A057023 -id:A057023 - 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)).

A339971 Odd part of A339821(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 15, 15, 15, 15, 3, 3, 3, 3, 9, 9, 9, 9, 15, 15, 15, 15, 45, 45, 45, 45, 1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 15, 15, 15, 15, 3, 3, 3, 3, 9, 9, 9, 9, 15, 15, 15, 15, 45, 45, 45, 45, 9, 9, 9, 9, 27, 27, 27, 27, 45, 45, 45, 45, 135, 135, 135, 135, 27, 27, 27, 27, 81, 81, 81, 81, 135

Offset: 0

Antti Karttunen, Dec 26 2020

nonn

Cf. A000010, A000265, A003961, A019565, A053575, A057023, A336466, A339821, A339822, A339970.

Cf. A339972 (quadrisection).

A000265(n) = (n>>valuation(n,2));
A339971(n) = { my(m=1, p=2); while(n>0, p = nextprime(1+p); if(n%2, m *= A000265(p-1)); n >>= 1); (m); };

If 4n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A057023(e1) * A057023(e2) * ... * A057023(ek).
a(n) = A053575(A019565(2n)) = A336466(A003961(A019565(n))) = A000265(A339821(n)).
a(n) = A339821(n) / A000079(A339822(n)).

A058500 Primes of the form p*2^k + 1, where p is an odd prime and k > 0.

Original entry on oeis.org

7, 11, 13, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 149, 167, 173, 179, 193, 227, 233, 263, 269, 293, 317, 347, 353, 359, 383, 389, 449, 467, 479, 503, 509, 557, 563, 569, 587, 593, 641, 653, 719, 769, 773, 797, 809, 839, 857, 863, 887, 929, 977

Offset: 1

Labos Elemer, Dec 20 2000

nonn

			719 is a term because 719 = 2*359 + 1 and 359 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from T. D. Noe)

Cf. A057023, A006093.

Cf. A074781 (this sequence and the Fermat primes), A147545.

mx = 1000; Select[ Sort@ Flatten@ Table[Prime[p] 2^k + 1, {p, 2, PrimePi[ mx/2]}, {k, Log2[ mx/Prime[ p]]}], PrimeQ] (* or *)
fQ[n_] := Block[{m = n -1}, PrimeQ[m/2^IntegerExponent[m, 2]]]; Select[
Prime@ Range@ PrimePi@ mx, fQ] (* Robert G. Wilson v, Feb 09 2018 *)

isoka(p) = isprime(p) && (pp=p-1) && isprime(pp/2^valuation(pp, 2)); \\ Michel Marcus, Feb 09 2018

Revised definition from T. D. Noe, Nov 03 2008

A340084 a(n) = gcd(n-1, A336466(n)); Odd part of A340081(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 11, 1, 1, 1, 1, 3, 7, 1, 15, 1, 1, 1, 1, 1, 9, 1, 1, 1, 5, 1, 21, 1, 1, 1, 23, 1, 3, 1, 1, 3, 13, 1, 1, 1, 1, 1, 29, 1, 15, 1, 1, 1, 1, 5, 33, 1, 1, 3, 35, 1, 9, 1, 1, 3, 1, 1, 39, 1, 1, 1, 41, 1, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 1, 1, 3, 1, 1, 1, 25, 1, 51, 1, 1

Offset: 1

Antti Karttunen, Dec 29 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A003958, A003961, A019565, A057023, A336466, A339899, A340081, A340085, A340086.

Cf. also A049559.

Array[GCD[#1 - 1, #2] & @@ {#, Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]]} &, 105] (* Michael De Vlieger, Dec 29 2020 *)

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A340084(n) = { my(u=A336466(n)); gcd(n-1, u); };

a(n) = gcd(n-1, A336466(n)).
a(n) = A000265(A340081(n)) = A336466(n) / A340085(n).
For n >= 2, a(n) = A000265(n-1) / A340086(n).
For n >= 1, a(A000040(n)) = A057023(n).
For n >= 0, a(A019565(2*n)) = A339899(n).

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).

A057025 Smallest prime of form (2n+1)*2^m+1 for some m.

Original entry on oeis.org

2, 7, 11, 29, 19, 23, 53, 31, 137, 1217, 43, 47, 101, 109, 59, 7937, 67, 71, 149, 79, 83, 173, 181

Offset: 0

Henry Bottomley, Jul 24 2000

nonn

next term a(23) = 47*2^583+1 > 10^177. Sequence then continues: 197, 103, 107, 881, 229, 1889, 977, 127, 131, 269, 139, 569, 293, 151, 617, 317, 163, 167, 1361, 349, 179, 23297, 373, 191, 389, 199, 809, ...

If no such prime exists for any m then 2n+1 is called a Sierpiński number. One could use a(n) = 0 for these cases. E.g., a(39278) = 0 because 78557 is a Sierpiński number. For the corresponding numbers m see A046067(n+1), n >= 0, where -1 entries corresponds to a(n) = 0. See also the Sierpiński links there. - Wolfdieter Lang, Feb 07 2013

			a(5)=23 because 2*5+1=11 and smallest prime of the form 11*2^m+1 is 23 (since 11+1=12 is not prime)

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A032373, A057023, A057026, A046067.

