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.

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A342188 Numbers k such that both k and k+1 are not exponentially squarefree numbers.

Original entry on oeis.org

80, 624, 2511, 5264, 6399, 7695, 7856, 10287, 13040, 14640, 15471, 15632, 18063, 19375, 20624, 20816, 23247, 23408, 25839, 27135, 28560, 28592, 31023, 31184, 33615, 35072, 36015, 36368, 38799, 38960, 39375, 40816, 41391, 44144, 46250, 46575, 46736, 49167, 51920
Offset: 1

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Author

Amiram Eldar, Mar 04 2021

Keywords

Comments

The numbers of terms not exceeding 10^k for k = 2, 3, ..., are 1, 2, 7, 72, 719, 7226, 72238, 722565, 7225651, ... Apparently this sequence has an asymptotic density 0.00007225...

Examples

			80 is a term since 80 = 2^4 * 5 and 81 = 3^4 both have a nonsquarefree exponent in their prime factorization.

Links

Crossrefs

Cf. A130897, A209061.
Similar sequences: A068140, A068781, A342187, A342189.

Programs

  • Mathematica
    expSqFQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], SquareFreeQ]; Select[Range[5*10^4], !expSqFQ[#] && !expSqFQ[# + 1] &]

A342189 Numbers k such that both k and k+1 are not exponentially 2^n-numbers.

Original entry on oeis.org

135, 296, 343, 351, 375, 512, 728, 999, 1160, 1215, 1375, 1431, 1592, 1624, 2079, 2240, 2295, 2375, 2456, 2624, 2727, 2888, 2943, 3104, 3159, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4616, 4671, 4832, 4887, 4913, 5048, 5144, 5319, 5480, 5535, 5696, 5831, 6183
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2021

Keywords

Comments

The numbers of terms not exceeding 10^k for k = 3, 4, ..., are 8, 76, 775, 7776, 77845, 778303, 7783285, 77832769, ... Apparently this sequence has an asymptotic density 0.0077832...

Examples

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17 both have an exponent in their prime factorization which is not a power of 2.

Links

Crossrefs

cf. A138302.
Similar sequences: A068140, A068781, A342187, A342188.

Programs

  • Mathematica
    exp2nQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], # == 2^IntegerExponent[#, 2] &]; Select[Range[10^4], ! exp2nQ[#] && ! exp2nQ[# + 1] &]
Showing 1-2 of 2 results.

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