A119487 -id:A119487 - cp's OEIS frontend

A152658 Beginnings of maximal chains of primes.

5, 13, 29, 37, 43, 61, 89, 109, 131, 139, 227, 251, 269, 277, 293, 359, 389, 401, 449, 461, 491, 547, 569, 607, 631, 743, 757, 773, 809, 857, 887, 947, 971, 991, 1069, 1109, 1151, 1163, 1187, 1237, 1289, 1301, 1319, 1373, 1427, 1453, 1481, 1499, 1549, 1601

Offset: 1

Klaus Brockhaus, Dec 10 2008

nonn

A sequence of consecutive primes prime(k), ..., prime(k+r), r >= 1, is called a chain of primes if i*prime(i) + (i+1)*prime(i+1) is prime (the linking prime for prime(i) and prime(i+1), cf. A119487) for i from k to k+r-1. A chain of primes prime(k), ..., prime(k+r) is maximal if it is not part of a longer chain, i.e. if neither (k-1)*prime(k-1) + k*prime(k) nor (k+r)*prime(k+r) + (k+r+1)*prime(k+r+1) is prime.

A chain of primes has two or more members; a prime is called secluded if it is not member of a chain of primes (cf. A152657).

			3*prime(3) + 4*prime(4) = 3*5 + 4*7 = 43 is prime and 4*prime(4) + 5*prime(5) = 4*7 + 5*11 = 83 is prime, so 5, 7, 11 is a chain of primes. 2*prime(2) + 3*prime(3) = 2*3 + 3*5 = 21 is not prime and 5*prime(5) + 6*prime(6) = 5*11 + 6*13 = 133 is not prime, hence 5, 7, 11 is maximal and prime(3) = 5 is the beginning of a maximal chain.

Klaus Brockhaus, Table of n, a(n) for n=1..10000

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A152657 (secluded primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes).

Cf. A105454 - Zak Seidov, Feb 04 2016

[ p: n in [1..253] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p) +n*p) ) and IsPrime((n)*p+(n+1)*NextPrime(p)) where p is NthPrime(n) ];

A105455 Numbers k such that kprime(k)+(k+1)prime(k+1)+(k+2)*prime(k+2) is prime.

Original entry on oeis.org

1, 6, 12, 20, 22, 24, 28, 30, 34, 56, 60, 142, 144, 148, 168, 192, 196, 230, 252, 260, 276, 282, 304, 322, 334, 344, 346, 352, 366, 374, 380, 386, 394, 404, 418, 424, 432, 440, 444, 470, 478, 484, 572, 590, 610, 612, 630, 642, 662, 684, 754, 766, 784, 790, 840, 842, 874, 886

Offset: 1

Zak Seidov, May 02 2005

nonn

			k=1: 1*prime(1) + 2*prime(2) + 3*prime(3) = 1*2 + 2*3 + 3*5 = 23 prime,
k=6: 6*prime(6) + 7*prime(7) + 8*prime(8) = 6*13 + 7*17 + 8*19 = 349 prime. - _Zak Seidov_, Feb 18 2016

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A033286, A105454, A152117, A119487.

[n: n in [1..1000] | IsPrime(n*NthPrime(n)+(n+1)*NthPrime(n+1)+(n+2)*NthPrime(n+2))]; // Vincenzo Librandi, Feb 06 2016

bb={};Do[If[PrimeQ[n Prime[n]+(n+1) Prime[n+1]+(n+2) Prime[n+2]], bb=Append[bb, n]], {n, 1, 400}];bb
Select[Range@ 900, PrimeQ[# Prime[#] + (# + 1) Prime[# + 1] + (# + 2) Prime[# + 2]] &] (* Michael De Vlieger, Feb 05 2016 *)

lista(nn) = {for(n=1, nn, if(ispseudoprime(n*prime(n)+(n+1)*prime(n+1)+(n+2)*prime(n+2)), print1(n, ", "))); } \\ Altug Alkan, Feb 05 2016

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    m, p, q, r = 1, 2, 3, 5
    while True:
        t = m*p + (m+1)*q + (m+2)*r
        if isprime(t): yield m
        m, p, q, r = m+1, q, r, nextprime(r)
print(list(islice(agen(), 58))) # Michael S. Branicky, May 17 2022

A152117 a(n) = n(n-th prime) + (n+1)((n+1)-th prime).

Original entry on oeis.org

8, 21, 43, 83, 133, 197, 271, 359, 497, 631, 785, 977, 1135, 1307, 1553, 1851, 2101, 2371, 2693, 2953, 3271, 3647, 4045, 4561, 5051, 5407, 5777, 6157, 6551, 7327, 8129, 8713, 9247, 9941, 10651, 11245, 12003, 12707, 13433, 14259, 14941, 15815, 16705

Offset: 1

Klaus Brockhaus, Dec 10 2008

nonn

a(n) = A033286(n) + A033286(n+1).

