A053176 -id:A053176 - cp's OEIS frontend

A350704 Composite numbers that have no Sophie Germain prime factors.

49, 91, 119, 133, 169, 217, 221, 247, 259, 289, 301, 323, 329, 343, 361, 403, 413, 427, 469, 481, 497, 511, 527, 553, 559, 589, 611, 629, 637, 679, 703, 707, 721, 731, 749, 763, 767, 793, 799, 817, 833, 871, 889, 893, 923, 931, 949, 959, 961, 973, 1003, 1027, 1037, 1043

Offset: 1

Karl-Heinz Hofmann, Feb 11 2022

nonn

A157342 is a subsequence. First differs at a(14) = 343.

A350705 is a subsequence too.

			a(2) = 91 = 7 * 13 and {7, 13} are not in A005384.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A157342, A005384, A053176, A350705, A350706.

Select[Range[1000], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 1]], !PrimeQ[2*#1 + 1] &] &] (* Amiram Eldar, Feb 12 2022 *)

isok(m) = if ((m>1) && !isprime(m), !#select(x->isprime(2*x+1), factor(m)[,1])); \\ Michel Marcus, Feb 11 2022

from sympy import primefactors, isprime
print([n for n in range(2,1044) if not isprime(n) and all(not isprime(p*2+1) for p in primefactors(n))])

A350705 Composite numbers that have no Sophie Germain prime and no "safe prime" factors.

Original entry on oeis.org

169, 221, 247, 289, 323, 361, 403, 481, 527, 559, 589, 629, 703, 731, 793, 817, 871, 923, 949, 961, 1027, 1037, 1139, 1147, 1159, 1207, 1241, 1261, 1273, 1313, 1333, 1339, 1343, 1349, 1369, 1387, 1417, 1501, 1591, 1649, 1651, 1717, 1751, 1781, 1807, 1843, 1849, 1853

Offset: 1

Karl-Heinz Hofmann, Feb 14 2022

nonn

Prime factors of the terms have to be in A059500.

			a(2) = 221 = 13 * 17 and {13, 17} are neither in A005384 nor in A005385, but they are in A059500.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Subsequence of A350704 and A350706.

Cf. A005384, A005385, A053176, A059500.

Select[Range[2000], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 1]], ! PrimeQ[2*#1 + 1] && ! PrimeQ[(#1 - 1)/2] &] &] (* Amiram Eldar, Feb 15 2022 *)

isok(m) = if ((m>1) && !isprime(m), my(f=factor(m)[,1]); !#select(x->isprime(2*x+1), f) && !#select(x->isprime((x-1)/2), f)); \\ Michel Marcus, Feb 14 2022

from sympy import primefactors, isprime
print([n for n in range(2,1854) if not isprime(n) and all(not isprime(p*2+1) and not isprime((p-1)//2) for p in primefactors(n))])

A367035 Numbers k such that the greatest prime less than 2*k is less than twice the greatest prime less than k.

Original entry on oeis.org

8, 14, 18, 20, 32, 33, 38, 39, 44, 48, 60, 61, 62, 63, 68, 72, 73, 74, 80, 81, 98, 102, 103, 104, 105, 108, 109, 110, 111, 128, 138, 140, 150, 151, 152, 153, 158, 164, 165, 168, 182, 183, 194, 198, 200, 212, 213, 214, 215, 224, 228, 230, 242, 243, 258, 259, 260, 264, 265, 266, 267, 268, 269, 270

Offset: 1

Robert Israel, Dec 15 2023

nonn

Numbers k such that A049711(2 * k) > 2 * A049711(k).

			a(3) = 18 is a term because the greatest prime < 18 is 17, the greatest prime < 2*18 = 36 is 31, and 31 < 2 * 17.

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

Includes k+1 for k in A053176. Disjoint from A006254.

Cf. A049711, A151799.

select(k -> prevprime(2*k) < 2*prevprime(k), [$3..300]);

isok(k) = precprime(2*k-1) < 2*precprime(k-1); \\ Michel Marcus, Dec 16 2023

A059688 Length of Cunningham chain containing prime(n) either as initial, internal or final term.

Original entry on oeis.org

5, 2, 5, 2, 5, 0, 0, 0, 5, 2, 0, 0, 3, 0, 5, 2, 2, 0, 0, 0, 0, 0, 3, 6, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 6, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Labos Elemer, Feb 06 2001

nonn

The length of a chain is measured by the total number of terms including the end points. a(n)=0 means that prime(n) is neither Sophie Germain nor a safe prime (i.e. it is in A059500).

