A319823 Primes p such that min(d(p-1), d(p+1)) is larger than the corresponding values of all previous primes, where d(n) is the number of divisors of n (A000005).

2, 3, 5, 7, 17, 19, 41, 197, 199, 449, 701, 881, 3079, 4159, 18089, 40699, 51679, 90271, 388961, 403649, 446081, 906751, 1276001, 12227489, 37487449, 53308529, 59522849, 109245401, 285258401, 459712639, 1381951999, 2560742911, 2969200961, 8505402751

Offset: 1

Author

Amiram Eldar, Sep 28 2018

Keywords

• nonn

Comments

Problem 104 in Sierpinski's book is to prove that this sequence is infinite.

The corresponding values of min(d(p-1), d(p+1)) are 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 16, 18, 20, 28, 32, 36, 40, 48, 50, 56, 64, 80, 96, 128, 144, 160, 168, 192, 216, 256, 288, 320, 336, 384, ...

References

• W. Sierpinski, 250 Problems in Elementary Number Theory, New York: American Elsevier, 1970, problem #104, pp. 9, 58-59.

Crossrefs

Cf. A000005, A000040, A145339.

Programs

Mathematica

s={}; f[p_] := Min[DivisorSigma[0,p-1], DivisorSigma[0,p+1]]; p=2; fm=0; Do[f1 = f[p]; If[f1>fm, AppendTo[s,p]; fm=f1]; p=NextPrime[p], {k, 1, 100}]; s

PARI

f(p)=min(numdiv(p-1),numdiv(p+1));
fm=0;forprime(p=1, 1000, f1=f(p); if(f1>fm, print1(p,", "); fm=f1))