A078827 -id:A078827 - cp's OEIS frontend

A078826 Number of distinct primes contained as binary substrings in binary representation of n.

0, 0, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 2, 4, 3, 2, 1, 2, 1, 3, 2, 2, 4, 6, 2, 2, 4, 5, 3, 6, 3, 3, 1, 1, 2, 3, 1, 3, 3, 4, 2, 3, 2, 5, 4, 5, 6, 7, 2, 3, 2, 3, 4, 5, 5, 7, 3, 3, 6, 8, 3, 7, 4, 3, 1, 1, 1, 3, 2, 3, 3, 5, 1, 2, 3, 5, 3, 5, 4, 5, 2, 3, 3, 6, 2, 2, 5, 7, 4, 5, 5, 5, 6, 8, 7, 8, 2, 3, 3, 3, 2, 5, 3, 5, 4

Offset: 0

Reinhard Zumkeller, Dec 08 2002

A143792(n) <= a(n) for n > 0. - Reinhard Zumkeller, Sep 08 2008

For n > 1: number of primes in n-th row of A165416, lengths in n-th row of A225243. - Reinhard Zumkeller, Jul 17 2015, Aug 14 2013

			n=7 -> '111' contains 2 different binary substrings which are primes: '11' (11b or b11) and '111' itself, therefore a(7)=2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004676, A035232, A078827, A078822, A007088, A001221.

Cf. A143792, A165416, A225243.

a078826 n | n <= 1 = 0
          | otherwise = length $ a225243_row n
-- Reinhard Zumkeller, Aug 14 2013

a[n_] := (bits = IntegerDigits[n, 2]; lg = Length[bits]; Reap[Do[If[PrimeQ[p = FromDigits[bits[[i ;; j]], 2]], Sow[p]], {i, 1, lg-1}, {j, i+1, lg}]][[2, 1]] // Union // Length); a[0] = a[1] = 0; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, May 23 2013 *)

A078830 Numbers having in binary representation exactly one binary substring representing a prime.

Original entry on oeis.org

2, 3, 4, 8, 9, 16, 32, 33, 64, 65, 128, 129, 256, 512, 513, 1024, 1025, 2048, 2049, 4096, 4097, 8192, 8193, 16384, 16385, 32768, 32769, 65536, 131072, 131073, 262144, 262145, 524288, 524289, 1048576, 1048577, 2097152, 2097153, 4194304, 4194305

Offset: 1

Reinhard Zumkeller, Dec 08 2002

A078827(a(n)) = 1; subsequence of A078829 and of A078831;

for k>2 also floor(a(k)/2) belongs to the sequence.

			p=2 -> '10' is the only prime contained exactly once in 2^k, k>0.

Cf. A078827, A078829, A007088, A000079, A048578.

Extended by Ray Chandler, Nov 03 2008

A078831 Numbers n with unique occurrence of all binary substrings representing primes in binary representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 19, 24, 25, 32, 33, 35, 48, 49, 64, 65, 67, 96, 97, 128, 129, 131, 192, 193, 256, 257, 259, 384, 385, 512, 513, 515, 768, 769, 1024, 1025, 1027, 1536, 1537, 2048, 2049, 2051, 3072, 3073, 4096, 4097, 4099, 6144, 6145

Offset: 1

Reinhard Zumkeller, Dec 08 2002

A078826(a(n)) = A078827(a(n)); A078830 is a subsequence.

			n=12 -> '1100' contains two substrings representing distinct primes: '10'=2 and '11'=3, therefore 12 is a term.

Ray Chandler, Table of n, a(n) for n = 1..110

Cf. A078826, A078827, A078830, A007088.

Extended by Ray Chandler, Nov 03 2008

A078828 Product of all primes contained as binary substrings in binary representation of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 10, 6, 63, 2, 2, 20, 330, 6, 390, 126, 1323, 2, 34, 4, 114, 20, 100, 660, 159390, 6, 6, 780, 12870, 126, 237510, 2646, 861273, 2, 2, 68, 102, 4, 740, 228, 2394, 20, 820, 200, 141900, 660, 42900, 318780, 157317930, 6, 102, 12, 342, 780, 206700

Offset: 0

Reinhard Zumkeller, Dec 08 2002

			n=7: product of the A078827(7)=3 primes as binary substrings in binary representation of 7 -> '111': a(7) = '11'*'11'*'111' = 3*3*7 = 63.

Cf. A078827, A078822, A007088.

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A078826 Number of distinct primes contained as binary substrings in binary representation of n.

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A078830 Numbers having in binary representation exactly one binary substring representing a prime.

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A078831 Numbers n with unique occurrence of all binary substrings representing primes in binary representation of n.

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A078828 Product of all primes contained as binary substrings in binary representation of n.

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