cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A372690 Numbers k such that k and k+1 are both numbers whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 2, 5, 6, 7, 10, 13, 14, 21, 22, 23, 26, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 46, 53, 54, 55, 56, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 87, 88, 93, 94, 101, 102, 103, 104, 105, 106, 109, 110, 113, 114, 118, 119, 122, 127, 128, 129, 130, 133
Offset: 1

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Author

Amiram Eldar, May 10 2024

Keywords

Comments

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 6, 44, 449, 4450, 44462, 444471, 4444647, 44446255, 444461038, 4444607360, ... . Apparently, the asymptotic density of this sequence exists and equals 0.44446... .

Examples

			1 is a term since the number of divisors of 1 is 1 = 2^0, and the number of divisors of 1 + 1 = 2 is 2 = 2^1.
54 is a term since the number of divisors of 54 is 8 = 2^3, and the number of divisors of 54 + 1 = 55 is 4 = 2^2.
		

Crossrefs

Subsequence of A007674 and A036537.
A372691 is a subsequence.

Programs

  • Mathematica
    pow2Q[n_] := n == 2^IntegerExponent[n, 2]; q[n_] := q[n] = pow2Q[DivisorSigma[0, n]]; Select[Range[150], q[#] && q[# + 1] &]
  • PARI
    is(n) = {my(d = numdiv(n)); d >> valuation(d, 2) == 1;}
    lista(kmax) = {my(is1 = is(1), is2); for(k = 2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}