A023512 -id:A023512 - cp's OEIS frontend

A023506 Exponent of 2 in prime factorization of prime(n) - 1.

0, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 5, 2, 1, 1, 2, 4, 1, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 6, 2, 1, 1, 1, 1, 2, 3, 1, 4, 1, 8, 1, 2, 1, 2, 3, 1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 1, 2, 1, 1, 2, 2, 4, 3, 1, 2, 1, 4, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 3, 1

Offset: 1

Clark Kimberling

Also the number of steps to reach an integer starting with prime(n)/2 and iterating the map x->x*ceiling(x). - Benoit Cloitre, Sep 06 2002
Also exponent of 2 in -1 + prime(n)^s for odd exponents s because (-1 + prime(n)^s)/(prime(n) - 1) is odd. - Labos Elemer, Jan 20 2004
First occurrence of 0,1,2,3,4,...: 1, 2, 3, 13, 7, 25, 44, 116, 55, 974, 1581, 2111, 1470, 4289, 10847, 15000, 6543, 91466, 62947, 397907, 498178, ..., for primes 2, 3, 5, 41, 17, 97, 193, 641, 257, 7681, 13313, 18433, 12289, 40961, 114689, 163841, 65537, 1179649, 786433, 5767169, 7340033, .... - Robert G. Wilson v, May 28 2009
By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 1 (mod 2^k) within all the primes is 1/2^(k-1), for k >= 1. This is also the asymptotic density of terms in this sequence that are >= k. Therefore, the asymptotic density of the occurrences of k in this sequence is d(k) = 1/2^(k-1) - 1/2^k = 1/2^k, and the asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2. - Amiram Eldar, Mar 14 2025

			For n = 25, prime(25) = 97, A006093(25) = 96 = 2*2*2*2*2*3, so a(25) = 5.

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

Cf. A007814, A000010, A000040, A006093, A023512, A057023, A057773 (partial sums).

Subsequence of A001511 (except 1st term).

[Valuation(NthPrime(n)-1, 2): n in [1..110]]; // Bruno Berselli, Aug 05 2013

A023506:= x -> padic[ordp](ithprime(x)-1,2):
seq(A023506(x),x=1..1000); # Robert Israel, May 06 2014

f[n_] := Block[{fi = First@ FactorInteger[ Prime@n - 1]}, If[ fi[[1]] == 2, fi[[2]], 0]]; Array[f, 105] (* Robert G. Wilson v, May 28 2009 *)
Table[IntegerExponent[Prime[n] - 1, 2], {n, 110}] (* Bruno Berselli, Aug 05 2013 *)

A023506(n) = {local(m,r);r=0;m=prime(n)-1;while(m%2==0,m=m/2;r++);r} \\ Michael B. Porter, Jan 26 2010

forprime(p=2, 700, print1(valuation(p-1,2),", ")); \\ Bruno Berselli, Aug 05 2013

from sympy import prime
def A023506(n): return (~(m:=prime(n)-1)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

a(n) = A007814(A000010(A000040(n))) = A007814(A006093(n)).

A091282 Exponent of 2 in prime(n)^2 - 1.

Original entry on oeis.org

0, 3, 3, 4, 3, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 3, 3, 3, 4, 4, 5, 3, 4, 6, 3, 4, 3, 3, 5, 8, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 3, 7, 7, 3, 4, 3, 6, 3, 3, 4, 5, 5, 3, 9, 4, 3, 5, 3, 4, 3, 3, 3, 4, 4, 3, 3, 5, 3, 3, 6, 4, 5, 3, 3, 8, 3, 3, 5, 4, 3, 3, 5, 5, 4, 3, 7, 4, 3, 5, 3, 6, 4, 3, 3, 4, 3, 4, 3, 3, 3, 3, 3, 4, 3

Offset: 1

Labos Elemer, Jan 20 2004

nonn

Also, exponent of 2 in -1+prime(n)^s if s is an exponent of the form 4k+2 (previous definition). - Michel Marcus, Dec 20 2013

Primes that give records for this sequence can be found in A233930. - Michel Marcus, Dec 20 2013

			a(1)=0 since -1+2^s is always odd...

