A014311 -id:A014311 - cp's OEIS frontend

A280997 Primes that have exactly 3 ones in both their binary and ternary expansions.

13, 37, 41, 67, 97, 131, 193, 577, 1033, 1153, 2053, 4129, 8209, 18433, 32771, 32801, 32833, 65539, 133121, 525313, 557057, 1049089, 4194433, 167772161, 268435459

Offset: 1

K. D. Bajpai, Jan 12 2017

Sequence is likely to be finite. If it exists, a(26) > 10^200. - Robert Israel, Jan 12 2017

			37 is in the sequence because it is a prime and its binary expansion 100101 and ternary expansion 1101 both have exactly 3 ones.
131 is in the sequence because it is a prime and its binary expansion 10000011 and ternary expansion 11212 both have exactly 3 ones.

Cf. A000040, A001363, A007088, A014311, A066196.

Subset of A281004.

A:= NULL:
for a from 2 to 100 do
  for b from 1 to a-1 do
    p:= 2^a + 2^b + 1;
    if numboccur(1, convert(p,base,3)) = 3 and isprime(p) then
      A:= A, p
    fi
od od:
A; # Robert Israel, Jan 12 2017

Select[Prime[Range[500000]], Count[IntegerDigits[#, 3], 1] == Count[IntegerDigits[#, 2], 1] == 3 &]
Select[Prime[Range[300000]],DigitCount[#,2,1]==DigitCount[#,3,1]==3&] (* The program generates the first 23 terms of the sequence. *) (* Harvey P. Dale, Jul 20 2025 *)

A362950 Bitmaps of king + rook vs. king checkmate patterns on a standard 8 X 8 board.

Original entry on oeis.org

65541, 65541, 65545, 65553, 65569, 65601, 65665, 66561, 131077, 131081, 131082, 131089, 131090, 131105, 131106, 131137, 131138, 131201, 131202, 262149, 262164, 262180, 262212, 262276, 327681, 524297, 524298, 524328, 524360, 524424, 1048593, 1048594

Offset: 1

M. F. Hasler, Jun 13 2023

For each of the 216 possible checkmate positions with K, R and k on one of the squares numbered from 0 to 63, the sequence lists the value 2^K + 2^R + 2^k, see examples.

With the usual rules of chess, this allows the unique determination of the square on which each piece is located, except for one single exception where the black king is in the corner and the white K & R are both at the distance of one (empty) square from it, also on the border: the position where K & R change places has the same bitmap. Therefore we list this value twice, a(1) = a(2) = 2^0 + 2^2 + 2^16 = 65541, and we do the same for the three values corresponding to the equivalent position in the other corners.

			The first term, a(1) = 2^0 + 2^2 + 2^16 = 65541, corresponds to the following position (omitting the first 4 empty ranks = rows):
      ........
      K.......
      ........
      k.R.....
where the black king "k" is in the corner on square a1 numbered 0, the white rook "R" is giving check from square c1 numbered 2, and the white king "K" is on square a3 numbered 16.
This is the only bitmap (up to 4-fold symmetry) that is ambiguous in that it represents two distinct positions: The white king and rook can exchange places, which yields a different checkmate position with the same bitmap. Therefore this term is listed twice, also as a(2). (The duplication of just the initial term (which is the only duplicate among the displayed data) does not affect the search results negatively, in case anyone searches for the sequence without duplicates.)
The next term, a(3) = 65545 = 2^0 + 2^3 + 2^16, corresponds to the position where the rook "R" is one square farther to the right (on square d1 numbered 3). There is no ambiguity here, for an exchange of "R" and "K", or any other two or three pieces on the same three squares, would not be a checkmate position.

M. F. Hasler, Table of n, a(n) for n = 1..216
OEIS Index to sequences related to chess.

Subsequence of A014311 (numbers with Hamming weight 3).

Cf. A000120 (Hamming weight), A007088 (binary numbers).

