A206581 -id:A206581 - cp's OEIS frontend

A192869 Thin primes: odd primes p such that p+1 is a prime (or 1) times a power of two.

3, 5, 7, 11, 13, 19, 23, 31, 37, 43, 47, 61, 67, 73, 79, 103, 127, 151, 157, 163, 191, 193, 211, 223, 271, 277, 283, 313, 331, 367, 383, 397, 421, 457, 463, 487, 523, 541, 547, 607, 613, 631, 661, 673, 691, 733, 751, 757, 787, 823, 877, 907, 991, 997, 1051

Offset: 1

Charles R Greathouse IV, Jul 11 2011

nonn

Broughan & Qizhi conjecture that a(n) << n (log n)^2, matching the lower bound they proved.

Sequence A206581 excludes the Mersenne primes (A000043), which are included here under the "or 1" case. - T. D. Noe, Mar 07 2012

D. R. Heath-Brown, "Artin's conjecture for primitive roots", Quarterly Journal of Mathematics 37:1 (1986) pp. 27-38.
N. M. Timofeev, "The Hardy-Ramanujan and Halasz inequalities for shifted primes", Mathematical Notes 57:5 (1995), pp. 522-535.

T. D. Noe, Table of n, a(n) for n = 1..1000
Kevin Broughan and Zhou Qizhi, Flat primes and thin primes, Bulletin of the Australian Mathematical Society 82:2 (2010), pp. 282-292.
Qizhi Zhou, Multiply perfect numbers of low abundancy, PhD thesis (2010)

Subsequence of A192868.

onePrimeQ[n_] := n == 1 || PrimeQ[n]; Select[Prime[Range[2, 1000]], onePrimeQ[(# + 1)/2^IntegerExponent[# + 1, 2]] &] (* T. D. Noe, Mar 06 2012 *)

is(n)=n%2&&isprime(n)&&(isprime((n+1)>>valuation(n+1,2)) || n+1==1<

a(n) >> n (log n)^2.

A273401 Numbers n such that n and n + 1 have exactly the same number of odd divisors.

Original entry on oeis.org

1, 5, 6, 10, 11, 12, 13, 19, 22, 23, 28, 37, 40, 43, 46, 47, 49, 52, 54, 58, 61, 65, 67, 69, 73, 77, 79, 82, 84, 88, 96, 103, 106, 110, 112, 114, 119, 129, 132, 136, 140, 148, 151, 154, 155, 157, 163, 166, 172, 178, 182, 185, 186, 191, 192, 193, 203, 204, 211, 215, 216, 219, 220, 221

Offset: 1

Juri-Stepan Gerasimov, May 26 2016

nonn

If A001227(n) = A001227(n*2^m) for m >= 0 then:
1) A001227(n) is equal to number of ways to write 2n - 1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers;
2) A001227(n) is equal to number of distinct values of k if k/(2n-1) + 1 divides (k/(2n - 1))^(k/(2n - 1)) + k, (k/(2n - 1))^k + k/(2n - 1) and k^(k/(2n - 1)) + k/(2n - 1).

			5 and 6 have both two odd divisors: (1 and 5) and (1 and 3) respectively; so 5 is a term in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001227, A206581 (primes in a(n)).

A001227:= n -> numtheory:-tau(n)/(1+padic:-ordp(n,2)):
R:= map(A001227,[$1..1000]):
ListTools:-SearchAll(0,A001227[2..-1]-A001227[1..-2]); # Robert Israel, May 27 2016

Select[Range@ 221, First@ Differences@ Map[Count[Divisors@ #, ?OddQ] &, {#, # + 1}] == 0 &] (* _Michael De Vlieger, Jun 26 2016 *)
SequencePosition[Table[Count[Divisors[n],?OddQ],{n,250}],{x,x_}] [[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 06 2019 *)

lista(nn) = for (n=1, nn, if (sumdiv(n, d, d%2) == sumdiv(n+1, d, d%2), print1(n, ", "))); \\ Michel Marcus, May 27 2016

is(n)=numdiv(n>>valuation(n,2))==numdiv((n+1)>>valuation(n+1,2)) \\ Charles R Greathouse IV, Jul 15 2016

A326817 Numbers k such that phi(k) > phi(k+1) > phi(k+2) > phi(k+3) where phi is the Euler totient function (A000010).

Original entry on oeis.org

823, 943, 3133, 4387, 4873, 5443, 5563, 5863, 7213, 7753, 7873, 8383, 9007, 10333, 10693, 11113, 11503, 12043, 12763, 13483, 13843, 13921, 14623, 14683, 16573, 16663, 16963, 16993, 17113, 17983, 19003, 19093, 19303, 20083, 20143, 20953, 21613, 21733, 22513

Offset: 1

Kritsada Moomuang, Oct 20 2019

nonn

			823 is in the sequence since phi(823) = 822, phi(824) = 408, phi(825) = 400, phi(826) = 348, and 822 > 408 > 400 > 348.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 106, entry 823.

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

Cf. A000010, A078776, A161962, A161963, A206581, A327880.

