A057326 -id:A057326 - cp's OEIS frontend

A231815 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p-1 and r = 2q-1.

30, 51319, 3882139, 289022911, 674910259, 991523479, 1893583519, 4550912389, 9761467669, 16721570539, 28685399311, 72886214809, 77372307511, 82720376839, 98685849571, 173850108931, 220038912319, 229352039821, 240313142749, 257401051861, 428178002569

Offset: 1

Jaroslav Krizek, Nov 13 2013

nonn

Squarefree numbers of the form p*q*r, where p < q < r = primes with q = 2*p - 1 and r = 2*q - 1; that is, r = 4*p - 3.

These numbers are divisible by the arithmetic mean of their proper divisors.

			3882139 = 79*157*313; 157 = 2*79 - 1; 313 = 2*157 - 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005117, A057326, A000040, A231814, A231816.

Cf. A057326 (first member of a prime triple in a 2p-1 progression).

t = {}; p = 1; Do[While[p = NextPrime[p]; ! (PrimeQ[p2 = 2 p - 1] && PrimeQ[p3 = 2 p2 - 1])]; AppendTo[t, p*p2*p3], {30}]; t (* T. D. Noe, Nov 15 2013 *)
3#-10#^2+8#^3&/@Select[Prime[Range[600]],AllTrue[{2#-1,4#-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 02 2016 *)

A336059 Numbers p such that p, 2p-1, 3p-2, 4p-3 are primes.

Original entry on oeis.org

331, 1531, 3061, 4261, 4951, 6841, 10831, 15391, 18121, 23011, 25411, 26041, 31771, 33301, 40111, 41491, 45061, 49831, 53881, 59341, 65851, 70141, 73771, 78541, 88741, 95461, 96931, 109471, 111721, 112621, 117721, 131311, 133201, 134731, 135301, 150151, 165901

Offset: 1

Jeppe Stig Nielsen, Jul 07 2020

The subset p, 2p-1, 4p-3 is a Cunningham chain of the 2nd kind, cf. A057326.

Wikipedia, Cunningham chain

Cf. A005382, A057326, A067257, A237189, A336060.

Select[Range[10^5], AllTrue[{#, 2# - 1, 3# - 2, 4# - 3}, PrimeQ] &] (* Amiram Eldar, Jul 07 2020 *)

a(n) = A237189(n) + 1.

A174568 Numbers n such that phi(n) + sigma(n) = sigma(n + phi(n)).

Original entry on oeis.org

2, 3, 7, 19, 31, 37, 79, 97, 99, 135, 139, 157, 198, 199, 211, 229, 271, 287, 307, 331, 337, 350, 367, 379, 439, 499, 539, 547, 577, 601, 607, 619, 661, 671, 691, 727, 811, 829, 877, 923, 937, 967, 997, 1009, 1069, 1171, 1237, 1254, 1279, 1297, 1399, 1429

Offset: 1

Michel Lagneau, Mar 22 2010

nonn

A005382 is included in this sequence : if p and 2p-1 primes, phi(p) = p-1, sigma(p)=p+1 and sigma(2p-1)=2p => phi(p) +sigma(p) = sigma(p+phi(p)). See the similar sequence A005384.

			2 is in the sequence because phi(2) + sigma(2) = 1 + 3 = 4, and sigma(2 + phi(2)) = sigma(3) = 4;
99 is in the sequence because phi(99) + sigma(99) = 60 + 156 = 216, and sigma(99 + phi(99)) = sigma(159) = 216.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A005382, A005384, A057326, A057327, A057328, A057330, A005603.

[n: n in [1..1500] | (EulerPhi(n) + SumOfDivisors(n)) eq (SumOfDivisors(n + EulerPhi(n)))]; // Vincenzo Librandi, Jul 15 2015

with(numtheory):for n from 1 to 3000 do :if phi(n)+sigma(n) = sigma(n+phi(n)) then print(n):else fi:od:

Select[Range[1500],EulerPhi[#]+DivisorSigma[1,#]==DivisorSigma[1, #+ EulerPhi[ #]]&] (* Harvey P. Dale, Jul 05 2018 *)

A289109 Primes p that remain prime through 3 iterations of function f(x) = 6x - 1.

Original entry on oeis.org

239, 269, 439, 569, 599, 829, 1429, 3389, 6379, 7159, 7649, 8779, 8969, 10799, 10939, 12919, 13729, 13879, 15649, 17159, 18149, 19379, 21649, 22669, 23929, 24799, 25679, 26849, 28219, 30389, 30689, 33749, 34759, 36109, 36209, 36899, 40759, 47659, 49639, 52369

Offset: 1

K. D. Bajpai, Jun 24 2017

nonn

All the terms are congruent to 9 (mod 10). The iteration of f(x) on a term of this sequence then produces primes congruent to 3, 7, 1 (mod 10), followed by a nontrivial multiple of 5.

