cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A277022 Primorial base representation of n is rewritten as a base-2 number with each nonzero digit k replaced by a run of k 1's (followed by one extra zero if not the rightmost run of 1's) and with each 0 kept as 0.

Original entry on oeis.org

0, 1, 2, 5, 6, 13, 4, 9, 10, 21, 22, 45, 12, 25, 26, 53, 54, 109, 28, 57, 58, 117, 118, 237, 60, 121, 122, 245, 246, 493, 8, 17, 18, 37, 38, 77, 20, 41, 42, 85, 86, 173, 44, 89, 90, 181, 182, 365, 92, 185, 186, 373, 374, 749, 188, 377, 378, 757, 758, 1517, 24, 49, 50, 101, 102, 205, 52, 105, 106, 213, 214, 429, 108, 217, 218, 437, 438, 877, 220
Offset: 0

Views

Author

Antti Karttunen, Sep 26 2016

Keywords

Examples

			9 = "111" in primorial base (A002110(0) + A002110(1) + A002110(2) = 9) is converted to three 1-bits, with separating zeros, in binary as "10101" = A007088(21), thus a(9) = 21.
91 = "3001" in primorial base (91 = 3*A002110(3) + A002110(0)) is converted to binary number "1110001" = A007088(113), thus a(91) = 113. Note how two of the zeros come from the primorial base representation and the third zero is an extra separating zero inserted after each run of 1-bits apart from the rightmost 1-run.
120 = "4000" in primorial base (120 = 4*A002110(3)) is converted to the binary number "1111000" = A007088(120), thus a(120) = 120.

Links

Crossrefs

Cf. A000035, A000079, A000120, A000225, A005940, A007088, A049345, A156552, A267263, A276084, A276086, A276088, A276093, A276150.
Cf. A277018 (terms sorted into ascending order).
Cf. A277021 (a left inverse).
Differs from analogous A277012 for the first time at n=24, where a(24) = 60, while A277012(24) = 8.

Formula

a(0) = 0; for n >= 1, a(n) = A000225(A276088(n))*A000079(A276084(n)) + A000079(A276088(n))*a(A276093(n)).
a(n) = A156552(A276086(n)).
Other identities. For all n >= 0:
A277021(a(n)) = n.
A005940(1+a(n)) = A276086(n).
A000035(a(n)) = A000035(n). [Preserves the parity of n.]
A000120(a(n)) = A276150(n).
A069010(a(n)) = A267263(n).

A277008 Numbers k such that in the binary expansion of k no run of 1-bits is longer than 1 + the total number of 0-bits anywhere to the right of that run.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 52, 53, 54, 56, 57, 58, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 80, 81, 82, 84, 85, 86, 88, 89, 90, 92, 96, 97, 98, 100, 101, 102, 104, 105, 106, 108, 109, 112, 113, 114, 116, 117, 118, 120, 128
Offset: 0

Views

Author

Antti Karttunen, Sep 25 2016

Keywords

Comments

Numbers k for which A277007(k) = 0.
Indexing starts from zero as a(0) = 0 is a special case in this sequence.

Links

Crossrefs

Complement: A277009.
Positions of zeros in A277007.
Sequence A277012 sorted into ascending order.
Cf. A005940, A276077, A276078.
Subsequence of A277018 from which this differs for the first time at n=41, where a(41)=64, skipping the value 60 present in A277018.

Programs

Formula

Other identitities:
A276077(A005940(1+a(n))) = 0 for all n.

A277017 Number of maximal runs of 1-bits (in binary expansion of n) such that the length of run >= A000040(1 + the total number of zeros to the right of that run).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0
Offset: 0

Views

Author

Antti Karttunen, Sep 26 2016

Keywords

Comments

a(n) = number of 1-runs in binary expansion of n that exceed the length allotted to that run by primorial base coding used in A277022. If a(n) = 0, then n is in the range of A277022.

Examples

			For n=3, "11" in binary, the only maximal run of 1-bits is of length 2, and 2 >= prime(0+1) (where 0 is the total number of zeros to the right of it), thus a(3) = 1.
For n=59, "111011" in binary, both the length of run "11" at the least significant end exceeds the limit (see case n=3 above), and also the length of run "111" >= prime(1 + the total number of 0's to the right of it) = prime(2) = 3, thus a(59) = 1+1 = 2.
For n=60, "111100" in binary, the length of only run of 1's is 4, and 4 < prime(2+1) = 5, thus a(60) = 0.

Links

Crossrefs

Cf. A277018 (positions of zeros), A277019 (of nonzeros).
Cf. A000040, A005940, A129251, A277021, A277022.
Differs from similar A277007 for the first time at n=60, where a(60)=0, while A277007(60)=1.

Programs

  • Scheme
    (define (A277017 n) (let loop ((e 0) (n n) (z 0) (r 0)) (cond ((zero? n) (+ e (if (>= r (A000040 (+ 1 z))) 1 0))) ((even? n) (loop (+ e (if (>= r (A000040 (+ 1 z))) 1 0)) (/ n 2) (+ 1 z) 0)) (else (loop e (/ (- n 1) 2) z (+ 1 r))))))

Formula

a(n) = A129251(A005940(1+n)).

A277019 Numbers not in range of A277022.

Original entry on oeis.org

3, 7, 11, 14, 15, 19, 23, 27, 29, 30, 31, 35, 39, 43, 46, 47, 51, 55, 59, 61, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 93, 94, 95, 99, 103, 107, 110, 111, 115, 119, 123, 124, 125, 126, 127, 131, 135, 139, 142, 143, 147, 151, 155, 157, 158, 159, 163, 167, 171, 174, 175, 179, 183, 187, 189, 190, 191, 195, 199, 203, 206, 207, 211, 215
Offset: 1

Views

Author

Antti Karttunen, Sep 26 2016

Keywords

Comments

Numbers such that at least one run of 1-bits in their binary expansion has length >= A000040(1 + the total number of 0-bits anywhere right of that run).
Numbers n for which A277017(n) > 0.
Numbers n for which A129251(A005940(1+n)) > 0.

Examples

			3 ("11" in binary, A007088) is present as the length of that only run of 1's is 2, and 2 >= prime(1+0), where 0 is the total number of 0's to the right of that run.
60 ("111100" in binary) is NOT present as 4 < 5 = prime(2+1).

Links

Crossrefs

Complement: A277018.
Positions of nonzeros in A277017.
Cf. A000040, A005940, A129251.
Subsequence of A277009 from which this differs for the first time at n=20, where a(20)=61, skipping the value 60 present in A277009.
Showing 1-4 of 4 results.

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