			5*(fifth prime) + 6*(sixth prime) = 5*11 + 6*13 = 55 + 78 = 133.

Klaus Brockhaus, Table of n, a(n) for n=1..1000

Cf. A000040 (prime numbers), A033286 (n*(n-th prime)), A033287 (first differences of A033286), A119487 (primes in this sequence).

[ n*NthPrime(n)+(n+1)*NthPrime(n+1): n in [1..43] ];

Total/@Partition[Times@@@Table[{n,Prime[n]},{n,50}],2,1] (* Harvey P. Dale, Aug 13 2019 *)

a(n) = n*prime(n) + (n+1)*prime(n+1); \\ Michel Marcus, Feb 05 2016

A152735 Count of links in n-th maximal chain of primes.

Original entry on oeis.org

2, 3, 1, 1, 2, 4, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Klaus Brockhaus, Dec 16 2008

nonn

One less than count of members of n-th maximal chain of primes. For definitions see A152658.

			The consecutive primes 5, 7, 11 form the first maximal chain of primes (see example in A152658); it has three members, two links. Therefore a(1) = 2.

Klaus Brockhaus, Table of n, a(n) for n=1..1000

Cf. A152658 (beginnings of maximal chains of primes), A152657 (secluded primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes).

{n=1; while(n<560, c=0; while(isprime(n*prime(n)+(n+1)*prime(n+1)), c++; n++); if(c>0, print1(c, ",")); n++)}

A152962 Beginning of the first maximal chain of primes with n links (n+1 members).

Original entry on oeis.org

29, 5, 13, 61, 417037, 2153, 4580041

Offset: 1

Klaus Brockhaus, Dec 16 2008

For definitions see A152658.
The prime indices of the terms are 10, 3, 6, 18, 35153, 325, 321150.
Is this sequence finite, i.e. is the length (count of members) of chains of primes bounded?

			First maximal chain of primes with one link (two members) is 29, 31; the linking prime is 631.

Carlos Rivera, The prime puzzles & problems connection, Puzzle 463

Cf. A152658 (beginnings of maximal chains of primes), A152735 (count of links in n-th maximal chain of primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes), A152865, A152866, A152867, A152868, A152869, A152963, A152964.

A152963 Beginnings of maximal chains of primes with seven members (six links).

Original entry on oeis.org

2153, 10316897, 114286307, 220073429, 406733477

Offset: 1

Klaus Brockhaus, Dec 16 2008

For definitions see A152658, of which this is a subsequence.

			First maximal chain of primes with seven members is 2153, 2161, 2179, 2203, 2207, 2213, 2221; the six linking primes are 1404211, 1417019, 1435117, 1448687, 1456393, 1465441.

Cf. A152658 (beginnings of maximal chains of primes), A152962 (beginning of the first maximal chain of primes with n links), A152735 (count of links in n-th maximal chain of primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes), A152865, A152866, A152867, A152868, A152869, A152964.

A152964 Beginnings of maximal chains of primes with eight members (seven links).

Original entry on oeis.org

4580041, 608744947, 2784514741, 3336893489, 3743034679

Offset: 1

Klaus Brockhaus, Dec 16 2008

For definitions see A152658, of which this is a subsequence.

Terms were computed by Farideh Firoozbakht (see C. Rivera link), who remarks that there are no other terms with prime indices up to 372*10^6, i.e., terms below approx. 8*10^9. The prime indices of the given terms are 321150, 31757516, 134558368, 159849354, 178323284.

			First maximal chain of primes with eight members is 4580041, 4580077, 4580117, 4580131, 4580141, 4580143, 4580201, 4580209; the seven linking primes are 2941776475777, 2941810043411, 2941836545827, 2941853413757, 2941866427879, 2941894857 521, 2941925214169.

Carlos Rivera, The prime puzzles & problems connection, Puzzle 463

Cf. A152962 (beginning of the first maximal chain of primes with n links), A152658 (beginnings of maximal chains of primes), A152735 (count of links in n-th maximal chain of primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes), A152865, A152866, A152867, A152868, A152869, A152963.

A105454 Numbers k such that kprime(k)+(k+1)prime(k+1) is prime.

Original entry on oeis.org

3, 4, 6, 7, 8, 10, 12, 14, 15, 18, 19, 20, 21, 24, 25, 26, 29, 32, 34, 35, 49, 50, 54, 57, 59, 60, 62, 72, 77, 79, 87, 89, 94, 101, 104, 111, 115, 132, 134, 137, 138, 140, 141, 142, 148, 154, 161, 162, 164, 167, 168, 180, 181, 182, 183, 186, 190, 192, 195, 203, 204

Offset: 1

Zak Seidov, May 02 2005

nonn

Or, numbers k such that A152117(k) is prime. - Zak Seidov, Feb 05 2016

			4*prime(4)+5*prime(5) = 4*7+5*11 = 83 prime.

