A095062 -id:A095062 - cp's OEIS frontend

A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 7, 7, 0, 3, 0, 3, 0, 11, 0, 0, 3, 13, 61, 0, 0, 0, 0, 3, 3, 0, 3, 7, 0, 45, 3, 0, 0, 0, 0, 7, 7, 35, 0, 7, 35, 3, 0, 3, 3, 0, 3, 15, 0, 0, 3, 15, 3, 0, 0, 7, 3, 0, 45, 3, 0, 0, 3, 3, 7, 7, 0, 3, 0, 3, 0, 3, 0, 0, 7, 7, 0, 0, 0, 67, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Diving index of an odd number n is the first integer u > 1 where Sum_{i=1..u} J(i/n) results -1 and zero if never. Here J(i/n) is Jacobi symbol of i and n, which reduces to a Legendre symbol L(i/n) when n is a prime.

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

a(n)=A095105(n)+1 modulo A002145(n). Cf. A095106, A095108 (same sequence with zeros removed), A095269.

A095105 Length of maximal Dyck path prefix in the Legendre-vector of the n-th 4k+3 prime (A002145(n)).

Original entry on oeis.org

2, 6, 10, 2, 22, 30, 2, 46, 58, 2, 70, 78, 82, 102, 6, 6, 130, 2, 150, 2, 166, 10, 190, 198, 2, 12, 60, 238, 250, 262, 270, 2, 2, 310, 2, 6, 358, 44, 2, 382, 418, 430, 438, 6, 6, 34, 478, 6, 34, 2, 502, 2, 2, 562, 2, 14, 598, 606, 2, 14, 2, 646, 658, 6, 2, 718

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

a(n)=A095104(n)-1 modulo A002145(n). Cf. A095107, A095270.

A095106 Sum of diving indices of all 4k+3 primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 3, 3, 17, 94, 158, 213, 571, 1987, 3163, 27993, 45176, 205432, 707079, 2319521, 7603409, 25967812, 83981455, 310267532

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095104, A095107, A095109.

A095107 Sum of maximal Dyck path prefix lengths of all 4k+3 primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

2, 6, 10, 54, 106, 346, 1410, 3994, 12828, 46242, 158252, 542568, 1865796, 6822592, 25120772, 88926198, 327561126, 1212380122, 4358259840, 16111686770, 59936356888

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095105, A095106, A095110.

A095270 Length of maximal Motzkin path prefix in the Jacobi-vector of 4n+3.

Original entry on oeis.org

2, 6, 10, 14, 2, 22, 26, 30, 34, 38, 2, 46, 9, 54, 58, 62, 2, 70, 74, 78, 82, 86, 2, 94, 9, 102, 6, 110, 2, 118, 6, 6, 130, 134, 2, 142, 7, 150, 7, 158, 2, 166, 170, 174, 10, 182, 2, 190, 33, 198, 7, 18, 2, 214, 9, 12, 60, 230, 2, 238, 242, 6, 250, 254, 2, 262

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A004767, A095269, A095110, A095105.

a(n) = A095269(n)-1 modulo A004767(n).

A095272 a(n) = (A095102(n)-3)/4.

Original entry on oeis.org

0, 1, 2, 5, 7, 11, 14, 17, 19, 20, 25, 32, 37, 41, 47, 49, 59, 62, 65, 67, 77, 89, 95, 104, 107, 109, 119, 125, 140, 149, 151, 161, 164, 179, 185, 187, 209, 215, 221, 227, 229, 242, 245, 247, 257, 259, 265, 272, 275, 287, 305, 307, 319, 329, 349

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Complement of A095273 in A095278, subset of A095274.

A095273 a(n) = (A095103(n)-3)/4.

Original entry on oeis.org

4, 10, 16, 26, 31, 34, 40, 44, 52, 55, 56, 70, 76, 82, 86, 91, 94, 110, 115, 116, 121, 122, 124, 130, 136, 142, 146, 154, 157, 160, 170, 172, 181, 184, 196, 202, 205, 206, 214, 220, 226, 236, 241, 254, 262, 271, 280, 290, 292, 296, 314, 320, 322

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Complement of A095272 in A095278, subset of A095275.

