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Search: id:A105551
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| A105551 |
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Number of distinct prime factors of n^3 + n^2 + 71. |
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+0
2
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| 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 3, 3, 2, 1, 2, 1, 1, 3, 2, 3, 1, 2, 1, 1, 4, 2, 1, 2, 3, 2, 2, 2, 1, 3, 1, 1, 3, 2, 1, 2, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 3, 2, 2, 1, 3, 1
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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This cubic equation with small positive coefficients is strangely rich in primes and semiprimes. The first 44 consecutive values, for n = 0, 1, 2, ..., 43, are all either prime (23 of them) or semiprime (21 of them), before the first 3-almost prime value is encountered.
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REFERENCES
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Abel, U. and Siebert, H. "Sequences with Large Numbers of Prime Values." Am. Math. Monthly 100, 167-169, 1993.
Forman, R. "Sequences with Many Primes." Amer. Math. Monthly 99, 548-557, 1992.
Garrison, B. "Polynomials with Large Numbers of Prime Values." Amer. Math. Monthly 97, 316-317, 1990.
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LINKS
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Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
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FORMULA
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a(n) = A001221(n^3 + n^2 + 71).
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EXAMPLE
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a(0) = 1 because 0^3 + 0^2 + 71 = 71 is prime.
a(1) = 1 because 1^3 + 1^2 + 71 = 73 is prime.
a(2) = 1 because 2^3 + 2^2 + 71 = 83 is prime.
a(3) = 1 because 3^3 + 3^2 + 71 = 107 is prime.
a(4) = 1 because 3^3 + 3^2 + 71 = 151 is prime.
a(5) = 2 because 3^3 + 3^2 + 71 = 221 = 13 * 17 is the first semiprime.
a(44) = 3 because 44^3 + 44^2 + 71 = 87191 = 13 * 19 * 353 is the first 3-almost prime for nonnegative integers n.
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MATHEMATICA
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f[n_] := Plus @@ Last /@ FactorInteger[n]; Table[ f[n^3 + n^2 + 71], {n, 0, 104}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 21 2005)
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CROSSREFS
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Cf. A000040, A001358, A005846, A007635, A007641, A048988, A050265, A050268, A050267, A050266.
Sequence in context: A099563 A099564 A126389 this_sequence A073772 A164562 A058188
Adjacent sequences: A105548 A105549 A105550 this_sequence A105552 A105553 A105554
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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), May 03 2005
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 21 2005
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