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

Cf. A007814, A023506, A023512, A091283, A091284.

IntegerExponent[Prime[Range[100]]^2 - 1, 2] (* Amiram Eldar, Jun 06 2024 *)

a(n) = valuation(prime(n)^2-1, 2); \\ Michel Marcus, Dec 20 2013

a(n) = A023506(n) + A023512(n). - Amiram Eldar, Jun 06 2024

Definition modified by Michel Marcus, Dec 20 2013

A057024 Largest odd factor of (n-th prime+1); k when n-th prime is written as k*2^m-1 [with k odd].

Original entry on oeis.org

3, 1, 3, 1, 3, 7, 9, 5, 3, 15, 1, 19, 21, 11, 3, 27, 15, 31, 17, 9, 37, 5, 21, 45, 49, 51, 13, 27, 55, 57, 1, 33, 69, 35, 75, 19, 79, 41, 21, 87, 45, 91, 3, 97, 99, 25, 53, 7, 57, 115, 117, 15, 121, 63, 129, 33, 135, 17, 139, 141, 71, 147, 77, 39, 157, 159, 83, 169, 87

Offset: 1

Henry Bottomley, Jul 24 2000

a(n) = 1 if and only if prime(n) is a Mersenne prime. - Ely Golden, Feb 06 2017

			a(5)=3 because 5th prime is 11 and 11=3*2^2-1.

Cf. A000040, A000265, A007814, A023512, A028815, A057023.

A057024:= func< n | (NthPrime(n)+1)/2^Valuation(NthPrime(n)+1, 2) >;
[A057024(n): n in [1..100]]; // G. C. Greubel, Aug 06 2024

Table[Max[Select[Divisors[Prime[n]+1],OddQ]],{n,100}] (* Daniel Jolly, Nov 15 2014 *)

a(n) = (prime(n)+1)/2^valuation(prime(n)+1, 2); \\ Michel Marcus, Feb 05 2017

def a(n):
    x=nth_prime(n)+1
    return x/2**((int(x)&int(-x)).bit_length()-1)
index=1
while(index<=10000):
    print(str(index)+" "+str(a(index)))
    index+=1
# Ely Golden, Feb 06 2017

a(n) = A000265(A000040(n) + 1) = A000265(A028815(n)).
a(n) = (A000040(n) + 1)/A007814(A000040(n) + 1).
a(n) = A028815(n)/A023512(n).

A227990 3^a(n) is the highest power of 3 dividing prime(n)+1.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 1, 3, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 1, 3, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 2

Offset: 1

Bruno Berselli, Aug 05 2013

This is the 3-adic valuation of prime(n)+1.

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 3^k-1 (mod 3^k) within all the primes is 1/(2*3^(k-1)), for k >= 1. This is also the asymptotic density of terms in this sequence that are >= k. Therefore, the asymptotic density of the occurrences of k in this sequence is d(k) = 1/(2*3^(k-1)) - 1/(2*3^k) = 1/3^k, and the asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 3/4. - Amiram Eldar, Mar 14 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Bruno Berselli)

Cf. A007949, A008864, A023512 (2-adic valuation of prime(n)+1), A099584 (3-adic valuation of prime(n)-1), A227991 (associated powers of 3).

[Valuation(NthPrime(n)+1, 3): n in [1..100]];

Table[IntegerExponent[Prime[n] + 1, 3], {n, 100}]

forprime(p=2, 700, print1(valuation(p+1,3),", "));

a(n) = A007949(A008864(n)).

A068504 Highest power of 2 dividing prime(n)+1.

