A320674
Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [m == 0 (mod prime(i))] for i = 1..k (where prime(i) denotes the i-th prime number and [] is an Iverson bracket).
Original entry on oeis.org
2, 4, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608
Offset: 1
The initial terms, alongside their binary representation and the prime divisors encoded therein, are:
n a(n) bin(a(n)) First prime divisors
-- -------- -------------------------- --------------------
1 2 10 2
2 4 100 2
3 6 110 2, 3
4 8 1000 2
5 10 1010 2, 5
6 12 1100 2, 3
7 16 10000 2
8 20 10100 2, 5
9 24 11000 2, 3
...
71 33554434 10000000000000000000000010 2, 97
...
33554434 is in the sequence because its binary expansion 10000000000000000000000010 of length 26 has a 1 in the 1st place and in the 25th place from the left and 0 elsewhere. As it is divisible by the 1st and 25th prime and by no other prime with index <= 26, 33554434 in the sequence. - _David A. Corneth_, Oct 20 2018
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selQ[n_] := With[{bb = IntegerDigits[n, 2]}, (Prime /@ Flatten[Position[bb, 1]]) == FactorInteger[n][[All, 1]]];
Select[Range[2, 200000], selQ] (* Jean-François Alcover, Nov 01 2018 *)
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is(n) = my (b=binary(n)); b==vector(#b, k, n%prime(k)==0)
A320675
Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [gcd(m, i)=1] for i = 1..k (where [] is an Iverson bracket).
Original entry on oeis.org
1, 2, 3, 7, 10, 20, 27, 31, 40, 127, 138, 219, 245, 276, 552, 650, 682, 1364, 2047, 2728, 8191, 10922, 13515, 14043, 32747, 112347, 131071, 524287, 2796202, 3459945, 5592404, 7124187, 8388607, 8530050, 10660010, 11184808, 16645111, 17060100, 21320020, 33554431
Offset: 1
The first terms, alongside their binary representation and the coprime numbers encoded therein, are:
n a(n) bin(a(n)) First numbers coprime
-- ---- --------- ---------------------
1 1 1 1
2 2 10 1
3 3 11 1, 2
4 7 111 1, 2, 3
5 10 1010 1, 3
6 20 10100 1, 3
7 27 11011 1, 2, 4, 5
8 31 11111 1, 2, 3, 4, 5
9 40 101000 1, 3
10 127 1111111 1, 2, 3, 4, 5, 6, 7
A358970
Nonnegative numbers m such that if 2^k appears in the binary expansion of m, then k+1 divides m.
Original entry on oeis.org
0, 1, 2, 6, 8, 12, 36, 60, 128, 136, 168, 261, 288, 520, 530, 540, 630, 640, 1056, 2052, 2088, 2100, 2184, 2208, 2304, 2340, 2520, 2580, 4134, 8232, 8400, 8820, 9240, 10248, 10920, 16440, 16560, 16920, 16950, 17010, 17040, 17190, 17280, 18480, 18600, 18720
Offset: 1
60 = 2^5 + 2^4 + 2^3 + 2^2 and 60 is divisible by 5+1, 4+1, 3+1 and 2+1, so 60 belongs to the sequence.
42 = 2^5 + 2^3 + 2^1 and 42 is not divisible by 3+1, so 42 does not belong to the sequence.
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Select[Range[20000], Function[n, AllTrue[Position[Reverse@ IntegerDigits[n, 2], 1][[All, 1]], Divisible[n, #] &]]] (* Michael De Vlieger, Dec 12 2022 *)
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is(n) = { my (r=n, k); while (r, r-=2^k=valuation(r,2); if (n%(k+1), return (0););); return (1); }
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def ok(n): return all(n%(k+1) == 0 or not n&(1<Michael S. Branicky, Dec 07 2022
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from itertools import count, islice
def A358970_gen(startvalue=0): # generator of terms >= startvalue
return filter(lambda n:not any(n%i for i,b in enumerate(bin(n)[:1:-1],1) if b=='1'),count(max(startvalue,0)))
A358970_list = list(islice(A358970_gen(),20)) # Chai Wah Wu, Dec 12 2022
Showing 1-3 of 3 results.
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