files / problems / ulam-sequence / compute / spectral_verify.json

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{
  "N": 20000,
  "a_N": 268553,
  "known_exceptions": [
    2,
    3,
    47,
    69
  ],
  "steinerberger_2015": {
    "alpha": 2.5714474995,
    "N": 20000,
    "a_N": 268553,
    "cos_negative": 19996,
    "cos_positive": 4,
    "cos_tiny": 0,
    "positive_or_tiny": [
      {
        "a": 2,
        "cos": 0.41733070262870486
      },
      {
        "a": 3,
        "cos": 0.13918577174048308
      },
      {
        "a": 47,
        "cos": 0.09314945116148635
      },
      {
        "a": 69,
        "cos": 0.07005004608158771
      }
    ],
    "exceptions_match_published": true,
    "mean_cos": -0.7934496466528408
  },
  "mccranie_lo": {
    "alpha": 2.57144749846,
    "N": 20000,
    "a_N": 268553,
    "cos_negative": 19996,
    "cos_positive": 4,
    "cos_tiny": 0,
    "positive_or_tiny": [
      {
        "a": 2,
        "cos": 0.41733070073849493
      },
      {
        "a": 3,
        "cos": 0.13918577483011416
      },
      {
        "a": 47,
        "cos": 0.09314949982895969
      },
      {
        "a": 69,
        "cos": 0.07005011766532054
      }
    ],
    "exceptions_match_published": true,
    "mean_cos": -0.7934372742188972
  },
  "mccranie_hi": {
    "alpha": 2.5714474985,
    "N": 20000,
    "a_N": 268553,
    "cos_negative": 19996,
    "cos_positive": 4,
    "cos_tiny": 0,
    "positive_or_tiny": [
      {
        "a": 2,
        "cos": 0.41733070081119533
      },
      {
        "a": 3,
        "cos": 0.13918577471128218
      },
      {
        "a": 47,
        "cos": 0.09314949795713706
      },
      {
        "a": 69,
        "cos": 0.07005011491209678
      }
    ],
    "exceptions_match_published": true,
    "mean_cos": -0.7934377504584651
  },
  "mccranie_mid": {
    "alpha": 2.57144749848,
    "N": 20000,
    "a_N": 268553,
    "cos_negative": 19996,
    "cos_positive": 4,
    "cos_tiny": 0,
    "positive_or_tiny": [
      {
        "a": 2,
        "cos": 0.41733070077484513
      },
      {
        "a": 3,
        "cos": 0.13918577477069818
      },
      {
        "a": 47,
        "cos": 0.09314949889305546
      },
      {
        "a": 69,
        "cos": 0.07005011628869448
      }
    ],
    "exceptions_match_published": true,
    "mean_cos": -0.7934375123426204
  },
  "packet_steinerberger": [
    {
      "ell": 0,
      "mean_cos": 1.0
    },
    {
      "ell": 1,
      "mean_cos": -0.7934496466528408
    },
    {
      "ell": 2,
      "mean_cos": 0.2868473958738794
    },
    {
      "ell": 3,
      "mean_cos": 0.2538408422630102
    },
    {
      "ell": 4,
      "mean_cos": -0.5747248086210118
    },
    {
      "ell": 5,
      "mean_cos": 0.5756075230617902
    },
    {
      "ell": 6,
      "mean_cos": -0.3418187843711579
    },
    {
      "ell": 7,
      "mean_cos": 0.060083393962196155
    },
    {
      "ell": 8,
      "mean_cos": 0.11213426499059276
    }
  ],
  "packet_mccranie_mid": [
    {
      "ell": 0,
      "mean_cos": 1.0
    },
    {
      "ell": 1,
      "mean_cos": -0.7934375123426204
    },
    {
      "ell": 2,
      "mean_cos": 0.2868071796854297
    },
    {
      "ell": 3,
      "mean_cos": 0.25390763478995454
    },
    {
      "ell": 4,
      "mean_cos": -0.5748053250116508
    },
    {
      "ell": 5,
      "mean_cos": 0.5756918154918569
    },
    {
      "ell": 6,
      "mean_cos": -0.3419071840492687
    },
    {
      "ell": 7,
      "mean_cos": 0.06017889966476571
    },
    {
      "ell": 8,
      "mean_cos": 0.11204123170092464
    }
  ],
  "note": "Finite cosine sign pattern on an independently generated prefix. Does not prove a limiting measure, nor that the density exists."
}