A353020 Every term is the absolute difference of a prime and a nonprime that is also the sum of their indices.

4, 7, 9, 14, 15, 16, 17, 25, 26, 27, 28, 29, 35, 43, 44, 45, 46, 47, 55, 56, 57, 58, 64, 71, 73, 74, 75, 76, 77, 78, 79, 83, 85, 86, 87, 96, 99, 107, 109, 110, 111, 112, 113, 117, 123, 125, 133, 139, 140, 141, 142, 143, 148, 151, 152, 153, 154, 155, 156, 157, 158

Offset: 1

Author

Tamas Sandor Nagy, Apr 17 2022

Keywords

• nonn

Comments

More than one pair of a prime and a nonprime may correspond to some of the terms.

Sequence A352707 is read by ascending antidiagonals from array T(n,k). Present sequence is those differences T(n,k) which are on their own antidiagonal number n + k.

.

Array T(n,k) = abs(prime(n)-nonprime(k))

n\k| 1 4 6 8 9 ...

-----------------------

2 | 1 2 4 6 7 ...

3 | 2 1 3 5 6 ...

5 | 4 1 1 3 4 ...

7 | 6 3 1 1 2 ...

11 | 10 7 5 3 2 ...

13 | 12 9 7 5 4 ...

17 | 16 13 11 9 8 ...

19 | 18 15 13 11 10 ...

23 | 22 19 17 15 14 ...

. | . . . . .

Examples

			7 is a term because the 5th prime 11 minus the 2nd nonprime 4 equals 7 and that is also 5 + 2.
9 is a term because the 2nd prime 3 minus the 7th nonprime 12 equals -9 whose absolute value 9 is also 2 + 7.
4 is a term because the 3rd prime 5 minus the 1st nonprime 1 equals 4 that is also 3 + 1. The absolute value of the 1st prime 2 minus the 3rd nonprime 6 also equals 4 that is 1 + 3, so this pair, too, makes 4 a term of this sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A000040, A018252, A352707 (table T).

Programs

Maple

M:= 1000:
P:= NULL: C:= NULL: np:= 0: nc:= 0:
for x from 1 while np < M do
  if isprime(x) then
    np:= np+1; P:= P,x
  elif nc < M then
    nc:= nc+1; C:= C,x
  fi
od:
filter:= proc(x) local i;
  ormap(i -> abs(P[i] - C[x-i]) = x, [$1..x-1]);
end proc:
select(filter, [$1..M]); # Robert Israel, Jul 29 2025

Formula

Sums x + y for which x + y = abs(prime(x) - nonprime(y)), for some x and y.

Extensions

More terms from Hugo Pfoertner, Apr 17 2022