Smallest square-free k-gonal number with n prime factors

[Andrew Plewe]

sums of which sequences form the various figurate numbers. My hope is this
	-Andrew Plewe-
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Dear D. Sloane,

You're right.  The flaws in this, below, illustrate your point well.

Sorted (by hand) by molecular weight

HMDB02106       1.00783 Hydrogen ion
HMDB01362       1.00783 Hydrogen
HMDB02386       11.00930        Boron
HMDB02714       16.03130        Methane
HMDB01039       17.00274        Hydroxide
HMDB00051       17.02655        Ammonia
HMDB02111       18.01060        Water
HMDB00662       18.99840        Fluorine
HMDB00588       22.98980        Sodium
HMDB00547       23.98504        Magnesium
HMDB02084       26.00307        Cyanide
HMDB02175       27.97690        Silicon
HMDB01361       27.99490        Carbon
HMDB01371       28.00610        Nitrogen
HMDB03378       29.99799        Nitric oxide
HMDB01426       30.01056        Formaldehyde
HMDB01315       30.97380        Phosphorus
HMDB00164       31.04220        Methylamine
HMDB02983       31.04220        Methylamine Hydrochloride
HMDB00598       31.97210        Sulfur
HMDB01377       31.98980        Oxygen
HMDB02168       31.98983        Superoxide
HMDB01875       32.02621        Methanol
HMDB03338       33.02150        Hydroxylamine
HMDB00983       33.98770        Hydrogen Sulfide
HMDB00492       34.96885        Chlorine
HMDB02162       35.97670        Hydrochloric acid
HMDB02306       35.97670        Hydrochloric acid
HMDB00586       38.96371        Potassium
HMDB00464       39.96260        Calcium
HMDB02078       43.00580        Cyanate
HMDB01967       43.98980        Carbon Dioxide
HMDB00990       44.02621        Acetaldehyde
HMDB01536       45.02146        Formamide
HMDB00087       45.05785        Dimethylamine
HMDB00142       46.00548        Formic acid
HMDB00108       46.04186        Ethanol
HMDB01382       47.00070        Nitrite
HMDB03227       48.00337        Methanethiol
HMDB02503       50.94400        Vanadium
HMDB01050       50.96377        Hypochlorite
HMDB00599       51.94050        Chromium
HMDB01333       54.93800        Manganese
HMDB00692       55.93490        Iron
HMDB02457       57.93530        Nickel
HMDB01659       58.04186        Acetone
HMDB03366       58.04190        Propanal
HMDB00608       58.93320        Cobalt
HMDB01842       59.04835        Guanidine
HMDB01869       59.03710        Acetamide
HMDB03656       59.03710        Acetaldehyde oxime
HMDB00906       59.07350        Trimethylamine
HMDB03551       60.00855        Carbamate
HMDB03344       60.02110        Glycolaldehyde
HMDB00042       60.02113        Acetic acid
HMDB00294       60.03236        Urea
HMDB00820       60.05751        Propyl alcohol
HMDB00863       60.05751        Isopropyl alcohol
HMDB00595       60.99257        Hydrogen Carbonate
HMDB00149       61.05276        Ethanolamine
HMDB02179       61.98782        Peroxynitrite
HMDB02878       61.98782        Nitrate
HMDB03538       62.00040        Carbonic acid
HMDB02303       62.01902        Dimethylsulfide
HMDB01887       62.03678        Ethylene glycol
HMDB00657       62.92960        Copper
HMDB01853       62.99560        Nitrate
HMDB01303       63.92910        Zinc
HMDB03276       65.95979        Hydrogen sulfide
HMDB02077       67.96650        Chlorite
HMDB01525       68.03745        Imidazole
HMDB04101       71.06092        beta-Aminopropionitrile
HMDB01167       72.02113        Pyruvaldehyde
HMDB03543       72.05750        Butanal
HMDB00474       72.05751        Butanone
HMDB01106       73.05276        3-Aminopropionaldehyde
HMDB02134       73.05276        Aminoacetone
HMDB01522       73.06400        Methylguanidine
HMDB02501       73.92120        Germanium
HMDB00119       74.00039        Glyoxylic acid
HMDB00237       74.03678        Propionic acid
HMDB03052       74.03678        Lactaldehyde
HMDB03453       74.03680        3-Hydroxypropanal
HMDB04327       74.07320        1-Butanol
HMDB00002       74.08440        1,3-Diaminopropane
HMDB00123       75.03203        Glycine
HMDB00925       75.06841        Trimethylamine oxide
HMDB00115       76.01604        Glycolic acid
HMDB01881       76.05243        1,2-Propanediol
HMDB02808       76.05243        1,3-Propanediol
HMDB02991       77.02992        Cysteamine
HMDB02151       78.01390        Dimethyl sulfoxide
HMDB01505       78.04695        Benzene
HMDB02500       78.91834        Bromine
HMDB00926       79.04220        Pyridine
HMDB01349       79.91652        Selenium
HMDB00240       79.95681        Sulfite
HMDB01033       80.96464        Hydrogen Sulfite
HMDB03008       81.97246        Bisulfite
HMDB03929       83.04830        5-Aminoimidazole
HMDB02036       83.96140        Chloric acid
HMDB04363       84.03240        Imidazolone
HMDB02039       85.05276        2-Pyrrolidinone
HMDB00549       86.03678        4-Deoxytetronic acid
HMDB02523       86.03678        Oxolan-3-one
HMDB03407       86.03680        Diacetyl
HMDB01080       87.06841        4-Aminobutyraldehyde
HMDB03642       87.90560        Strontium
HMDB01880       87.97966        Hydrogen Oxalate
HMDB00243       88.01604        Pyruvic acid
HMDB00039       88.05243        Butyric acid
HMDB01873       88.05243        Isobutyric acid
HMDB03243       88.05243        Acetoin
HMDB01414       88.10005        Putrescine
HMDB02435       89.01129        (hydroxyimino)- Acetic acid
HMDB00056       89.04768        b-Alanine
HMDB00161       89.04768        L-Alanine
HMDB00271       89.04768        Sarcosine
HMDB02329       89.99531        Oxalic acid
HMDB00190       90.03169        L-Lactic acid
HMDB00700       90.03169        Hydroxypropionic acid
HMDB01051       90.03169        Glyceraldehyde
HMDB01311       90.03169        D-Lactic acid
HMDB01882       90.03169        Dihydroxyacetone
HMDB03156       90.06808        2,3-Butanediol
HMDB03692       90.06810        (S,S)-Butane-2,3-diol
HMDB00131       92.04734        Glycerol
HMDB03012       93.05785        Aniline
HMDB04983       94.00890        Dimethyl sulfone
HMDB00228       94.04186        Phenol
HMDB01429       94.95510        Phosphate
HMDB00979       96.95955        Hydrogen Sulfate
HMDB00973       96.96907        Hydrogen phosphate
HMDB02105       96.96907        Dihydrogen Phosphate
HMDB01302       97.90540        Molybdenum
HMDB02934       97.96738        Sulfuric acid
HMDB01448       97.96740        Sulfate