A152117 -id:A152117 - cp's OEIS frontend

A119487 Primes of the form iprime(i) + (i+1)prime(i+1).

43, 83, 197, 271, 359, 631, 977, 1307, 1553, 2371, 2693, 2953, 3271, 4561, 5051, 5407, 6551, 8713, 9941, 10651, 22573, 23333, 27689, 31051, 33203, 34123, 37507, 52639, 60919, 64399, 79279, 82699, 93559, 112061, 119131, 136033, 146921, 197959

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 23 2006

Primes in A152117; also called linking primes, cf. A152658. - Klaus Brockhaus, Dec 11 2008

			The third prime is 5 and the fourth is 7. Therefore 5*3 + 7*4 = 15 + 28 = 43 which is a prime.

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

Cf. A119488.

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A152658 (beginnings of maximal chains of primes). - Klaus Brockhaus, Dec 11 2008

[ q: n in [1..133] | IsPrime(q) where q is n*p+(n+1)*NextPrime(p) where p is NthPrime(n) ] // Klaus Brockhaus, Dec 11 2008

P:=proc(n) local i,j; for i from 1 by 1 to n do j:=ithprime(i)*i +ithprime(i+1)*(i+1); if isprime(j) then print(i); fi; od; end: P(200);

A152658 Beginnings of maximal chains of primes.

Original entry on oeis.org

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

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) ];

A145650 Linking prime for the first and second member of maximal chains of primes that have at least three members.

Original entry on oeis.org

43, 197, 1307, 2371, 4561, 9941, 22573, 33203, 214507, 227611, 306853, 332993, 389167, 505907, 695059, 758441, 810023, 1072657, 1202987, 1404211, 1567487, 1621621, 2407309, 2773681, 2854331, 2932511, 3013601, 3206773, 3851423

Offset: 1

Enoch Haga, Oct 15 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.
A145651 gives the linking prime for the second and third member of maximal chains of primes that have at least three members.
Suggested by J. M. Bergot in Puzzle 463 of Carlos Rivera's Prime Puzzles & Problems Connection

			Primes 13, 17, 19, 23 have prime indices 6, 7, 8, 9. 6*13 + 7*17 = 197 is prime; 7*17 + 8*19 = 271 is prime; 8*19 + 9*23 = 359 is prime. Neither 5*11 + 6*13 = 133 nor 9*23 + 10*29 = 497 is prime, so 13, 17, 19, 23 is maximal. Hence 6*13 + 7*17 = 197, the linking prime for 13 and 17, is in the sequence.

Carlos Rivera, Puzzle 463

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A119487 (primes in A152117, linking primes), A152658 (beginnings of maximal chains of primes), A145651.

[ n*p+(n+1)*q: n in [1..520] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p)+n*p) ) and IsPrime(n*p+(n+1)*q) and IsPrime((n+1)*q+(n+2)*r) where r is NextPrime(q) where q is NextPrime(p) where p is NthPrime(n) ]; // Klaus Brockhaus, Dec 11 2008

{n=1; while(n<520, c=0; while(isprime(b=n*prime(n)+(n+1)*prime(n+1)), c++; n++; if(c==1, a=b)); if(c>1, print1(a, ",")); n++)}

Edited by Klaus Brockhaus, Dec 10 2008

A145651 Linking prime for the second and third member of maximal chains of primes that have at least three members.

Original entry on oeis.org

83, 271, 1553, 2693, 5051, 10651, 23333, 34123, 219389, 230933, 312007, 338017, 395309, 512891, 699437, 763999, 815257, 1078127, 1208791, 1417019, 1577561, 1629083, 2420609, 2787947, 2868787, 2944429, 3038639, 3222101, 3868201

Offset: 1

Enoch Haga, Oct 15 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.
A145650 gives the linking prime for the first and second member of maximal chains of primes that have at least three members.
Suggested by J. M. Bergot in Puzzle 463 of Carlos Rivera's Prime Puzzles & Problems Connection

			Primes 13, 17, 19, 23 have prime indices 6, 7, 8, 9. 6*13 + 7*17 = 197 is prime; 7*17 + 8*19 = 271 is prime; 8*19 + 9*23 = 359 is prime. Neither 5*11 + 6*13 = 133 nor 9*23 + 10*29 = 497 is prime, so 13, 17, 19, 23 is maximal. Hence 7*17 + 8*19 = 271, the linking prime for 17 and 19, is in the sequence.