/* helper functions */ SYM(S, ROT=concat(Vecrev(matrix(8,,i,j,2^(i*8+j-9)))))=concat([ S=if(k, TRANS(S, ROT), concat(S, TRANS(S)))| k<-[0..3]])
TRANS(S, FLIP=[2^(k+7-k%8*2) | k<-[0..31]])=[vecsum(vecextract(FLIP, p)) | p<-S]
A362950=vecsort( SYM( concat([ [1+2^R+2^16 | R<-[2..7]] , [1+2^R+2^17 | R<-[2..7]], [2+2^R+2^17 | R<-[3..7]], concat([[ (1+2^16)<1] | K <- [2..3] ])  ])))

A367110 Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.

Original entry on oeis.org

1, 4, 2, 8, 5, 9, 1, 5, 4, 5, 8, 5, 2, 6, 3, 8, 1, 2, 3, 9, 9, 6, 8, 5, 4, 8, 4, 4, 4, 0, 0, 5, 3, 7, 9, 5, 2, 7, 8, 1, 6, 8, 8, 7, 5, 0, 9, 0, 6, 1, 3, 3, 0, 6, 8, 3, 9, 7, 1, 8, 9, 5, 2, 9, 7, 7, 5, 3, 6, 5, 9, 5, 0, 0, 3, 9, 7, 4, 4, 5, 2, 9, 6, 8, 0, 0, 5, 1, 1, 6, 3, 5, 7, 0, 8, 6, 2, 2, 7, 2, 7, 1, 9, 1, 5

Offset: 1

Tengiz Gogoberidze, Dec 16 2023

For 1 bit equal to 1 the sum is 2, for 2 bits equal to 1 the sum is 1.52899956069688841838263949451... (see A179951).

			1.4285915458526381...

Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Cf. A179951, A014311.

RealDigits[iSum[1, 3, 105, 2]][[1]] (* Amiram Eldar, Dec 16 2023, using Baillie's irwinSums.m *)

Equals Sum_{m>=2} Sum_{j=1..m-1} Sum_{i=0..j-1} 1/(2^i + 2^j + 2^m).
Equals 2 * Sum_{j>=2} Sum_{i=1..j-1} 1/(2^i + 2^j + 1).
Equals Sum_{k>=1} 1/A014311(k).

A367680 Number of integer compositions x1+x2+...+xk of n such that each xj has exactly j bits set.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 1, 3, 2, 1, 2, 4, 2, 4, 6, 2, 4, 5, 10, 7, 10, 12, 8, 6, 11, 14, 16, 13, 16, 16, 14, 14, 30, 32, 19, 35, 28, 23, 27, 38, 36, 47, 44, 42, 55, 52, 51, 85, 88, 74, 84, 84, 72, 81, 102, 110, 122, 115, 108, 132, 137, 136, 179, 195, 164, 160, 181

Offset: 0

Arnauld Chevallier, Nov 26 2023

			There are 6 such compositions for n = 14:
  14 = 1 + 6 + 7 (1 + 110 + 111)
  14 = 2 + 5 + 7 (10 + 101 + 111)
  14 = 2 + 12 (10 + 1100)
  14 = 4 + 3 + 7 (100 + 11 + 111)
  14 = 4 + 10 (100 + 1010)
  14 = 8 + 6 (1000 + 110)
Therefore a(14) = 6.

Cf. A000079, A000120, A018900, A014311, A014313, A023688, A023689, A023690, A023691.

a(n) = my(nb=0); forpart(v=n, if (vecsort(apply(hammingweight, Vec(v))) == [1..#v], nb++)); nb; \\ Michel Marcus, Nov 28 2023

cp's OEIS Frontend

A280997 Primes that have exactly 3 ones in both their binary and ternary expansions.

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Maple

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A362950 Bitmaps of king + rook vs. king checkmate patterns on a standard 8 X 8 board.

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PARI

A367110 Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.

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Mathematica

Formula

A367680 Number of integer compositions x1+x2+...+xk of n such that each xj has exactly j bits set.

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Author

Keywords

Examples

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Programs

PARI