A225243 Irregular triangle read by rows, where row n contains the distinct primes that are contained in the binary representation of n as substrings; first row = [1] by convention.

1, 2, 3, 2, 2, 5, 2, 3, 3, 7, 2, 2, 2, 5, 2, 3, 5, 11, 2, 3, 2, 3, 5, 13, 2, 3, 7, 3, 7, 2, 2, 17, 2, 2, 3, 19, 2, 5, 2, 5, 2, 3, 5, 11, 2, 3, 5, 7, 11, 23, 2, 3, 2, 3, 2, 3, 5, 13, 2, 3, 5, 11, 13, 2, 3, 7, 2, 3, 5, 7, 13, 29, 2, 3, 7, 3, 7, 31, 2, 2, 2, 17

Offset: 1

Author

Reinhard Zumkeller, Aug 14 2013

Keywords

• nonn
• base
• tabf

Comments

Row n = primes in row n of tables A165416 or A119709.

Examples

			.   n   T(n,*)              |  in binary
.  ---  --------------------|-------------------------------------------
.   1:  1                   |  00001:  .
.   2:  2                   |  00100:  ___10
.   3:  3                   |  00011:  ___11
.   4:  2                   |  00100:  __10_
.   5:  2  5                |  00101:  ___10 _11__
.   6:  2  3                |  00110:  ___10 __11_
.   7:  3  7                |  00111:  __11_ __111
.   8:  2                   |  01000:  _10__
.   9:  2                   |  01001:  _10__
.  10:  2  5                |  01010:  _10__ _101_
.  11:  2  3  5 11          |  01011:  _10__ ___11 _101_ 01011
.  12:  2  3                |  01100:  ___10 _11__
.  13:  2  3  5 13          |  01101:  __10_ _11__ __101 01101
.  14:  2  3  7             |  01110:  ___10 _11__ _111_
.  15:  3  7                |  01111:  _11__ _111_
.  16:  2                   |  10000:  10___
.  17:  2 17                |  10001:  10___ 10001
.  18:  2                   |  10010:  10___
.  19:  2  3 19             |  10011:  10___ ___11 10011
.  20:  2  5                |  10100:  10___ 101__
.  21:  2  5                |  10101:  10___ 101__
.  22:  2  3  5 11          |  10110:  10___ __11_ 101__ 10110
.  23:  2  3  5  7 11 23    |  10111:  10___ __11_ 101__ __111 1011_ 10111
.  24:  2  3                |  11000:  _10__ 11___
.  25:  2  3                |  11001:  _10__ 11___ .
		

Links

• Reinhard Zumkeller, Rows n = 1..1024 of table, flattened
• Michael De Vlieger, Plot of pi(p) such that T(n,k) = p for n = 1..4096.

Crossrefs

Cf. A078826 (row lengths), A078832 (left edge), A078833 (right edge), A004676, A007088.

Programs

Haskell

a225243 n k = a225243_tabf !! (n-1) !! (k-1)
a225243_row n = a225243_tabf !! (n-1)
a225243_tabf = [1] : map (filter ((== 1) . a010051')) (tail a165416_tabf)

Mathematica

Array[Union@ Select[FromDigits[#, 2] & /@ Rest@ Subsequences@ IntegerDigits[#, 2], PrimeQ] &, 34] /. {} -> {1} // Flatten (* Michael De Vlieger, Jan 26 2022 *)

Python

from sympy import isprime
from itertools import count, islice
def primess(n):
    b = bin(n)[2:]
    ss = (int(b[i:j], 2) for i in range(len(b)) for j in range(i+2, len(b)+1))
    return sorted(set(k for k in ss if isprime(k)))
def agen():
    yield 1
    for n in count(2):
        yield from primess(n)
print(list(islice(agen(), 82))) # Michael S. Branicky, Jan 26 2022