A059305 -id:A059305 - cp's OEIS frontend

A345867 Total number of 0's in the binary expansions of the first n primes.

1, 1, 2, 2, 3, 4, 7, 9, 10, 11, 11, 14, 17, 19, 20, 22, 23, 24, 28, 31, 35, 37, 40, 43, 47, 50, 52, 54, 56, 59, 59, 64, 69, 73, 77, 80, 83, 87, 90, 93, 96, 99, 100, 105, 109, 112, 115, 116, 119, 122, 125, 126, 129, 130, 137, 142, 147, 151, 156, 161, 165, 170

Offset: 1

Alois P. Heinz, Jun 26 2021

			a(3) = 2: 2 = 10_2, 3 = 11_2, 5 = 101_2, so there are two 0's in the binary expansions of the first three primes.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Partial sums of A035103.

Cf. A000040, A059305, A095375, A328659.

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)
      +add(1-i, i=Bits[Split](ithprime(n))))
    end:
seq(a(n), n=1..100);

Accumulate[DigitCount[Prime[Range[100]], 2, 0]] (* Paolo Xausa, Feb 26 2024 *)

from sympy import prime, primerange
from itertools import accumulate
def f(n): return (bin(n)[2:]).count('0')
def aupton(nn): return list(accumulate(map(f, primerange(2, prime(nn)+1))))
print(aupton(62)) # Michael S. Branicky, Jun 26 2021

a(n) = Sum_{i=1..n} A035103(i).
a(n) = a(n-1) for n in { A059305 }.
a(n) = A328659(n) - A095375(n).

A360357 Numbers k such that k and k+1 are both products of primes of nonprime index (A320628).

Original entry on oeis.org

1, 7, 13, 28, 37, 46, 52, 73, 91, 97, 103, 106, 112, 148, 151, 172, 181, 193, 196, 202, 223, 226, 232, 256, 262, 292, 298, 301, 316, 337, 343, 346, 361, 376, 388, 397, 427, 448, 457, 463, 466, 478, 487, 502, 511, 523, 541, 556, 568, 592, 601, 607, 613, 622, 631

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

There are no 3 consecutive integers that are products of primes of nonprime index since 1 out of 3 consecutive integers is divisible by 3 which is a prime-indexed prime (A006450).

If a Mersenne prime (A000668) is a prime of nonprime index, then it is in this sequence. Of the first 10 Mersenne primes 6 are in this in sequence: A000668(k) for k = 2, 5, 7, 8, 9, 10 (see A059305).

			7 = prime(4) is a term since 4 is nonprime, 7 + 1 = 8 = prime(1)^3, and 1 is also nonprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000668, A006450, A059305, A320628.

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], ! PrimeQ[PrimePi[#]] &]; seq = {}; q1 = q[1]; n = 2; c = 0; While[c < 55, q2 = q[n]; If[q1 && q2, c++; AppendTo[seq, n - 1]]; q1 = q2; n++]; seq

is(n) = {my(p = factor(n)[,1]); for(i = 1, #p, if(isprime(primepi(p[i])), return(0))); 1;}
lista(nmax) = {my(q1 = is(1), q2); for(n = 2, nmax, q2 = is(n); if(q1 && q2, print1(n-1, ", ")); q1 = q2); }

A365160 Least k such that A000668(n) + k is prime, where A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

2, 4, 6, 4, 18, 30, 22, 12, 16, 30, 40, 30, 888, 486, 2056, 696, 310, 718, 4692, 1600, 2788, 4290, 4326, 4150, 18088, 22096, 16342, 72816, 181720, 4200, 58416

Offset: 1

Robert P. P. McKone, Aug 24 2023

The distance between the n-th Mersenne prime and the next prime.

			A000668(6) = 131071, the next prime is 131101, so a(6) = 131101 - 131071 = 30.

Cf. A000040, A000668 (Mersenne primes), A001223, A059305, A074626, A365161.

m[n_] := m[n] = (2^MersennePrimeExponent[n] - 1); a[k_, n_] := a[k, n] = m[n] + k; l[k_, n_] := l[k, n] = PrimeQ[a[k, n]]; Table[k = 1; Monitor[Parallelize[While[True, If[l[k, n], Break[]]; k++]; k], {n, k}], {n, 1, 20}]

a(n) = A001223(A059305(n)). - Michel Marcus, Aug 25 2023

a(n) = A074626(n) - A000668(n). - Amiram Eldar, Aug 10 2024

a(28) from Michael S. Branicky, Aug 11 2024
a(29) from Minfeng Wang, Oct 29 2024
a(30)-a(31) from Minfeng Wang, Oct 29 2024

A365161 Least k such that A000668(n) - k is prime, where A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

1, 2, 2, 14, 12, 8, 18, 18, 30, 20, 170, 24, 114, 56, 156, 2510, 1824, 12, 3980, 3630, 16902, 284, 7712, 20022, 12930, 9698, 16232, 1058, 256016, 23712, 26298

Offset: 1

Robert P. P. McKone, Aug 24 2023

The distance between the n-th Mersenne prime and the previous prime.

