A095062 -id:A095062 - cp's OEIS frontend

A095079 Primes with two 0-bits in their binary expansion.

19, 43, 53, 79, 103, 107, 109, 367, 379, 431, 439, 443, 463, 487, 491, 499, 751, 863, 887, 983, 1013, 1279, 1471, 1531, 1663, 1759, 1783, 1787, 1789, 1951, 1979, 1999, 2011, 2027, 2029, 3067, 3581, 3823, 4027, 5119, 6079, 6911, 7039, 7103

Offset: 1

Antti Karttunen, Jun 01 2004

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Cf. A095059.

Select[Prime[Range[1000]], DigitCount[#, 2, 0] == 2 &]

{ forprime(p=2,8000,
  v=binary(p); s=0;
  for(k=1,#v, s+=if(v[k]==0,+1,0));
  if(s==2,print1(p,", "))
) }

from sympy import isprime
from itertools import combinations, count, islice
def agen(): # generator of terms
    for d in count(2):
        b = (1<<(d+2))-1
        for i, j in combinations(range(d), 2):
            if isprime(t:=b-(1<<(d-i))-(1<<(d-j))):
                yield t
print(list(islice(agen(), 43))) # Michael S. Branicky, Dec 27 2023

A095085 Fib000 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with three zeros.

Original entry on oeis.org

5, 13, 29, 47, 73, 89, 97, 107, 131, 149, 157, 173, 191, 199, 233, 241, 251, 293, 317, 419, 461, 479, 487, 521, 547, 563, 631, 673, 683, 691, 733, 751, 809, 827, 877, 911, 919, 937, 953, 971, 1013, 1021, 1039, 1063, 1097, 1123, 1249, 1259

Offset: 1

Antti Karttunen, Jun 01 2004

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

Intersection of A000040 and A101345.

Cf. A014417, A095065.

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return str(a(n))[-3:]=="000"
print([n for n in primerange(1, 1261) if ok(n)]) # Indranil Ghosh, Jun 08 2017

A095086 Fib001 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros and final 1.

Original entry on oeis.org

19, 43, 53, 61, 103, 137, 163, 179, 197, 229, 239, 263, 281, 307, 331, 349, 383, 433, 467, 509, 569, 577, 619, 653, 739, 773, 797, 823, 839, 857, 883, 907, 941, 967, 1009, 1051, 1061, 1069, 1103, 1129, 1153, 1171, 1187, 1213, 1229, 1289

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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

Intersection of A000040 and A095098. Cf. A014417, A095066.

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return str(a(n))[-3:]=="001"
print([n for n in primerange(1, 1301) if ok(n)]) # Indranil Ghosh, Jun 08 2017

A095089 Fib101 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends as one, zero, one.

Original entry on oeis.org

17, 59, 67, 101, 127, 211, 271, 313, 347, 373, 389, 449, 457, 491, 499, 593, 601, 643, 661, 677, 787, 821, 881, 983, 991, 1033, 1093, 1109, 1237, 1279, 1321, 1381, 1423, 1499, 1559, 1567, 1601, 1609, 1669, 1753, 1787, 1847, 1889, 1931, 1999

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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

Intersection of A000040 and A134860. Cf. A014417, A095069.

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return str(a(n))[-3:]=="101"
print([n for n in primerange(1, 2001) if ok(n)]) # Indranil Ghosh, Jun 08 2017

A095090 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is a Motzkin-path (A095100).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 33, 60, 108, 202, 360, 703, 1328, 2519, 4779, 9103, 17501, 33473, 64761

Offset: 1

Antti Karttunen and Jun 01 2004

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

Cf. A095092, A095100.

is(m) = {my(s=0); for(i=1, m-1, if((s+=kronecker(i, m))<0, return(0))); 1; }
a(n) = {my(c=0); forstep(m=2^n+3-(n==1), 2^(n+1), 4, c+=is(m)); c; } \\ Jinyuan Wang, Jul 20 2020

a(n) = 2^(n-2) - A095091(n) for n > 1.

A095108 Diving index of the n-th diving 4k+3 prime (A095103(n)).

Original entry on oeis.org

3, 3, 3, 7, 7, 3, 3, 11, 3, 13, 61, 3, 3, 3, 7, 45, 3, 7, 7, 35, 7, 35, 3, 3, 3, 3, 15, 3, 15, 3, 7, 3, 45, 3, 3, 3, 7, 7, 3, 3, 3, 7, 7, 67, 3, 7, 3, 7, 3, 7, 57, 7, 3, 7, 15, 7, 45, 7, 23, 3, 3, 11, 7, 3, 89, 13, 55, 3, 45, 35, 3, 7, 7, 3, 3, 3, 13, 7, 15, 3, 19, 3, 7, 3, 7, 3, 99

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

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

Nonzero terms of A095104. Cf. A095271.

A095281 Upper Wythoff primes, i.e., primes in A001950.

Original entry on oeis.org

2, 5, 7, 13, 23, 31, 41, 47, 73, 83, 89, 107, 109, 149, 151, 157, 167, 191, 193, 227, 233, 251, 269, 277, 293, 311, 337, 353, 379, 397, 421, 431, 439, 463, 479, 523, 541, 547, 557, 599, 607, 617, 641, 659, 683, 691, 701, 709, 719, 727, 733, 743

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

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

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

Intersection of A000040 & A001950. Complement of A095280 in A000040. Cf. A095081, A095083, A095084, A095290.

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A095281_gen(): # generator of terms
    return filter(isprime,((n+isqrt(5*n**2)>>1)+n for n in count(1)))
A095281_list = list(islice(A095281_gen(),30)) # Chai Wah Wu, Aug 16 2022

A095290 Number of lower Wythoff primes (A095280) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 0, 1, 3, 5, 8, 14, 30, 40, 86, 162, 289, 541, 1017, 1881, 3527, 6652, 12641, 23855, 45455, 86753, 165844, 317363, 609942, 1171377, 2253588, 4343268, 8381084, 16198859, 31329311, 60683252, 117637523, 228259189

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095291(n). Cf. A095060, A095291.

A095296 Number of A095286-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 16, 21, 48, 69, 175, 229, 529, 768, 1850, 2860, 6276, 10252, 23248, 36563, 81622, 133739, 300311, 491193, 1091809, 1816561, 4062176, 6772098, 15021634, 25284670, 56134342, 94895078, 209889612

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 0.5, 1, 0.6, 0.714286, 0.384615, 0.695652, 0.488372, 0.64, 0.50365, 0.686275, 0.493534, 0.606651, 0.476427, 0.610561, 0.500963, 0.583868, 0.502795, 0.601734, 0.496874, 0.581618, 0.498624, 0.584595, 0.498259, 0.57642, 0.498269, 0.578057, 0.499347, 0.573186, 0.498736, 0.571734, 0.498567, 0.568309

Ratios a(n)/A095335(n) converge as: 1, 1, 1, 1.5, 1.25, 0.625, 0.842105, 0.954545, 1.116279, 1.014706, 1.100629, 0.974468, 0.985102, 0.909953, 0.966562, 1.003861, 0.984008, 1.011245, 1.00445, 0.987575, 0.991822, 0.994512, 0.988408, 0.993061, 0.99389, 0.9931, 0.99673, 0.997392, 0.997286, 0.994955, 0.995265, 0.994285, 0.996248

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095297(n). Cf. A095298.

A095320 Primes in whose binary expansion the number of 1-bits is > number of 0-bits minus 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269

Offset: 1

Antti Karttunen, Jun 04 2004

Differs from primes (A000040) first time at n=55, where a(55)=263, while A000040(55)=257, as 257 whose binary expansion is 100000001, with 2 1-bits and 7 0-bits is the first prime excluded from this sequence. Note that 129 (10000001 in binary, 2 1-bits and 6 0-bits) is not prime.

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

Complement of A095321 in A000040. Subset: A095316.

Cf. A095330.

Select[Prime[Range[60]],DigitCount[#,2,1]>DigitCount[#,2,0]-3&] (* Harvey P. Dale, Jul 24 2013 *)

forprime(p=2,269,v=binary(p);s=0;for(k=1,#v,    s+=if(v[k]==1,+1,-1));if(s>-3,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

cp's OEIS Frontend

A095079 Primes with two 0-bits in their binary expansion.

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A095085 Fib000 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with three zeros.

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A095086 Fib001 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with two zeros and final 1.

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Python

A095089 Fib101 primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends as one, zero, one.

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Python

A095090 Number of 4k+3 integers in range ]2^n,2^(n+1)] whose Jacobi-vector is a Motzkin-path (A095100).

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PARI

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A095108 Diving index of the n-th diving 4k+3 prime (A095103(n)).

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A095281 Upper Wythoff primes, i.e., primes in A001950.

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Python

A095290 Number of lower Wythoff primes (A095280) in range ]2^n,2^(n+1)].

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A095296 Number of A095286-primes in range ]2^n,2^(n+1)].

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A095320 Primes in whose binary expansion the number of 1-bits is > number of 0-bits minus 3.

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