			239 is prime and 6 * 239 - 1 = 1433, which is also prime. 6 * 1433 - 1 = 8597, which is also prime. 6 * 8597 = 51581, which is also prime. 6 * 51581 - 1 = 309485 = 5 * 11 * 17 * 331, which is composite, but the previous three primes are enough for 239 to be in the sequence.
241 is not in the sequence because 6 * 241 - 1 = 1445 = 5 * 17^2, which is composite.

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

Cf. A057326, A057327, A057328, A057329, A057330, A158015.

filter:= x -> andmap(isprime, [x,6*x-1,36*x-7,216*x-43]):
select(filter, [seq(i,i=9..60000,10)]); # Robert Israel, May 10 2020

Select[Prime[Range[15000]], And @@ PrimeQ[NestList[6 # - 1 &, #, 3]] &]

forprime(p= 1, 100000, if(isprime(6*p-1) && isprime(36*p-7) && isprime(216*p-43) , print1(p, ", ")));

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 3, 2, 509, 2, 89, 16651, 15514861, 85864769, 26089808579, 665043081119, 554688278429, 758083947856951, 95405042230542329, 69257563144280941

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one). a(1)-a(8) computed by Gilles Sadowski.

			a(1)=11 because 2, 3, 5 and 7 are included in longer chains than one prime long; and 11 (although included in a <2p+1> chain) has no prime connection through <2p-1>.
a(2)=3 because 3 begins (through 2p+1>) the first complete two primes chain: 3-> 7 (even if 3 and 7 are also part of two others chains, but through <2p-1>).
a(3)=2 because (although 2 begins also a five primes chain through <2p+1>) it begins, through <2p-1>, the first complete three primes chain encountered: 2->3->5.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

a(n) = min(A005602(n), A005603(n)). - R. J. Mathar, Jul 23 2008

a(8)-a(13) via A005602, A005603 from R. J. Mathar, Jul 23 2008

a(14)-a(15) via A005602, A005603 from Jason Yuen, Sep 03 2024

A110092 Smallest prime ending (through <2+1> or <2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

17, 59, 73, 4079, 47, 2879, 1065601

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may not be involved in a chain of the other kind (i.e. the chain may not be connected to another one).

			a(1)=17 because 2, 3, 5, 7, 11 and 13 are part of longer chains whatever the operator; 17 is the first completely isolated prime.
a(2)=59 because it ends the first two primes chain not connected to another one: 29->59.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Terms computed by Gilles Sadowski.

A110093 Smallest prime ending (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 7, 5, 4079, 47, 2879, 1065601

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); but the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one).

			a(1)=11 because 2, 3, 5 and 7 are not ending chains; or are part of chains longer than one prime; 11, although is part of a five primes <2p+1> chain, is isolated through <2p-1>.
a(2)=7 because 7 ends through <2p+1> the first two primes chain: 3->7 (even if both primes are also part of <2p-1> chains).

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Terms computed by Gilles Sadowski.

A379144 a(n) is the number of iterations of the function x --> 2*x - 1 such that x remains prime, starting from A005382(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1

Offset: 1

Ctibor O. Zizka, Dec 16 2024

nonn

Cunningham chain of the second kind of length i is a sequence of prime numbers (p_1, ..., p_i) such that p_(r + 1) = 2*p_r - 1 for all 1 =< r < i. This sequence tells the length of the Cunningham chain of the second kind for primes from A005382.

			n = 1: A005382(1) = 2 --> 3 --> 5 --> 9, 9 is not a prime, thus a(1) = 2.
n = 3: A005382(3) = 7 --> 13 --> 25, 25 is not a prime, thus a(3) = 1.

Cf. A000040, A005382, A005383, A057326, A064812, A110581, A307390.

s[n_] := -2 + Length[NestWhileList[2*# - 1 &, n, PrimeQ[#] &]]; Select[Array[s, 5000], # > 0 &] (* Amiram Eldar, Dec 16 2024 *)

a(A110581(n)) = 1.

a(A057326(n)) = 2.

cp's OEIS Frontend

A231815 Squarefree numbers (A005117) of the form p*q*r with prime factors p, q, r with q = 2*p-1 and r = 2*q-1.

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A336059 Numbers p such that p, 2p-1, 3p-2, 4p-3 are primes.

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A174568 Numbers n such that phi(n) + sigma(n) = sigma(n + phi(n)).

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A289109 Primes p that remain prime through 3 iterations of function f(x) = 6x - 1.

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PARI

A110089 Smallest prime beginning (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

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A110092 Smallest prime ending (through <*2+1> or <*2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

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A110093 Smallest prime ending (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

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Extensions

A379144 a(n) is the number of iterations of the function x --> 2*x - 1 such that x remains prime, starting from A005382(n).

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A231815 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p-1 and r = 2q-1.

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

A110092 Smallest prime ending (through <2+1> or <2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

A110093 Smallest prime ending (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.