A095275 a(n) = (A095101(n)-3)/4.

Original entry on oeis.org

4, 10, 12, 16, 22, 24, 26, 28, 30, 31, 34, 36, 38, 40, 44, 46, 48, 50, 51, 52, 54, 55, 56, 58, 61, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 91, 94, 96, 100, 102, 105, 106, 108, 110, 112, 114, 115, 116, 118, 120, 121, 122, 124, 126, 128, 130, 132

Offset: 0

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095101. Complement of A095274. Subset: A095273.

A095280 Lower Wythoff primes, i.e., primes in A000201.

Original entry on oeis.org

3, 11, 17, 19, 29, 37, 43, 53, 59, 61, 67, 71, 79, 97, 101, 103, 113, 127, 131, 137, 139, 163, 173, 179, 181, 197, 199, 211, 223, 229, 239, 241, 257, 263, 271, 281, 283, 307, 313, 317, 331, 347, 349, 359, 367, 373, 383, 389, 401, 409, 419, 433

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Contains all primes p whose Zeckendorf-expansion A014417(p) ends with an even number of 0's.

For generalizations and conjectures, see A184774.

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & A000201. Complement of A095281 in A000040. Cf. A095080, A095083, A095084, A095290, A184792, A184793, A184794, A184796.

R:= NULL: count:= 0:
for n from 1 while count < 100 do
  p:= floor(n*phi);
  if isprime(p) then R:= R,p; count:= count+1 fi
od:
R; # Robert Israel, Jan 17 2023

Mathematica
```
(See A184792.)
```

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A095280_gen(): # generator of terms
    return filter(isprime,((n+isqrt(5*n**2)>>1) for n in count(1)))
A095280_list = list(islice(A095280_gen(),30)) # Chai Wah Wu, Aug 16 2022

A095282 Primes whose binary-expansion ends with an even number of 1's.

Original entry on oeis.org

2, 3, 11, 19, 43, 47, 59, 67, 79, 83, 107, 131, 139, 163, 179, 191, 211, 227, 239, 251, 271, 283, 307, 331, 347, 367, 379, 419, 431, 443, 463, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 719, 739, 751, 787, 811, 827, 859

Offset: 1

Antti Karttunen, Jun 04 2004

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & (complement of A079523). Complement of A095283 in A000040. Cf. A027699, A095292.

q:= proc(n) local i, l, r; l, r:= convert(n, base, 2), 0;
      for i to nops(l) while l[i]=1 do r:=r+1 od; is(r, even)
    end:
select(q, [ithprime(i)$i=1..200])[];  # Alois P. Heinz, Dec 15 2019

been1Q[n_]:=Module[{c=Split[IntegerDigits[n,2]][[-1]]},c[[1]]==1&&EvenQ[ Length[ c]]]; Join[{2},Select[Prime[Range[150]],been1Q]] (* Harvey P. Dale, Dec 14 2019 *)

is(n)=valuation(n+1,2)%2==0 && isprime(n) \\ Charles R Greathouse IV, Oct 09 2013

cp's OEIS Frontend

A095104 Diving index of the n-th 4k+3 prime (A002145(n)).

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A095105 Length of maximal Dyck path prefix in the Legendre-vector of the n-th 4k+3 prime (A002145(n)).

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A095106 Sum of diving indices of all 4k+3 primes in range ]2^n,2^(n+1)].

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A095107 Sum of maximal Dyck path prefix lengths of all 4k+3 primes in range ]2^n,2^(n+1)].

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A095270 Length of maximal Motzkin path prefix in the Jacobi-vector of 4n+3.

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Formula

A095272 a(n) = (A095102(n)-3)/4.

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A095273 a(n) = (A095103(n)-3)/4.

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A095275 a(n) = (A095101(n)-3)/4.

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A095280 Lower Wythoff primes, i.e., primes in A000201.

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Comments

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Programs

Maple

Mathematica

Python

A095282 Primes whose binary-expansion ends with an even number of 1's.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI