A353553 Initial term of a set of consecutive primes {p1, p2, p3, p4} such that Sum_{k=p1..p2} d(k) = Sum_{k=p2..p3} d(k) = Sum_{k=p3..p4} d(k), where d(k) is the number of divisors function A000005.

105373, 147073, 432031, 663959, 699367, 1144511, 1856287, 2126611, 2235661, 2271383, 2661931, 2699087, 2887763, 3798019, 5093917, 5722261, 5802521, 6158093, 7504307, 7512749, 7605791, 8279203, 9166301, 9585881, 11195881, 11267293, 11644901, 11925019, 11978419, 12359857

Offset

1,1

Comments

It doesn't matter if the primes are included in the sums or excluded as long as the symmetry is taken into account (d(prime) is always 2).

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Example

For n=1, the primes are a(1) = p1 = 105373, p2 = 105379, p3 = 105389, and p4 = 105397; the composites between them have the following numbers of divisors:
  -----------------------------------------------------------------
  Between p1 and p2       Between p2 and p3       Between p3 and p4
  -----------------------------------------------------------------
                             d(105380) = 24
                           + d(105381) = 10
                           + d(105382) =  4          d(105390) = 24
                           + d(105383) =  4        + d(105391) = 12
     d(105374) = 16        + d(105384) = 16        + d(105392) = 20
   + d(105375) = 16        + d(105385) =  8        + d(105393) = 12
   + d(105376) = 24        + d(105386) = 16        + d(105394) =  4
   + d(105377) =  4        + d(105387) =  4        + d(105395) =  8
   + d(105378) = 32        + d(105388) =  6        + d(105396) = 12
  -----------------------------------------------------------------
           sum = 92                sum = 92                sum = 92
  -----------------------------------------------------------------
Sums are equal. Thus 105373 is a term.

Mathematica

f[p_, i_, j_] := Sum[DivisorSigma[0, k], {k, p[[i]], p[[j]]}]; Select[Partition[Prime[Range[3*10^5]], 4, 1], f[#, 1, 2] == f[#, 2, 3] == f[#, 3, 4] &][[;; , 1]] (* Amiram Eldar, May 15 2022 *)

Prog

(Python)
from sympy import sieve as p, divisor_count as d
def sum_of_taus(a, b): return sum(d(ab) for ab in range(a+1, b))
print([p[k] for k in range (1, 100000) if sum_of_taus(p[k], p[k+1]) == sum_of_taus(p[k+1], p[k+2]) == sum_of_taus(p[k+2], p[k+3])])
(PARI) a353553(limit) = {my(p1=3, p2=5, p3=7, ns1=sum(k=p1+1, p2-1, numdiv(k)), ns2=sum(k=p2+1, p3-1, numdiv(k)), ns3); forprime(p4=11, limit, ns3=sum(k=p3+1, p4-1, numdiv(k)); if(ns3==ns2 && ns3==ns1, print1(p1, ", ")); p1=p2; p2=p3; p3=p4; ns1=ns2; ns2=ns3)};
a353553(13000000) \\ Hugo Pfoertner, May 17 2022

Crossrefs

Cf. A000005, A000040, A133760.

Cf. A353552 (sum of two sets are equal), A353554 (sum of four sets are equal).

Sequence in context: A203822 A013889 A380232 * A025308 A025290 A242479

Adjacent sequences: A353550 A353551 A353552 * A353554 A353555 A353556

Keyword

nonn

Author

Karl-Heinz Hofmann, May 15 2022

Status

approved