Original entry on oeis.org

1, 4, 2, 8, 4, 2, 2, 4, 8, 2, 32, 2, 2, 4, 16, 2, 4, 2, 4, 8, 2, 16, 4, 2, 2, 2, 8, 4, 2, 2, 128, 4, 2, 4, 2, 8, 2, 4, 8, 2, 4, 2, 64, 2, 2, 8, 4, 32, 4, 2, 2, 16, 2, 4, 2, 8, 2, 16, 2, 2, 4, 2, 4, 8, 2, 2, 4, 2, 4, 2, 2, 8, 16, 2, 4, 128, 2, 2, 2, 2, 4, 2, 16, 2, 8, 4, 2, 2, 2, 16, 4, 32, 8, 4, 4, 8, 2

Offset: 1

Benoit Cloitre, Mar 11 2002

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

Cf. A006519, A008864, A023512.

f:= n -> 2^padic:-ordp(ithprime(n)+1,2):
map(f, [$1..100]); # Robert Israel, Jan 13 2017

a[n_] := 2^IntegerExponent[Prime[n] + 1, 2]; Array[a, 100] (* Amiram Eldar, Jun 04 2022 *)

a(n) = 2^valuation(prime(n)+1, 2); \\ Michel Marcus, Nov 24 2013

from sympy import prime
def A068504(n): return (m:=prime(n)+1)&-m # Chai Wah Wu, Jul 09 2022

a(n) = A006519(A008864(n)). - Michel Marcus, Nov 24 2013

a(n) = 2^A023512(n). - Michel Marcus, Nov 24 2013

A239114 Exponent of 2 in prime factorization (i.e., 2-adic valuation) of odd nonprimes A014076(n) + 1.

Original entry on oeis.org

1, 1, 4, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 6, 1, 1, 2, 1, 1, 1, 3, 2, 1, 5, 2, 1, 4, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 1, 2, 5, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 1, 8, 2, 1, 1, 2, 1, 2, 3, 1, 5, 1, 2, 3, 1, 2, 1, 4, 1, 1, 2, 6, 1, 2, 1, 3, 1

Offset: 1

K. G. Stier, Mar 10 2014

nonn

Sequence is counterpart to A023512, i.e., merging these two sequences gives the ruler function A001511.

			a(13) = 3, because the 13th odd nonprime is 55, and the largest power of 2 dividing 55+1 is 3.

K. G. Stier, Table of n, a(n) for n = 1..7739

Cf. A023512, A001511, A014076.

lista(nn) = {forstep(n=1, nn, 2, if (! isprime(n), print1(valuation(n+1, 2), ", ")););} \\ Michel Marcus, Mar 13 2014

a(n) = A001511((A014076(n)+1)/2)

A241541 Exponent of 11 in prime factorization of (2^n + 3^n + 5^n + 7^n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Zak Seidov, Apr 25 2014

nonn

11^a(n) is the largest power of 11 dividing (2^n + 3^n + 5^n + 7^n);
(2^n + 3^n + 5^n + 7^n) is divisible by 11^2 = 121 for n == 5 mod 10.
Among first 10000 nonzero terms there are {8182, 1652, 150, 14, 1, 1} terms with values {2, 3, 4, 5, 6, 7}, respectively.
Record values are a(5) = 2, a(45) = 3, a(595) = 5, a(40525) = 7, a(6482565) = 8, a(97435855) = 9, a(927694285) = 10, a(11789738455) = 11, a(129687123005) = 12, a(508958242255) = 13, a(11921425066695) = 14, a(74689992601115) = 15, a(1110371356919045) = 16, a(20886240847078255) = 17, a(229748649317860805) = 18, etc. - Charles R Greathouse IV, Apr 25 2014

			at n = 5, 2^n + 3^n + 5^n + 7^n = 20207 = 11^2*167,
at n = 15, 2^n + 3^n + 5^n + 7^n = 4778093469743 = 11^2*587*67271509.

Zak Seidov, Table n, a(-5 + 10*n) for n = 1..10^4.

Cf. A006519, A007814, A023512.

