A357167 Numbers k such that k and k+2 are both odd numbers whose prime factors are all prime-indexed primes.

1, 3, 9, 15, 25, 31, 81, 83, 121, 123, 125, 153, 155, 177, 241, 275, 277, 295, 367, 459, 545, 561, 603, 615, 633, 737, 773, 991, 1003, 1023, 1087, 1199, 1201, 1203, 1215, 1375, 1383, 1395, 1409, 1411, 1413, 1445, 1681, 1845, 1851, 2025, 2075, 2099, 2125, 2319, 2417

Offset

1,2

Comments

Numbers k such that both k and k+2 are in A076610.

Since 2 is not a prime-indexed prime, all the terms of A076610 are odd, so there are no 2 consecutive integers in A076610.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

3 is a term since 3 = prime(2) and 5 = prime(3) are both prime-indexed primes.
15 is a term since 15 = 3 * 5, 15 + 2 = 17, and 3 = prime(2), 5 = prime (3) and 17 = prime(7) are all prime-indexed primes.

Mathematica

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], PrimeQ[PrimePi[#]] &]; q[1] = True; Select[Range[1, 2500, 2], q[#] && q[# + 2] &]

Prog

(PARI) isokf(k) = my(f = factor(k)[, 1]); sum(i=1, #f, isprime(primepi(f[i]))) == #f; \\ A076610
isok(k) = (k % 2) && isokf(k) && isokf(k+2); \\ Michel Marcus, Sep 16 2022

Crossrefs

Cf. A006450, A076610.

Subsequences: A357168, A357169.

Sequence in context: A058972 A026222 A192720 * A099989 A209980 A085046

Adjacent sequences: A357164 A357165 A357166 * A357168 A357169 A357170

Keyword

nonn

Author

Amiram Eldar, Sep 16 2022

Status

approved