Cf. A152117, A119487.

[n: n in [1..250] | IsPrime(n*NthPrime(n)+(n+1)*NthPrime(n+1))]; // Vincenzo Librandi, Feb 06 2016

bb={};Do[If[PrimeQ[n Prime[n]+(n+1)Prime[n+1]], bb=Append[bb, n]], {n, 400}];bb
Select[Range[250],PrimeQ[# Prime[#]+(#+1)Prime[#+1]]&] (* Harvey P. Dale, Dec 08 2011 *)

isok(n) = isprime(n*prime(n)+(n+1)*prime(n+1)); \\ Michel Marcus, Feb 05 2016

lista(nn) = {for(n=1, nn, if(ispseudoprime(n*prime(n)+(n+1)*prime(n+1)), print1(n, ", ")));} \\ Altug Alkan, Feb 05 2016

A152657 Secluded primes.

Original entry on oeis.org

2, 3, 59, 83, 107, 127, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 239, 241, 263, 311, 313, 317, 331, 337, 347, 349, 353, 373, 379, 383, 419, 421, 431, 433, 439, 443, 467, 479, 487, 503, 509, 521, 523, 541, 563, 577, 587, 593, 599, 601, 617

Offset: 1

Klaus Brockhaus, Dec 10 2008

nonn

A prime p is called secluded if it is not member of a chain of primes. A sequence of consecutive primes prime(k), ..., prime(k+r), r >= 1, is called a chain of primes if i*prime(i) + (i+1)*prime(i+1)* is prime for i from k to k+r-1.

			16*prime(16) + 17*prime(17) = 16*53 + 17*69 = 1851 = 3*617 is not prime; 17*prime(17) + 18*prime(18) = 17*59 + 18*61 = 2101 = 11+191 is not prime. Hence prime(17) = 59 is secluded.

Klaus Brockhaus, Table of n, a(n) for n=1..10000

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A152658 (beginnings of maximal chains of primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes).

[ p: n in [1..113] | (n eq 1 or not IsPrime((n-1)*NthPrime(n-1)+k)) and not IsPrime(k+(n+1)*NthPrime(n+1)) where k is n*p where p is NthPrime(n) ];

A119488 Primes of the form prime(i+1)(i+1) - prime(i)i, for increasing values of i.

Original entry on oeis.org

13, 23, 41, 83, 103, 89, 103, 113, 227, 229, 547, 373, 419, 263, 373, 787, 419, 433, 593, 563, 577, 739, 487, 811, 823, 683, 1013, 599, 1153, 641, 827, 1571, 1223, 863, 883, 719, 1567, 1187, 1279, 1999, 1361, 1373, 1951, 1297, 2477, 1091, 1399, 1117, 2897, 1459

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 23 2006

Some terms are repeated: e.g. 157*37 - 151*36 = 197*45 - 193*44 = 373.

The first numbers that are repeated 3 times are 96553, 104597, 109793, 139303, etc.

			The fourth prime is 7 and the third is 5.
Therefore 7*4 - 5*3 = 28 - 15 = 13 that is a prime.

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

Cf. A119487.

[m: i in [1..300] | IsPrime(m) where m is NthPrime(i+1)*(i+1)-NthPrime(i)*i]; // Bruno Berselli, Jun 06 2017

P:=proc(n) local i, j; j:=ithprime(n+1)*(n+1)-ithprime(n)*n;
if isprime(j) then j; fi; end: a:=seq(P(i), i=1..10000);

Select[#[[2]] - #[[1]] &/@ Partition[Table[n Prime[n], {n, 300}], 2, 1], PrimeQ] (* Harvey P. Dale, Jun 05 2017 *)

cp's OEIS Frontend

A152658 Beginnings of maximal chains of primes.

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A105455 Numbers k such that k*prime(k)+(k+1)*prime(k+1)+(k+2)*prime(k+2) is prime.

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A152117 a(n) = n*(n-th prime) + (n+1)*((n+1)-th prime).

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A152735 Count of links in n-th maximal chain of primes.

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A152962 Beginning of the first maximal chain of primes with n links (n+1 members).

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A152963 Beginnings of maximal chains of primes with seven members (six links).

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A152964 Beginnings of maximal chains of primes with eight members (seven links).

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A105454 Numbers k such that k*prime(k)+(k+1)*prime(k+1) is prime.

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A105455 Numbers k such that kprime(k)+(k+1)prime(k+1)+(k+2)*prime(k+2) is prime.

A152117 a(n) = n(n-th prime) + (n+1)((n+1)-th prime).

A105454 Numbers k such that kprime(k)+(k+1)prime(k+1) is prime.

A119488 Primes of the form prime(i+1)(i+1) - prime(i)i, for increasing values of i.