			For all of {2,5,11,23,47}, i.e. at positions {j}={1,3,5,9,15} a(j)=5. Similarly for indices of all terms in {89,...,5759} a(i)=6. No chains are intelligible with length = 1 because the minimal chain enclose one Sophie Germain and also one safe prime. Dominant values are 0 and 2.

C. K. Caldwell, Cunningham Chains
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A059500.

Offset and a(5) corrected by Sean A. Irvine, Oct 01 2022

A138887 Numbers that are not Sophie Germain primes.

Original entry on oeis.org

0, 1, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 0

Omar E. Pol, Apr 05 2008

Nonnegative integers that are not in A005384.

A156660(a(n)) = 0; A053176 is a subsequence. [From Reinhard Zumkeller, Feb 18 2009]

Cf. A001690, A005384, A053176, A062289, A138888, A138889, A138890, A138891.

A230225 Primes p such that 2p+1 and 2p+3 are not prime.

Original entry on oeis.org

31, 37, 59, 61, 71, 79, 101, 103, 107, 109, 149, 151, 163, 181, 211, 241, 257, 263, 271, 311, 313, 317, 331, 347, 367, 373, 389, 401, 421, 433, 449, 457, 461, 479, 499, 521, 541, 569, 571, 577, 587, 601, 619, 631, 661, 673, 677, 691, 701, 709, 727, 733, 751

Offset: 1

Vincenzo Librandi, Oct 12 2013

			31 is in the sequence because 2*31+1=63 and 2*31+3=65 are not prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005384, A023204, A053176, A089531, A126107, A163769, A230039, A230117.

[p: p in PrimesUpTo(2500)|not IsPrime(2*p+1) and not IsPrime(2*p+3)];

Select[Range[10^3], PrimeQ[#]&&!PrimeQ[2 # + 1]&&!PrimeQ[2 # + 3]&]
Select[Prime[Range[200]],NoneTrue[2#+{1,3},PrimeQ]&] (* Harvey P. Dale, Sep 19 2021 *)

A292084 a(n) = least prime p such that 2p + 1 equals (2n - 1)*q where q is a prime, or 0 if no such p exists.

Original entry on oeis.org

0, 7, 7, 17, 13, 71, 19, 37, 59, 47, 31, 149, 37, 67, 43, 263, 181, 227, 1091, 97, 61, 107, 67, 1433, 73, 127, 79, 137, 199, 383, 701, 157, 97, 167, 103, 461, 109, 487, 269, 197, 283, 2531, 127, 739, 311, 227, 139, 617, 2861, 643, 151, 257, 157, 1979, 163, 277

Offset: 1

Arkadiusz Wesolowski, Sep 08 2017

nonn

Conjecture: 7 <= a(n) <= 4*n^2 - 5*n + 1 for n > 1. This conjecture implies that for every odd k > 1 there exist two primes p and q < 2*k such that k = (2*p + 1)/q.

Every positive term belongs to A053176.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A053176, A292083.

lst:=[]; for n in [2..56] do q:=1; repeat q+:=2; p:=Truncate((2*n*q-q-1)/2); until IsPrime(p) and IsPrime(q); Append(~lst, p); end for; [0] cat lst;

a(n) = {if (n==1, return(0)); forprime(p=3, , q = (2*p+1)/(2*n-1); if ((denominator(q) == 1) && isprime(q), return (p)););} \\ Michel Marcus, Sep 16 2017

A043298 Numbers n such that B(6*n) has denominator 42 where B(2k) are the Bernoulli numbers.

Original entry on oeis.org

1, 19, 31, 43, 59, 67, 71, 79, 97, 109, 127, 139, 149, 157, 163, 167, 193, 197, 199, 211, 223, 227, 229, 269, 307, 317, 337, 349, 353, 361, 379, 383, 389, 401, 409, 421, 433, 439, 449, 457, 463, 479, 487, 499, 521, 523, 541, 547, 563, 569, 571, 587, 589, 599

Offset: 1

Benoit Cloitre, Mar 24 2002

Except for 1 and 361=19^2 terms listed are primes.

Most a(n) are primes p such that 2p+1 is composite A053176. Nonprime a(n) (except a(1) = 1) are the powers or the products of primes from a(n). For example, 361 = 19^2, 589 = 19*31, 961 = 31^2, 1333 = 31*43, 1849 = 43^2, 2071 = 19*109, 2077 = 31*67, 2201 = 31*71, 2449 = 31*79, 2537 = 43*59, 2641 = 19*139, 2881 = 43*67, 2983 = 19*157, 3053 = 43*71, 3173 = 19*167, ..., 6859 = 19^3. - Alexander Adamchuk, Jul 28 2006

Enrique Pérez Herrero, Table of n, a(n) for n=1..50000

Cf. A051225, A053176.

