A353554 Initial term of a set of consecutive primes {p1, p2, p3, p4, p5} such that Sum_{k=p1..p2} d(k) = Sum_{k=p2..p3} d(k) = Sum_{k=p3..p4} d(k) = Sum_{k=p4..p5} d(k), where d(k) is the number of divisors function A000005.

238820129, 2219617987, 3089392231, 4071864457, 4633981813, 4710405229, 4909907729, 5912801617, 5979418121, 6639163651, 7088972563, 7929458543, 8235321617, 8540714341, 8832705757, 10029168811, 10421237143, 10680661877, 11423715839, 12495445649, 12956275471, 13250783867

Offset

1,1

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..579

Example

For n=2, the primes are a(2) = p1 = 2219617987, p2 = 2219617991, p3 = 2219617993, p4 = 2219617997 and p5 = 2219617999; the composites k between them have the following numbers of divisors:
.
       p1 < k < p2   |   p2 < k < p3   |   p3 < k < p4   |   p4 < k < p5
     ----------------+-----------------+-----------------+----------------
          k     d(k) |      k     d(k) |      k     d(k) |      k     d(k)
     ---------- ---- + ---------- ---- + ---------- ---- + ---------- ----
     2219617988  24  | 2219617992  64  | 2219617994   8  | 2219617998  64
     2219617989  24  |                 | 2219617995  32  |
     2219617990  16  |                 | 2219617996  24  |
     ----------------+-----------------+-----------------+----------------
Sums:            64  |             64  |             64  |             64
     ----------------+-----------------+-----------------+----------------
.
Sums are equal. Thus 2219617987 is a term.

Mathematica

seq[len_, pmax_] := Module[{s = {}, c = 0, p1 = 2, p2 = 3, sums = Table[0, {4}]}, While[c < len && p1 < pmax, sums = Join[Rest[sums], {Sum[DivisorSigma[0, k], {k, p1 + 1, p2 - 1}] + 4}]; If[SameQ @@ sums, c++; AppendTo[s, NextPrime[p1, -3]]]; p1 = p2; p2 = NextPrime[p2]]; s]; seq[2, 3*10^9] (* Amiram Eldar, May 20 2022 *)

Crossrefs

Cf. A000005, A000040, A133760, A354444.

Cf. A353552 (sum of two sets are equal), A353553 (sum of three sets are equal).

Sequence in context: A203885 A335591 A234058 * A064869 A016824 A016860

Adjacent sequences: A353551 A353552 A353553 * A353555 A353556 A353557

Keyword

nonn

Author

Karl-Heinz Hofmann, May 18 2022

Extensions

More terms from Hugo Pfoertner, May 18 2022

Status

approved