			A000668(6) = 131071, the previous prime is 131063, so a(6) = 131071 - 131063 = 8.

Cf. A000040, A000668 (Mersenne primes), A001223, A059305, A073715, A365160.

m[n_] := m[n] = (2^MersennePrimeExponent[n] - 1); a[k_, n_] := a[k, n] = m[n] - k; l[k_, n_] := l[k, n] = PrimeQ[a[k, n]]; Table[k = 1; Monitor[Parallelize[While[True, If[l[k, n], Break[]]; k++]; k], {n, k}], {n, 1, 20}]

a(n) = A001223(A059305(n)-1). - Michel Marcus, Aug 25 2023

a(n) = A000668(n) - A073715(n). - Amiram Eldar, Aug 10 2024

a(29)-a(31) from Michael S. Branicky, Sep 01 2024

A378252 Least prime power > 2^n.

Original entry on oeis.org

2, 3, 5, 9, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659, 4294967311, 8589934609

Offset: 0

Gus Wiseman, Nov 30 2024

nonn

Prime powers are listed by A246655.

Conjecture: All terms except 9 are prime. Hence this is the same as A014210 after 9. Confirmed up to n = 1000.

Subtracting 2^n appears to give A013597 except at term 3.
For prime we have A014210.
For previous we have A014234.
For perfect power we have A357751.
For squarefree we have A372683.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, diffs A375708 and A375735.
A031218 gives the greatest prime power <= n.
A244508 counts prime powers between powers of 2.
Prime powers between primes are counted by A080101 and A366833.
Cf. A007918, A037035, A053707, A059305, A065514, A069584, A151800, A304521, A377051, A377282, A377283, A377289, A377432, A378249.

Table[NestWhile[#+1&,2^n+1,!PrimePowerQ[#]&],{n,0,20}]

a(n) = my(x=2^n+1); while (!isprimepower(x), x++); x; \\ Michel Marcus, Dec 03 2024

from itertools import count
from sympy import primefactors
def A378252(n): return next(i for i in count(1+(1<Chai Wah Wu, Dec 02 2024

A240981 Smallest perfect power b^e such that b^e+prime(n) is also a perfect power.

Original entry on oeis.org

25, 1, 4, 1, 16, 36, 8, 8, 4, 196, 1, 27, 8, 441, 81, 676, 841, 64, 1089, 125, 8, 49, 1681, 32, 128, 27, 25, 2197, 16, 8, 1, 125, 32, 2048, 2048, 361, 243, 6561, 49, 7396, 64, 8100, 25, 32, 6859, 125, 32, 289, 16, 27, 128, 4, 243, 1936, 32, 17161, 243, 729

Offset: 1

Zak Seidov, Aug 06 2014

nonn

a(n) = 1 if and only if prime(n) is a Mersenne prime (A000668). Thus A059305 gives the values of n for which a(n) = 1.
For all n, a(n) exists.
Subsequence of A103953.
For n > 1, a(n) <= ((prime(n)-1)/2)^2, since ((p-1)/2)^2 + p = ((p+1)/2)^2. - Jens Kruse Andersen, Aug 10 2014

			n=1: prime(n)=2, 25 = 5^2 and  25+5=27=3^3;
n=2: prime(n)=3, 1 = 1^2 and  1+3=4=2^2;
n=3: prime(n)=5, 4 = 2^2 and  4+5=9=3^2.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000668, A001597, A103953.

{forprime(p=2,200,if(ispower(1+p),print1(1","),n=4;while(!(ispower(n)&&ispower(n+p)),n++);print1(n",")))}

A352232 a(n) is the smallest positive integer k such that 1 + k * prime(n) is a power of two.

