A342093 Primes that can be represented as (1/2)*Sum_{i=0..m} binomial(m,i)*prime(i+k) for some k and m >= 2.

47, 61, 73, 103, 137, 157, 167, 179, 223, 257, 263, 337, 347, 383, 467, 563, 613, 719, 733, 757, 769, 877, 887, 1021, 1097, 1223, 1297, 1327, 1367, 1453, 1481, 1571, 1613, 1621, 1759, 1811, 1987, 1997, 2003, 2027, 2039, 2129, 2251, 2473, 2477, 2539, 2593, 2633, 2767, 2879, 3001, 3037, 3083, 3119

Offset

1,1

Comments

Each prime is just listed once, though it may arise in more than one way.

Links

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

Example

a(3) = 73 is a term because it is prime and Sum_{i=0..2} binomial(2,i)*prime(11+i) = 31+2*37+41 = 2*73.
a(10) = 257 arises in two ways:
  2*257 = Sum_{i=0..3} binomial(3,i)*prime(17+i) = 59+3*61+3*67+71
and Sum_{i=0..4} binomial(4,i)*prime(9+i) = 23+4*29+6*31+4*37+41.

Maple

N:= 10^4: # for terms <= N
S:= {}:
for m from 2 do
  for k from 2 do
     v:= add(binomial(m, i)*ithprime(i+k), i=0..m)/2;
     if v > N then break fi;
     if isprime(v) then
      S:= S union {v}; count:= count+1;
     fi;
  od;
  if k = 2 then break fi
od:
sort(convert(S, list));

Crossrefs

Contains A338273.

Sequence in context: A139909 A368077 A243430 * A126980 A354442 A354461

Adjacent sequences: A342090 A342091 A342092 * A342094 A342095 A342096

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Feb 28 2021

Status

approved