A058501 Primes p such that largest odd factor of p-1 is not a prime (i.e., is composite or 1).

Original entry on oeis.org

2, 3, 5, 17, 19, 31, 37, 43, 61, 67, 71, 73, 79, 101, 103, 109, 127, 131, 139, 151, 157, 163, 181, 191, 197, 199, 211, 223, 229, 239, 241, 251, 257, 271, 277, 281, 283, 307, 311, 313, 331, 337, 349, 367, 373, 379, 397, 401, 409, 419, 421, 431, 433, 439, 443

Offset: 1

Labos Elemer, Dec 20 2000

nonn

			127 is here because 127 - 1 = 126 = 2*63 and 63 is not a prime. 2 is here because 2 - 1 = 1 = 1*2^0 and 1 is not a prime.

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

Cf. A057023, A006093.

lofQ[n_]:=Module[{c=Select[Divisors[n-1],OddQ][[-1]]},!PrimeQ[c]]; Select[ Prime[ Range[100]],lofQ] (* Harvey P. Dale, Jul 24 2021 *)

Nonprimes in A057023.

A339972 Odd part of phi(A019565(8*n)).

Original entry on oeis.org

1, 3, 5, 15, 3, 9, 15, 45, 1, 3, 5, 15, 3, 9, 15, 45, 9, 27, 45, 135, 27, 81, 135, 405, 9, 27, 45, 135, 27, 81, 135, 405, 11, 33, 55, 165, 33, 99, 165, 495, 11, 33, 55, 165, 33, 99, 165, 495, 99, 297, 495, 1485, 297, 891, 1485, 4455, 99, 297, 495, 1485, 297, 891, 1485, 4455, 7, 21, 35, 105, 21, 63, 105, 315, 7, 21

Offset: 0

Antti Karttunen, Dec 26 2020

Compare also to the scatter plots of A339898 and A339901.

Antti Karttunen, Table of n, a(n) for n = 0..16383

Quadrisection of A339971.

Cf. A000265, A019565, A057023, A053575, A339821, A339898, A339901.

A000265(n) = (n>>valuation(n, 2));
A339972(n) = { my(m=1, p=5); while(n>0, p = nextprime(1+p); if(n%2, m *= A000265(p-1)); n >>= 1); (m); };

If 16n = 2^e1 + 2^e2 + ... + 2^ek [e1 ... ek distinct], then a(n) = A057023(e1) * A057023(e2) * ... * A057023(ek).

a(n) = A339971(4*n) = A000265(A339821(4*n)) = A053575(A019565(8*n)).

A099585 Remove all 3s from prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 2, 10, 4, 16, 2, 22, 28, 10, 4, 40, 14, 46, 52, 58, 20, 22, 70, 8, 26, 82, 88, 32, 100, 34, 106, 4, 112, 14, 130, 136, 46, 148, 50, 52, 2, 166, 172, 178, 20, 190, 64, 196, 22, 70, 74, 226, 76, 232, 238, 80, 250, 256, 262, 268, 10, 92, 280, 94, 292, 34, 310

Offset: 1

Ralf Stephan, Oct 24 2004

nonn

Equals A038502( A006093(n) ). prime(n)-1 = a(n) * 3^A099584(n).

Cf. A057023.

a(n) = (prime(n)-1)/3^valuation(prime(n)-1,3)

A356489 a(n) = A000265(rad(prime(n)-1)), rad = A007947.

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 1, 3, 11, 7, 15, 3, 5, 21, 23, 13, 29, 15, 33, 35, 3, 39, 41, 11, 3, 5, 51, 53, 3, 7, 21, 65, 17, 69, 37, 15, 39, 3, 83, 43, 89, 15, 95, 3, 7, 33, 105, 111, 113, 57, 29, 119, 15, 5, 1, 131, 67, 15, 69, 35, 141, 73, 51, 155, 39, 79, 165, 21, 173, 87, 11, 179

Offset: 1

Jianing Song, Aug 09 2022

			prime(8) = 19, so a(8) = A000265(rad(18)) = A000265(6) = 3.
prime(11) = 31, so a(11) = A000265(rad(30)) = A000265(30) = 15.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000265, A007947, A077063, A057023, A204455.

Array[#/2^IntegerExponent[#, 2] &[Times @@ FactorInteger[Prime[#] - 1][[All, 1]]] &, 72] (* Michael De Vlieger, Aug 09 2022 *)

a(n) = factorback(setminus(factorint(prime(n)-1)[, 1]~, [2]))

a(n) = A000265(rad(prime(n)-1)) = rad(A000265(prime(n)-1)).
a(n) = Product_{odd primes p dividing prime(n)-1} p.
a(n) = A000265(A077063(n)) = rad(A057023(n)) = A204455(prime(n)-1).

cp's OEIS Frontend

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

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A339971 Odd part of A339821(n).

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A058500 Primes of the form p*2^k + 1, where p is an odd prime and k > 0.

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A340084 a(n) = gcd(n-1, A336466(n)); Odd part of A340081(n).

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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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A057025 Smallest prime of form (2n+1)*2^m+1 for some m.

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A058501 Primes p such that largest odd factor of p-1 is not a prime (i.e., is composite or 1).

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A339972 Odd part of phi(A019565(8*n)).

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