Carlos Rivera, Puzzle 463

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A119487 (primes in A152117, linking primes), A152658 (beginnings of maximal chains of primes), A145650.

[ (n+1)*q+(n+2)*r: n in [1..520] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p)+n*p) ) and IsPrime(n*p+(n+1)*q) and IsPrime((n+1)*q+(n+2)*r) where r is NextPrime(q) where q is NextPrime(p) where p is NthPrime(n) ]; // Klaus Brockhaus, Dec 11 2008

{n=1; while(n<520, c=0; while(isprime(b=n*prime(n)+(n+1)*prime(n+1)), c++; n++; if(c==2, a=b)); if(c>1, print1(a, ",")); n++)}

Edited by Klaus Brockhaus, Dec 10 2008

A268467 Smallest prime that is the (sum, k*prime(k),k=m,..n+m-1) for some m, or 0 if no such m exists.

Original entry on oeis.org

2, 43, 23, 0, 1109, 1187, 929, 0, 4973, 1291, 11197, 0, 26099, 15583, 4423, 0, 42139, 10729, 21283, 0, 36899, 27179, 21563, 0, 24359, 33863, 27361, 0, 223423, 51239, 293467, 42043, 67699, 56503, 118361, 0, 80449, 94693, 136739, 0, 127837, 136991, 387913, 0, 304259, 192013, 321721, 0, 339517, 357683

Offset: 1

Zak Seidov, Feb 05 2016

nonn

Smallest prime that is the sum of n consecutive terms of A033286.
Apparently a(n) exists for any odd n.
Values of m = {1, 3, 1, 0, 7, 6, 4, 0, 9, 2, 12, 0, 17, 11, 2, 0, 17, 4, 8, 0, 11, 7, 4, 0, 3, 5, 2, 0, 27, 5, 30, 1, 5, 2, 10, 0, 3, 4, 8, 0, 5, 5, 22, 0, 15, 6, 14, 0, 13, 13, ...}. - Michael De Vlieger, Feb 05 2016

			n=1: m=1 and 1*prime(1) = 1*2 = 2 = a(1),
n=2: m=3 and 3*prime(3)+4*prime(4) = 3*5+4*7 = 43 = a(2),
n=3: m=1 and 1*prime(1)+2*prime(2)+3*prime(3) = 1*2+2*3+3*15 = 23 = a(3),
n=4: no solution => a(4) = 0,
n=5: m=7 and 7*prime(7)+..11*prime(11) = 119+152+207+290+341 = 1109 = a(5).

Cf. A000040, A033286, A105455, A268465, A105454, A152117, A119487.

Table[If[# == 0, 0, Sum[k Prime@ k, {k, #, n + # - 1}]] &@(SelectFirst[Range[10^3], PrimeQ@ Sum[k Prime@ k, {k, #, n + # - 1}] &] /. x_ /; MissingQ@ x -> 0), {n, 50}] (* Michael De Vlieger, Feb 05 2016, Version 10.2 *)

cp's OEIS Frontend

A119487 Primes of the form i*prime(i) + (i+1)*prime(i+1).

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

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A152657 Secluded primes.

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A145650 Linking prime for the first and second member of maximal chains of primes that have at least three members.

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A145651 Linking prime for the second and third member of maximal chains of primes that have at least three members.

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A119487 Primes of the form iprime(i) + (i+1)prime(i+1).

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

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