8, 10, 18, 52, 678
Offset: 1
The maximal antiruns of nonsquarefree numbers begin:
4 8
9 12 16 18 20 24
25 27
28 32 36 40 44
45 48
49
50 52 54 56 60 63
64 68 72 75
76 80
81 84 88 90 92 96 98
99
The a(n)-th rows are:
4 8
9 12 16 18 20 24
28 32 36 40 44
49
64 68 72 75
81 84 88 90 92 96 98
148 150 152
477 480 484 486 488 490 492 495
6345 6348 6350 6352 6354 6356 6358 6360 6363
For squarefree runs we have the triple (1,3,5), firsts of
A120992.
For prime runs we have the triple (1,2,3), firsts of
A175632.
For nonsquarefree runs we have
A373199 (assuming sorted), firsts of
A053797.
For composite antiruns we have the triple (1,2,7), firsts of
A373403.
Sorted positions of first appearances in
A373409.
Cf.
A007674,
A025157,
A049094,
A061399,
A068781,
A072284,
A077643,
A110969,
A251092,
A294242,
A373410,
A373412.
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t=Length/@Split[Select[Range[100000],!SquareFreeQ[#]&],#1+1!=#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
A077642
Number of squarefree integers in the closed interval [10^n, -1 + 2*10^n], i.e., among 10^n consecutive integers beginning with 10^n.
Original entry on oeis.org
1, 7, 61, 607, 6077, 60787, 607951, 6079284, 60792732, 607927092, 6079270913, 60792710227, 607927101577, 6079271018873, 60792710185938, 607927101853650, 6079271018542500, 60792710185398417, 607927101854027370, 6079271018540264581, 60792710185402679735, 607927101854026683706
Offset: 0
n=10: among numbers {10,...,19} seven are squarefree [10,11,13,14,15,17,19], so a(1)=7.
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with(numtheory): for n from 0 to 5 do ct:=0: for k from 10^n to 2*10^n-1 do if abs(mobius(k))>0 then ct:=ct+1 else ct:=ct: fi: od: a[n]:=ct: od: seq(a[n],n=0..5); # Emeric Deutsch, Mar 28 2005
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Table[Apply[Plus, Table[Abs[MoebiusMu[10^w+j]], {j, 0, -1+10^(w-1)}]], {w, 0, 6}]
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{ a(n) = sum(m=1,sqrtint(2*10^n-1), moebius(m) * ((2*10^n-1)\m^2 - (10^n-1)\m^2) ) } \\ Max Alekseyev, Oct 18 2008
A373124
Sum of indices of primes between powers of 2.
Original entry on oeis.org
1, 2, 7, 11, 45, 105, 325, 989, 3268, 10125, 33017, 111435, 369576, 1277044, 4362878, 15233325, 53647473, 189461874, 676856245, 2422723580, 8743378141, 31684991912, 115347765988, 421763257890, 1548503690949, 5702720842940, 21074884894536, 78123777847065
Offset: 0
Row-sums of the sequence of all positive integers as a triangle with row-lengths A036378:
1
2
3 4
5 6
7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
For indices of primes between powers of 2:
For primes between powers of 2:
For squarefree numbers between powers of 2:
Cf.
A000040,
A000120,
A014499,
A029837,
A029931,
A035100,
A069010,
A070939,
A112925,
A112926,
A211997.
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Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]
-
ip(n) = primepi(1<A007053
t(n) = n*(n+1)/2; \\ A000217
a(n) = t(ip(n+1)) - t(ip(n)); \\ Michel Marcus, May 31 2024
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