A353552 Primes p such that Sum_{k=PreviousPrime(p)..p} d(k) = Sum_{k=p..NextPrime(p)} d(k), where d(k) is the number of divisors function A000005.

1871, 2141, 2677, 2777, 2903, 2963, 3673, 4969, 5107, 5417, 6323, 7487, 10459, 11173, 11497, 11689, 14519, 18047, 18077, 19081, 19379, 20357, 20533, 20611, 21577, 22619, 25621, 32621, 34543, 35531, 36821, 39089, 39503, 40111, 40771, 44263, 44647, 44917, 51551, 52181

Offset

1,1

Comments

It doesn't matter if terms or the neighboring primes are included in the sums or excluded as long as the symmetry is taken into account.

If A133760(n) = A133760(n-1), then A000040(n) is a term.

Links

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

Example

a(2) = 2141 (a prime); previous prime is 2137; next prime is 2143.
  The numbers of divisors between are:
      + d(2138) =  4         + d(2142) = 24
      + d(2139) =  8
      + d(2140) = 12
  ------------------------------------------
            sum = 24               sum = 24    Sums are equal. Thus 2141 is a term.

Mathematica

Select[Prime[Range[5000]], Total[DivisorSigma[0, Range[NextPrime[#, -1], #]]] == Total[DivisorSigma[0, Range[#, NextPrime[#]]]] &] (* Amiram Eldar, May 11 2022 *)

Prog

(Python)
from sympy import sieve as A000040, divisor_count as A000005
def sotsb(a, b): return sum(A000005(n) for n in range(a+1, b))
    # sotsb stands for the "sum of taus strictly between (a and b)"
print([A000040[n] for n in range(2, 5400) if sotsb(A000040[n-1], A000040[n]) == sotsb(A000040[n], A000040[n+1])])
(PARI) isok(p) = isprime(p) && sum(k=precprime(p-1), p, numdiv(k)) == sum(k=p, nextprime(p+1), numdiv(k)); \\ Michel Marcus, May 11 2022

Crossrefs

Cf. A000005, A000040, A133760.

Cf. A353553 (sum of three sets are equal), A353554 (sum of four sets are equal).

Sequence in context: A306877 A073495 A073489 * A232040 A054817 A270244

Adjacent sequences: A353549 A353550 A353551 * A353553 A353554 A353555

Keyword

nonn

Author

Karl-Heinz Hofmann, May 10 2022

Status

approved