A393561 Smallest prime p such that p+1 starts a run of exactly n consecutive numbers having exactly the same number of divisors.

13, 229, 241, 13781, 298693, 80749767323

Offset

2,1

Links

Table of n, a(n) for n=2..7.

Example

The 2 consecutive integers following the prime 13, namely [14, 15], all have the same number of divisors, namely 4; 16 has 5 divisors and no smaller integer following a prime displays this property, hence a(2) = 13.
The 3 consecutive integers following the prime 229, namely [230, 231, 232], all have the same number of divisors, namely 8; 233 has 2 divisors and no smaller integer following a prime displays this property, hence a(3) = 229.
a(5) = A119730(1).
a(6) = A119740(1).

Mathematica

a[n_]:=Module[{p=2}, While[Length[Union[DivisorSigma[0, Range[p+1, p+n]]]]>1, p=NextPrime[p]]; p]; Array[a, 5, 2] (* James C. McMahon, Mar 26 2026 *)

Prog

(PARI) card(k)=my(n=numdiv(k+1), i=1); while(numdiv(k+i+1)==n, i++); i
a(n)=forprime(p=2, +oo, if(card(p)==n, return(p)))

Crossrefs

Cf. A119711, A119728, A119730, A119740.

Cf. A005237, A006601, A006558, A049051, A049052, A049053.

Cf. A067889, A391384.

Sequence in context: A160196 A218588 A158518 * A223548 A201706 A176722

Adjacent sequences: A393558 A393559 A393560 * A393562 A393563 A393564

Keyword

nonn,hard,more

Author

Jean-Marc Rebert, Feb 21 2026

Status

approved