A340464 -id:A340464 - cp's OEIS frontend

A340463 Primes p such that pq+rs+t*u is prime, where p,q,r,s,t,u are consecutive primes.

3, 17, 41, 47, 67, 107, 193, 197, 199, 211, 229, 239, 313, 331, 367, 461, 467, 503, 523, 571, 919, 929, 991, 1021, 1039, 1093, 1109, 1163, 1193, 1237, 1277, 1327, 1361, 1381, 1621, 1627, 1783, 1901, 2029, 2099, 2143, 2381, 2389, 2423, 2473, 2663, 2677, 2801, 2917, 2939, 2953, 2957, 2963, 3019

Offset: 1

J. M. Bergot and Robert Israel, Jan 08 2021

nonn

			a(3)=41 is a term because 41*43+47*53+59*61=7853 is prime, where 41,43,47,53,59,61 are consecutive primes.

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

Cf. A340464.

map(ithprime, select(i -> isprime(ithprime(i)*ithprime(i+1)+ithprime(i+2)*ithprime(i+3)+ithprime(i+4)*ithprime(i+5)), [$1..1000]));

Select[Partition[Prime[Range[500]],6,1],PrimeQ[#[[1]]#[[2]]+#[[3]]#[[4]]+#[[5]]#[[6]]]&][[;;,1]] (* Harvey P. Dale, Jan 10 2025 *)

from sympy import nextprime, isprime
def aupto(nn):
  alst, consec6 = [], [2, 3, 5, 7, 11, 13]
  p, q, r, s, t, u = consec6; prod = p*q+r*s+t*u
  while p <= nn:
    if isprime(prod): alst.append(p)
    consec6 = consec6[1:] + [nextprime(consec6[-1])]
    p, q, r, s, t, u = consec6; prod = p*q+r*s+t*u
  return alst
print(aupto(3019)) # Michael S. Branicky, Jan 08 2021

A340465 Primes of the form prime(i)prime(i+1)+prime(i+2)prime(i+3)+...+prime(k-1)*prime(k).

Original entry on oeis.org

41, 313, 2137, 6569, 7853, 10133, 10847, 12401, 13757, 14747, 17569, 17911, 24001, 24049, 27901, 31307, 38729, 43177, 43961, 44819, 51607, 69191, 81517, 88379, 104683, 107099, 130631, 137177, 138239, 145967, 154487, 154723, 158777, 162947, 175463, 184409, 192853, 196169, 232499, 243137, 261983

Offset: 1

J. M. Bergot and Robert Israel, Jan 08 2021

nonn

A prime that has more than one expression of the given form is included only once. The first such prime is a(14353) = 6858604873 = 1979*1987+...+7109*7121 = 19949*19961+...+20231*20233.

			a(1) = 2*3+5*7 = 41.
a(2) = 3*5+7*11+13*17 = 313.
a(3) = 17*19+23*29+31*37 = 2137.
a(4) = 5*7+11*13+17*19+23*29+31*37+41*43+47*53 = 6569.
a(5) = 41*43+47*53+59*61 = 7853.

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

Includes A340464.

S1:= [0,seq(ithprime(2*i)*ithprime(2*i+1),i=1..100)]:
P1:= ListTools:-PartialSums(S1):
S2:= [0,seq(ithprime(2*i-1)*ithprime(2*i),i=1..100)]:
P2:= ListTools:-PartialSums(S2):
M:= 2*max(S1):
S:= select(t -> t < M and isprime(t), {seq(seq(P1[i]-P1[j],j=i mod 2 + 1 .. i-2,2),i=1..101)} union {seq(seq(P2[i]-P2[j],j=i mod 2 + 1..i-2,2),i=1..101)} union {seq(P2[i],i=1..101,2)}):
sort(convert(S,list));

from sympy import isprime, nextprime, prime
def sp2(lst):
  ans = 0
  for i in range(0, len(lst), 2): ans += lst[i]*lst[i+1]
  return ans
def aupto(nn):
  alst, i = [], 1
  while True:
    consec2i = [prime(j+1) for j in range(2*i)]; sp = sp2(consec2i)
    if sp > nn: break
    while sp <= nn:
      if isprime(sp): alst.append(sp)
      consec2i = consec2i[1:] + [nextprime(consec2i[-1])]; sp = sp2(consec2i)
    i += 1
  return sorted(alst)
print(aupto(261983)) # Michael S. Branicky, Jan 08 2021

cp's OEIS Frontend

A340463 Primes p such that pq+rs+t*u is prime, where p,q,r,s,t,u are consecutive primes.

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A340465 Primes of the form prime(i)prime(i+1)+prime(i+2)prime(i+3)+...+prime(k-1)*prime(k).

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cp's OEIS Frontend

A340463 Primes p such that p*q+r*s+t*u is prime, where p,q,r,s,t,u are consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A340465 Primes of the form prime(i)*prime(i+1)+prime(i+2)*prime(i+3)+...+prime(k-1)*prime(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

A340463 Primes p such that pq+rs+t*u is prime, where p,q,r,s,t,u are consecutive primes.

A340465 Primes of the form prime(i)prime(i+1)+prime(i+2)prime(i+3)+...+prime(k-1)*prime(k).