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A123238
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31st row of 3-almost prime power sum array, with polynomial reducible over Z.
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32, 1425203755583399996168049966590022914305, 106693983773666239196515949681928803974352131646102553502213120, 1854935026981753007730687157925149063119367096762979743090469098299990573383681
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| This polynomial, A(31,n), is surprisingly reducible over Z and thus prime-free. A(31,n) = 31st row of 3-almost prime power sum array A(k,n) = 1 + SUM[i=1..k]n^(3almostprime(i)) = 1 + SUM[i=1..k]n^A014612(i). If we deem 3almostprime(0) = 1, the array is A(k,n) = SUM[i=0..k]n^A014612(i). This array is to 3-almost primes as the array of A123656-A123665 is to semiprimes and the array of A123650-A123652 and A123113 is to primes.
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FORMULA
| a(n) = 1 + n^8 + n^12 + n^18 + n^20 + n^27 + n^28 + n^30 + n^42 + n^44 + n^45 + n^50 + n^52 + n^63 + n^66 + n^68 + n^70 + n^75 + n^76 + n^78 + n^92 + n^98 + n^99 + n^102 + n^105 + n^110 + n^114 + n^116 + n^117 + n^124 + n^125 + n^130 = + / - (n^2 + 1) * (n^128 - n^126 + n^124 + n^123 - n^121 + n^119 - n^117 + 2n^115 + n^114 - 2n^113 + 2n&111 - 2n^109 + n^108 + 2n^107 - n^106 - 2n^105 + n^104 + 3n^103 - n^102 - 3n^101 + 2n^100 + 3n^99 - 2n^98 - 2n^97 + 3n^96 + 2n^95 - 3n^94 - 2n^93 + 3n^92 + 2n^91 - 2n^90 - 2n^89 + 2n^88 + 2n^87 - 2n^86 - 2n^85 + 2n^84 + 2n^83 - 2n^82 - 2n^81 + 2n^80 + 2n^79 - 2n^78 - 2n^77 + 3n^76 + 2n^75 - 2n^74 - n^73 + 2n^72 + n^71 - 2n^70 - n^69 + 3n^68 + n^67 - 2n^66 - n^65 + 3n^64 + n^63 - 3n^62 + 3n^60 - 3n^58 + 3n^56 - 3n^54 + 3n^52 - 2n^50 + 3n^48 - 3n^46 + 3n^44 + n^43 - 2n^42 - n^41 + 3n^40 + n^39 - 3n^38 - n^37 + 3n^36 + n^35 - 3n^34 - n^33 + 3n^32 + n^31 - 3n^30 - n^29 + 4n^28 + n^27 - 3n^26 + 3n^24 - 3n^22 + 3n^20 - 2n^18 + 3n^16 - 3n^14 + 3n^12 - 2n^10 + 2n^8 - n^6 + n^4 - n^2 + 1) = 10000 110000 001101 000100 001001 001100 000100 000000 000001 011000 010101 001000 000000 010100 001101 000000 000001 011000 000101 000001 000100 000001 (base n).
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EXAMPLE
| a(2) = 1 + 2^8 + 2^12 + 2^18 + 2^20 + 2^27 + 2^28 + 2^30 + 2^42 + 2^44 + 2^45 + 2^50 + 2^52 + 2^63 + 2^66 + 2^68 + 2^70 + 2^75 + 2^76 + 2^78 + 2^92 + 2^98 + 2^99 + 2^102 + 2^105 + 2^110 + 2^114 + 2^116 + 2^117 + 2^124 + 2^125 + 2^130 = 1425203755583399996168049966590022914305 = 5 * 313031 * 988231 * 8407441109 * 109596654684906089.
a(3) = 2^10 * 5 * 7^2 * 82571487271614599 * 8094234237339031517 * 636308925187787085503.
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CROSSREFS
| Cf. A014612 = 3-almost primes, A123656-A123665.
Sequence in context: A017417 A017549 A087927 * A034060 A032445 A135088
Adjacent sequences: A123235 A123236 A123237 * A123239 A123240 A123241
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 06 2006
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