Original entry on oeis.org

1, 3, 1, 93, 315, 15, 13797, 89, 9256395, 1, 1857283155, 25575, 381, 178481, 84973577874915, 4885260612740877, 18900352534538475, 1101298153654301589, 483939977, 7, 6958934353, 58261485282632731311141, 23, 2901803883615, 12550996041863657440561417875

Offset: 2

Alois P. Heinz, Mar 08 2022

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 2..471

Cf. A000040, A000079, A000225, A000668, A007814, A014664, A059305.

a:= n-> (p-> (2^numtheory[order](2, p)-1)/p)(ithprime(n)):
seq(a(n), n=2..28);

from sympy.ntheory import n_order, prime
def A352232(n): return (2**n_order(2,p:=prime(n))-1)//p # Chai Wah Wu, Mar 09 2022

a(n) = (2^A014664(n)-1)/prime(n).
A007814(a(n)*prime(n)+1) = A014664(n).
a(n) = 1 <=> n in { A059305 } <=> prime(n) in { A000668 }.
a(n)*prime(n) + 1 in { A000079 }.
a(n)*prime(n) in { A000225 }.

A023511 Least odd prime divisor of prime(n) + 1, or 1 if prime(n) + 1 is a power of 2.

Original entry on oeis.org

3, 1, 3, 1, 3, 7, 3, 5, 3, 3, 1, 19, 3, 11, 3, 3, 3, 31, 17, 3, 37, 5, 3, 3, 7, 3, 13, 3, 5, 3, 1, 3, 3, 5, 3, 19, 79, 41, 3, 3, 3, 7, 3, 97, 3, 5, 53, 7, 3, 5, 3, 3, 11, 3, 3, 3, 3, 17, 139, 3, 71, 3, 7, 3, 157, 3, 83, 13, 3, 5, 3, 3, 23, 11, 5, 3, 3, 199, 3, 5, 3, 211, 3, 7, 5, 3, 3

Offset: 1

Clark Kimberling

nonn

Note that a(n) = 1 for n= 2, 4, 11, 31, 1028, ... (A059305). - Michel Marcus, Oct 01 2013

Cf. A008864, A078701.

a(n) = my(p = prime(n) + 1, v = p/(2^valuation(p, 2))) ; if (v == 1, 1, factor(v)[1, 1]); \\ Michel Marcus, Oct 01 2013

a(n) = A078701(A008864(n)). - Michel Marcus, Jun 06 2019

A131483 Meissel_Lehmer recursion: a(n,m) = a(n,m-1)-a(Floor[n/Prime[m]],m-1).

Original entry on oeis.org

1, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1

Offset: 1

Roger L. Bagula, Oct 01 2007

			{1},
{0, -1},
{0, -1, -1},
{0, 0, 0, 0},
{0, 0, 0, 0, 0},
{0, 0, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560.

Cf. A000720, A006880, A007053, A075986, A059305.

a(1,1) = 1; a(n,m) = a(n,m-1)-a(Floor[n/Prime[m]],m-1);

A243979 Indices of Wagstaff primes.

Original entry on oeis.org

2, 5, 14, 124, 399, 4552, 15898, 203095, 37029521, 105973558438, 19140185454656173, 3827634977577891833517

Offset: 1

Omar E. Pol, Jun 18 2014

			For n = 3 the third Wagstaff prime is A000979(3) = 43 and 43 is also the 14th prime number, so a(3) = 14.

Andrew R. Booker, The Nth Prime Page.
Chris K. Caldwell, Wagstaff, The Top Twenty, The PrimePages.
Xavier Gourdon and Pascal Sebah, Counting primes.
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x).
Samuel S. Wagstaff, Jr., The Cunningham Project.
Kim Walisch, Fast C++ prime counting function implementation (primecount).
Wikipedia, Wagstaff prime.

Cf. A000040, A000720, A000978, A000979, A059305, A123176, A126614, A194810.

default(primelimit, 10^9); forprime(p=3, 31, q=(2^p+1)/3; if(isprime(q), print1(primepi(q)", "))) \\ Jens Kruse Andersen, Jun 22 2014

a(n) = A000720(A000979(n)).

A000040(a(n)) = A000979(n).

a(11) from Jens Kruse Andersen, Jun 22 2014

a(12) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 05 2024

cp's OEIS Frontend

A345867 Total number of 0's in the binary expansions of the first n primes.

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A360357 Numbers k such that k and k+1 are both products of primes of nonprime index (A320628).

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A365160 Least k such that A000668(n) + k is prime, where A000668(n) is the n-th Mersenne prime.

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A365161 Least k such that A000668(n) - k is prime, where A000668(n) is the n-th Mersenne prime.

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A378252 Least prime power > 2^n.

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A240981 Smallest perfect power b^e such that b^e+prime(n) is also a perfect power.

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A352232 a(n) is the smallest positive integer k such that 1 + k * prime(n) is a power of two.

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A023511 Least odd prime divisor of prime(n) + 1, or 1 if prime(n) + 1 is a power of 2.

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