A007921 Numbers that are not the difference of two primes.

7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, 103, 109, 113, 115, 117, 119, 121, 123, 127, 131, 133, 139, 141, 143, 145, 151, 153, 157, 159, 163, 167, 169, 173, 175, 181, 183, 185, 187, 193

Offset: 1

Author

R. Muller

Keywords

• nonn
• easy
• nice

Comments

Conjecturally, odd numbers k such that k+2 is composite.

Is this the same as A068780(2n-1) - 1? - J. Stauduhar, Aug 23 2012

A092953(a(n)) = 0. - Reinhard Zumkeller, Nov 10 2012

It seems that the sequence contains the squares of all primes except for 2 and 3. - Ivan N. Ianakiev, Aug 29 2013 [It does: For every prime p > 3, note that p^2 == 1 (mod 3), so p^2 cannot be q - r where q and r are primes. (If it were, then since p^2 is odd, q and r could not both be odd primes; r would have to be the even prime, 2, which would mean that p^2 = q - 2, so q = p^2 + 2 == 0 (mod 3), i.e., 3 would divide q, so q would not be prime -- a contradiction.) - Jon E. Schoenfield, May 03 2024]

Integers d such that A123556(d) = 1, that is, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has only one element. For each such d, the unique element of all the first largest APs with 1 element is A342309(d) = 2. - Bernard Schott, Jan 08 2023

If it exists, the least even term is > 10^12 (see 1st comment in A020483). - Bernard Schott, Jan 09 2023

References

• F. Smarandache, Properties of Numbers, 1972. (See Smarandache odd sieve.)

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
• C. Dumitrescu and V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
• F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 94.
• Wikipedia, Primes in arithmetic progression.
• Index entries for sequences related to primes in arithmetic progressions.
• Index entries for primes, gaps between.

Crossrefs

Cf. A048859.

Complement of A030173. Cf. A001223.

Cf. also A005408, A010051.

Cf. A123556, A342309.

Largest AP of prime numbers with k elements: this sequence (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A206042 (k=8), A206043 (k=9), A206044 (k=10), A206045 (k=11).

Programs

Haskell

a007921 n = a007921_list !! (n-1)
a007921_list = filter ((== 0) . a010051' . (+ 2)) [1, 3 ..]
-- Reinhard Zumkeller, Jul 03 2015

Maple

filter :=  d -> irem(d, 2) <> 0 and not isprime(2+d) : select(filter, [`$`(1 .. 200)]); # Bernard Schott, Jan 08 2023

Mathematica

Lim=200;nn=10;seq:=Complement[Range[Lim],Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]];Until[AllTrue[seq,OddQ],nn++];seq (* James C. McMahon, May 04 2024 *)

PARI

is(n)=n%2 && !isprime(n+2) \\ On Polignac's conjecture; Charles R Greathouse IV, Jun 28 2013

Python

from sympy import isprime
print([n for n in range(1, 200) if n%2 and not isprime(n + 2)]) # Indranil Ghosh, Jun 15 2017, after Charles R Greathouse IV