Table[IntegerExponent[2^n + 3^n + 5^n + 7^n, 11], {n, 0, 100}]

a(n)=valuation(2^n+3^n+5^n+7^n,11) \\ Charles R Greathouse IV, Apr 25 2014

a(n,e=8)=my(m=11^e, o=valuation(Mod(2,m)^n +Mod(3,m)^n +Mod(5,m)^n +Mod(7,m)^n, 11)); if(oCharles R Greathouse IV, Apr 25 2014

A363017 a(n) is the least integer k such that the k-th, (k+1)-th, ..., (k+n-1)-th primes are congruent to 3 mod 8.

Original entry on oeis.org

2, 94, 334, 4422, 23969, 303493, 303493, 606529, 28725046, 92865581, 397316305, 511883558, 848516256, 23738949809, 144899085865, 469694200388, 3800553021301, 8571139291304, 63858322306341, 90990757864814

Offset: 1

Léo Gratien, May 13 2023

a(n) is also the minimal rank where n consecutive 2's appear in A023512.

The sequence is infinite by Shiu's theorem.

			For n=2, a(2) = 94 because prime(94)+1 = 492 = 4*123, prime(95)+1 = 500 = 4*125 are the first two consecutive primes p such that p+1 is divisible by 4 and not by 8.

Cf. A057632, A023512.

Cf. A363016 (with 1 mod 4).

a(n) = primepi(A057632(n)). - Amiram Eldar, May 13 2023

a(19)-a(20) from Martin Ehrenstein, May 28 2023

A363016 a(n) is the least integer k such that the k-th, (k+1)-th, ..., (k+n-1)-th primes are congruent to 1 mod 4.

Original entry on oeis.org

3, 6, 24, 77, 378, 1395, 1395, 1395, 1395, 31798, 61457, 240748, 800583, 804584, 804584, 804584, 16118548, 16138563, 16138563, 56307979, 56307979, 56307979, 56307979, 56307979, 3511121443, 3511121443, 26284355567, 26284355567, 26284355567, 118027458557, 118027458557, 118027458557, 118027458557

Offset: 1

Léo Gratien, May 13 2023

nonn

a(n) is also the minimal rank where n consecutive 1's appear in A023512.

The sequence is infinite by Shiu's theorem.

			For n=3, a(3) = 24 because prime(24)+1=90, prime(25)+1=98, and prime(26)+1=102 are the first 3 consecutive primes p such that p+1 is divisible by 2 and not by 4.

D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373.

Cf. A000720, A057624, A023512.

Cf. A363017 (3 mod 8).

a(n) = A000720(A057624(n)). - Amiram Eldar, May 13 2023

A227471 Position of first 0 in the binary representation of prime(n), starting the count of positions at 1 for the least significant bit.

Original entry on oeis.org

1, 3, 2, 4, 3, 2, 2, 3, 4, 2, 6, 2, 2, 3, 5, 2, 3, 2, 3, 4, 2, 5, 3, 2, 2, 2, 4, 3, 2, 2, 8, 3, 2, 3, 2, 4, 2, 3, 4, 2, 3, 2, 7, 2, 2, 4, 3, 6, 3, 2, 2, 5, 2, 3, 2, 4, 2, 5, 2, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 2, 2, 4, 5, 2, 3, 8, 2, 2, 2, 2, 3, 2, 5, 2, 4, 3, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jul 12 2013

nonn

If no zero appears in the base-2 representation, the search "falls through" and addresses the (virtual) leading zero of that prime, see A035100. - R. J. Mathar, Jul 20 2013

Cf. A004676, A023512.

a(n) = 1+A023512(n). - Antti Karttunen, Jul 13 2013

cp's OEIS Frontend

A023506 Exponent of 2 in prime factorization of prime(n) - 1.

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A091282 Exponent of 2 in prime(n)^2 - 1.

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A057024 Largest odd factor of (n-th prime+1); k when n-th prime is written as k*2^m-1 [with k odd].

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A227990 3^a(n) is the highest power of 3 dividing prime(n)+1.

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A068504 Highest power of 2 dividing prime(n)+1.

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A239114 Exponent of 2 in prime factorization (i.e., 2-adic valuation) of odd nonprimes A014076(n) + 1.

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A241541 Exponent of 11 in prime factorization of (2^n + 3^n + 5^n + 7^n).