A143792 -id:A143792 - 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 *)

A143791 A positive integer k is included if no prime divisor p of k, when p is represented in binary, occurs within k represented in binary.

Original entry on oeis.org

1, 9, 21, 25, 33, 35, 49, 65, 69, 77, 81, 115, 121, 129, 133, 143, 145, 161, 169, 203, 209, 217, 253, 259, 261, 265, 273, 275, 289, 295, 297, 299, 301, 305, 319, 321, 323, 329, 341, 361, 377, 385, 391, 403, 415, 427, 437, 451, 481, 505, 513, 515, 517, 527, 529

Offset: 1

Leroy Quet, Sep 01 2008

This sequence contains no primes.

This sequence contains no even numbers (A014076). - Robert G. Wilson v, Sep 22 2008

			21 is binary is 10101. The prime divisors of 21 are 3 and 7. 3 is 11 in binary, which does not occur within 10101. 7 is 111 in binary, which also does not occur within 10101. So 21 is in the sequence.
On the other hand, 27 in binary is 11011. The only prime divisor of 27 is 3, which is 11 in binary. 11 does occur (twice) within 11011 like so: (11)0(11). So 27 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A143792.

f[n_] := Block[{nb = ToString@ FromDigits@ IntegerDigits[n, 2], psb = ToString@ FromDigits@ IntegerDigits[ #, 2] & /@ First@ Transpose@ FactorInteger@ n, c = 0, k = 1}, lmt = 1 + Length@ psb; While[ k < lmt, If[ StringCount[ nb, psb[[k]]] > 0, c++ ]; k++ ]; c]; f[1] = 0; Select[ Range@ 1000, f@# == 0 &] (* Robert G. Wilson v, Sep 22 2008 *)
npdQ[k_]:=Max[SequenceCount[IntegerDigits[k,2],IntegerDigits[#,2]]&/@FactorInteger[k][[;;,1]]]==0; Join[{1},Select[Range[600],npdQ]] (* Harvey P. Dale, Dec 03 2024 *)

a(7) and further terms from Robert G. Wilson v, Sep 22 2008

cp's OEIS Frontend

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

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