A396234 Smaller term of pairs of consecutive primes of the form p, p+6 where both primes eventually reach a value of 1 under iteration of the sums of the squares of their digits and the first iteration of both terms are also prime.

2333, 3323, 12143, 21143, 21341, 23893, 30097, 32303, 33023, 123401, 213223, 221233, 230303, 238093, 245783, 254873, 278543, 280933, 289033, 300233, 302411, 302983, 304211, 312401, 323003, 329083, 336397, 340121, 342101, 344963, 396443, 398023, 401231, 413201, 475283, 478253

Offset

1,1

Comments

Equivalently, consecutive happy sexy primes where both primes produce a second prime.

Conjecture: this sequence has an infinite number of terms.

All terms of this sequence go through a minimum of three iterations before going to the value of 1.

References

Liz Strachan, Numbers Are Forever, Constable, London, 2014, p. 92.

Links

Table of n, a(n) for n=1..36.

Example

2333 is a term because it is the smaller of a consecutive sexy prime pair (2333, 2339) where 2333 is a happy prime because 2333 --> 2^2 + 3^2 + 3^2 + 3^2 = 4 + 9 + 9 + 9 = 31 --> 3^2 + 1^2 = 10 --> 1^2 = 1 --> 1^2 = 1, and the first iteration produces the prime number 31. The larger term, p = 2339 is also a happy prime because 2339 --> 2^2 + 3^2 + 3^2 + 9^2 = 103 --> 1^2 + 0^2 + 3^2 = 10 --> 1^2 = 1 --> 1^2 = 1, and the first iteration produces the prime number 103.
Both the smaller and larger term of consecutive sexy prime pairs must be happy primes and both must produce another prime on their first iteration for the smaller term to be a term of this sequence.
The happy sexy prime p=3301 is not a term because the next term (p+6) is not a happy prime.
The smallest terms of prime triples are not terms because the sexy prime pairs are not consecutive primes.

Mathematica

s[n_] := Total[IntegerDigits[n]^2]; happyQ[n_] := NestWhile[s, n, UnsameQ, All] == 1; seq[lim_] := Module[{ps = Prime[Range[lim]], ind}, ind = Position[ps, _?(PrimeQ[s[#]] && happyQ[#] &), 1] // Flatten; ps[[Intersection[-1 + ind[[1 + Position[Differences[ind], 1] // Flatten]], Position[Differences[ps], 6] // Flatten]]]]; seq[50000] (* Amiram Eldar, May 22 2026 *)

Crossrefs

Cf. A035497, A007770, A046519, A023201, A078854.

Sequence in context: A203344 A381976 A178552 * A166581 A385037 A237387

Adjacent sequences: A396231 A396232 A396233 * A396235 A396236 A396237

Keyword

nonn,base

Author

Harry E. Neel, May 19 2026

Status

approved