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Search: id:A117065
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| A117065 |
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Primes that are not the sum of 3 pentagonal numbers. |
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+0
7
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| 19, 31, 43, 67, 89, 101, 113, 131, 229, 241, 277, 359, 383, 491, 523, 619, 631, 643, 701, 761, 1321, 1381, 1621, 2221, 2861
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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5 is the only prime pentagonal number; every greater pentagonal number A000326(n) = n(3n-1)/2 is either divisible by n/2 or (3n-1)/2. Every number is the sum of 5 pentagonal numbers, hence every prime is the sum of 5 pentagonal numbers. There are an infinite number of primes which are the sum of two pentagonal numbers, the subset of primes which are the sum of two pentagonal numbers in exactly two different ways begins {211, 853, 1259, 1427, 1571, 2297, 2351}.
The sum may include the pentagonal number 0. Hence this sequence does not have any primes that are the sum of two positive pentagonal numbers. The sequence is probably finite. There are no other primes < 59900. - T. D. Noe (noe(AT)sspectra.com), Apr 19 2006
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REFERENCES
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R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.
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FORMULA
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A000040 INTERSECT A003679.
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MATHEMATICA
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nn=201; pen=Table[n(3n-1)/2, {n, 0, nn-1}]; ps=Prime[Range[PrimePi[pen[[ -1]]]]]; Do[n=pen[[i]]+pen[[j]]+pen[[k]]; If[n<=pen[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps, _?(#==n&)]]], {i, nn}, {j, i, nn}, {k, j, nn}]; ps - T. D. Noe (noe(AT)sspectra.com), Apr 19 2006
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CROSSREFS
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Cf. A000040, A000326, A003679, A064826.
Sequence in context: A040068 A096787 A104006 this_sequence A006035 A104485 A141184
Adjacent sequences: A117062 A117063 A117064 this_sequence A117066 A117067 A117068
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 17 2006
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EXTENSIONS
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More terms from T. D. Noe (noe(AT)sspectra.com), Apr 19 2006
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