A284783 Numbers k such that k and k + 5040 have the same number of divisors.

11, 19, 22, 37, 38, 39, 41, 46, 47, 51, 55, 57, 58, 59, 61, 62, 65, 67, 68, 73, 74, 76, 78, 79, 87, 88, 91, 92, 99, 102, 104, 107, 113, 114, 115, 116, 118, 123, 124, 125, 127, 129, 131, 132, 133, 136, 138, 139, 142, 143, 146, 148, 149, 153, 155, 156, 157, 159

Offset: 1

Author

Amiram Eldar, Apr 02 2017

Keywords

• nonn

Comments

Claudia Spiro proved in 1981 that this sequence is infinite. Her work helped D. R. Heath-Brown to prove in 1984 that A005237 is also infinite.

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, p. 111.
• Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996, p. 332.
• Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, p. 69.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Claudia A. Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Multiplicative or Additive Function of n Divides a Polynomial Function of n, Ph. D. Thesis, University of Illinois, Urbana-Champaign, 1981.

Crossrefs

Cf. A000005, A005237.

Programs

Mathematica

Select[Range[160], DivisorSigma[0, #] == DivisorSigma[0, # + 5040] &]

PARI

for(n=1, 200, if(numdiv(n) == numdiv(n + 5040), print1(n,", "))) \\ Indranil Ghosh, Apr 04 2017

Python

from sympy.ntheory import divisor_count as D
print([n for n in range(1, 201) if D(n) == D(n + 5040)]) # Indranil Ghosh, Apr 04 2017