Subsequence of A328056.

aQ[n_] := AllTrue[Differences @ EulerPhi[n + Range[0, 3]], # < 0 &]; Select[Range[23000], aQ] (* Amiram Eldar, Oct 20 2019 *)

A328056 Numbers k such that phi(k) > phi(k+1) > phi(k+2) where phi is the Euler totient function (A000010).

Original entry on oeis.org

313, 523, 733, 823, 824, 943, 944, 973, 1153, 1363, 1573, 1753, 1783, 1813, 1993, 2143, 2413, 2473, 2623, 2803, 3043, 3133, 3134, 3253, 3313, 3463, 3703, 3883, 4093, 4123, 4303, 4387, 4388, 4513, 4723, 4873, 4874, 4933, 5113, 5143, 5353, 5443, 5444, 5563, 5564

Offset: 1

Kritsada Moomuang, Oct 03 2019

nonn

Contains all members k of A206581 such that k==103 (mod 210) except 103.- Robert Israel, Oct 16 2019

			313 is in the sequence since phi(313) = 312, phi(314) = 156, phi(315) = 144, and 312 > 156 > 144.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A078776, A161962, A161963, A206581, A327880.

Supersequence of A326817.

[k:k in [1..5600]| EulerPhi(k) gt EulerPhi(k+1) and EulerPhi(k+1) gt EulerPhi(k+2)]; // Marius A. Burtea, Oct 07 2019

R:= NULL: count:= 0:
q:= false: s:= 0:
for i from 1 while count < 100 do
  t:= numtheory:-phi(i);
  r:= q;
  q:= evalb(tRobert Israel, Oct 16 2019

Flatten[Position[Partition[EulerPhi[Range[5600]], 3, 1], ?(Max[Differences[#]] < 0 &)] // Quiet] (* _Amiram Eldar, Oct 06 2019 after Harvey P. Dale at A078776 *)

A272887 Number of ways to write prime(n) as (4x + 2)y + 4*x + 1 where x and y are nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, May 16 2016

nonn

Number of distinct values of k such that k/p_n + k divides (k/p_n)^(k/p_n) + k, (k/p_n)^k + k/p_n and k^(k/p_n) + k/p_n where p_n = prime(n) is n-th prime.
a(1) = 0, a(n+1) = number of odd divisors of 1+prime(n+1).
Conjectures:
1) a(Fermat prime(n)) >= n, i.e. a(A019434(1)=3) = 1, a(A019434(2)=5) = 2, a(A019434(3)=17) = 3, a(A019434(4)=257) = 4, a(A019434(5)=65537) = 12 > 5, ...
2) a(2^(2^n)+1) > n;
3) a(2^(2^n)+1) < a(2^(2^(n+1))+1).

			a(3) = 2 because (4*0+2)*2+4*0+1 = 5 for (x=0, y=2) and (4*1+2)*0+4*1+1 = 5 for (x=1, y=0) where 5 is the 3rd prime.

Cf. A000215 (Fermat numbers), A001227, A000668 (Mersenne primes n such that a(n)=1), A019434 (Fermat primes), A069283, A192869 (primes n such that a(n) = 1 or 2), A206581 (primes n such that a(n)=2), A254748.

{0}~Join~Table[Count[Divisors[Prime[n + 1] + 1], ?OddQ], {n, 120}] (* _Michael De Vlieger, Jun 24 2016 *)

a(n+1) = A001227(A000040(n+1) + 1).

More terms from Alois P. Heinz, May 17 2016

A276660 Primes of the form p*2^k - 1 where p is an odd prime and k >= 0.

Original entry on oeis.org

2, 5, 11, 13, 19, 23, 37, 43, 47, 61, 67, 73, 79, 103, 151, 157, 163, 191, 193, 211, 223, 271, 277, 283, 313, 331, 367, 383, 397, 421, 457, 463, 487, 523, 541, 547, 607, 613, 631, 661, 673, 691, 733, 751, 757, 787, 823, 877, 907, 991, 997, 1051, 1087, 1093, 1123

Offset: 1

Juri-Stepan Gerasimov, Sep 11 2016

nonn

			2 is in this sequence because 3*2^0 - 1 = 2 is prime.
5 is in this sequence because 3*2^1 - 1 = 5 is prime.
11 is in this sequence because 3*2^2 - 1 = 11 is prime.

Essentially the same as A192869 and A206581.

Cf. A038550, A058500.

is(n)=isprime((n+1)>>valuation(n+1,2)) && isprime(n) \\ Charles R Greathouse IV, Sep 11 2016

a(n) >> n (log n)^2. - Charles R Greathouse IV, Sep 11 2016

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A192869 Thin primes: odd primes p such that p+1 is a prime (or 1) times a power of two.

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A273401 Numbers n such that n and n + 1 have exactly the same number of odd divisors.

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A326817 Numbers k such that phi(k) > phi(k+1) > phi(k+2) > phi(k+3) where phi is the Euler totient function (A000010).

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A328056 Numbers k such that phi(k) > phi(k+1) > phi(k+2) where phi is the Euler totient function (A000010).

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A272887 Number of ways to write prime(n) as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers.

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A276660 Primes of the form p*2^k - 1 where p is an odd prime and k >= 0.

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A272887 Number of ways to write prime(n) as (4x + 2)y + 4*x + 1 where x and y are nonnegative integers.