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Search: id:A132399
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| A132399 |
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Number of ordered ways of writing n = i + j, where i is 0 or a prime and j is a triangular number (A000217) >= 0. |
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+0
5
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| 1, 1, 1, 3, 1, 2, 3, 1, 3, 1, 2, 2, 2, 3, 2, 2, 1, 4, 2, 2, 3, 2, 2, 4, 2, 1, 3, 1, 3, 3, 2, 2, 4, 2, 3, 2, 1, 2, 4, 3, 2, 4, 1, 3, 4, 2, 2, 6, 2, 2, 3, 2, 3, 4, 1, 2, 3, 3, 4, 4, 2, 1, 6, 1, 3, 3, 2, 3, 6, 3, 1, 4, 2, 4, 6, 1, 3, 4, 2, 4, 3, 3, 4, 5, 2, 3, 4, 1, 3, 7, 1, 2, 4, 2, 3, 5, 2, 4, 5, 2, 2, 3, 3, 4, 6
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Based on a posting by Zhi-Wei Sun to the Number Theory Mailing List, Mar 23 2008, where he conjectures that a(n) > 0 except for n = 216.
Note that A076768 contains 216 and the numbers n whose only representation has 0 instead of a prime; all other integers appear to be the sum of a prime and a triangular number. Except for n=216, there is no other n<2*10^9 for which a(n)=0.
It is clear that a(t)>0 for any triangular number t because we always have the representation t=t+0. Triangular numbers tend to have only a few representations. Hence by not plotting a(n) for triangular n, the plot (see link) more clearly shows how a(n) slowly increases as n increases. This is more evidence that 216 is the only exception.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..10000
T. D. Noe, Plot of A132399(n) for n to 10^6
Zhi-Wei Sun, Posing to Number Theory List (1)
Zhi-Wei Sun, Posting to Number Theory List (2)
Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers
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EXAMPLE
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0 = 0+0, so a(0) = 1,
3 = 3+0 = 2+1 = 0+3, so a(3) = 3.
8 = 7+1 = 5+3 = 2+6, so a(8) = 3.
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CROSSREFS
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Cf. A065397 (primes p whose only representation as the sum of a prime and a triangular number is p+0), A090302 (largest prime p for each n).
Sequence in context: A079722 A079723 A080511 this_sequence A081485 A100337 A036584
Adjacent sequences: A132396 A132397 A132398 this_sequence A132400 A132401 A132402
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KEYWORD
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nonn,easy
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AUTHOR
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njas, Mar 23 2008
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EXTENSIONS
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Corrected and extended by T. D. Noe (noe(AT)sspectra.com), Mar 26 2008
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