Do[s=1+Divisors[n]; s1=Flatten[Position[PrimeQ[s], True]]; s2=Part[s, s1]; If[Equal[s2, {2, 3, 7}], Print[n/6]], {n, 1, 10000}] (* Alexander Adamchuk, Jul 28 2006 *)
Select[Range[600],Denominator[BernoulliB[6#]]==42&] (* Harvey P. Dale, Jan 09 2024 *)

Corrected and extended by Ralf Stephan, Oct 21 2002

More terms from Alexander Adamchuk, Jul 28 2006

A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

Original entry on oeis.org

1, 2, 2, 2, 3, 5, 7, 13, 20, 31, 52, 83, 142, 242, 412, 742, 1308, 2294, 4040, 7327, 13253, 24255, 44306, 81700, 150401, 277335, 513705, 954847, 1780466, 3325109, 6224282, 11676337, 21947583, 41327438

Offset: 1

Labos Elemer, Feb 06 2001

			a(11)-a(10) = 21 means that between 1024 and 2048 exactly 21 primes introduce Cunningham chains: {1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003}.
Their lengths are 2, 3 or 4. Thus the complete chains spread over more than one binary size-zone: {1409, 2819, 5639, 11279}. The primes 1439 and 2879 also form a chain but 1439 is not at the beginning of that chain, 89 is.

C. K. Caldwell, Cunningham Chains
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Cf. A023272, A023302, A023330, A005602, A007700, A053176, A059452-A059456, A059500, A057331, A059688, A007053, A036378, A029837, A007053.

c = 0; k = 1; Do[ While[k <= 2^n, If[ PrimeQ[k] && !PrimeQ[(k - 1)/2] && PrimeQ[2k + 1], c++ ]; k++ ]; Print[c], {n, 1, 29}]

from itertools import count, islice
from sympy import isprime, primerange
def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
def agen():
    s = 1
    for n in count(2):
        yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022

Edited and extended by Robert G. Wilson v, Nov 23 2002
Title and a(30)-a(31) corrected, and a(32) from Sean A. Irvine, Oct 02 2022
a(33)-a(34) from Michael S. Branicky, Oct 09 2022

A072607 If D[n] is divisor-set of n, then in set of 1+D only 2 primes occur:{2,3}; also n is not squarefree.

Original entry on oeis.org

98, 338, 578, 686, 722, 1274, 1862, 1922, 2366, 2738, 3038, 3626, 3698, 4214, 4394, 4418, 4802, 5054, 5978, 6422, 6566, 6962, 7154, 7442, 7742, 8918, 8978, 9386, 9506, 9826, 9898, 10082, 10094, 10478, 10658, 10682, 12446, 12482, 12506, 13034, 13426

Offset: 1

Labos Elemer, Jun 24 2002

nonn

			n = 338 = 2*13*13 is not squarefree; D = {1,2,13,26,169,338}; 1 + D = {2,3,14,27,170,339} contains only two primes {2,3}. Such numbers are nonsquarefree even nontotient numbers (from A005277), present also in A051222. Their odd prime divisors seem to arise from A053176.

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

Cf. A000005, A005277, A002202, A067513, A051222, A053176.

di[x_] := Divisors[x] dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1 Do[s=Length[dp[n]]; If[Equal[s, 2]&&Equal[MoebiusMu[n], 0], Print[n]], {n, 1, 25000}]

cp's OEIS Frontend

A350704 Composite numbers that have no Sophie Germain prime factors.

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Mathematica

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A350705 Composite numbers that have no Sophie Germain prime and no "safe prime" factors.

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Mathematica

PARI

Python

A367035 Numbers k such that the greatest prime less than 2*k is less than twice the greatest prime less than k.

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Maple

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A059688 Length of Cunningham chain containing prime(n) either as initial, internal or final term.

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A138887 Numbers that are not Sophie Germain primes.

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A230225 Primes p such that 2*p+1 and 2*p+3 are not prime.

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Magma

Mathematica

A292084 a(n) = least prime p such that 2*p + 1 equals (2*n - 1)*q where q is a prime, or 0 if no such p exists.

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Magma

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A043298 Numbers n such that B(6*n) has denominator 42 where B(2k) are the Bernoulli numbers.

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Mathematica

A230225 Primes p such that 2p+1 and 2p+3 are not prime.

A292084 a(n) = least prime p such that 2p + 1 equals (2n - 1)*q where q is a prime, or 0 if no such p exists.