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A276076
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Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.
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34
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1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450, 3675, 7350, 11025, 22050, 6125, 12250, 18375, 36750, 55125, 110250, 343
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OFFSET
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0,2
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COMMENTS
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These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the digit in one-based position k of the factorial base representation of n. See the examples.
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LINKS
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FORMULA
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Other identities.
For all n >= 0:
For all n >= 1:
(End)
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EXAMPLE
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n A007623 polynomial encoded as a(n)
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0 0 0-polynomial (empty product) = 1
1 1 1*x^0 prime(1)^1 = 2
2 10 1*x^1 prime(2)^1 = 3
3 11 1*x^1 + 1*x^0 prime(2) * prime(1) = 6
4 20 2*x^1 prime(2)^2 = 9
5 21 2*x^1 + 1*x^0 prime(2)^2 * prime(1) = 18
6 100 1*x^2 prime(3)^1 = 5
7 101 1*x^2 + 1*x^0 prime(3) * prime(1) = 10
and:
23 321 3*x^2 + 2*x + 1 prime(3)^3 * prime(2)^2 * prime(1)
= 5^3 * 3^2 * 2 = 2250.
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MATHEMATICA
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a[n_] := Module[{k = n, m = 2, r, p = 2, q = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, q *= p^r; p = NextPrime[p]; m++]; q]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)
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PROG
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(Scheme, two versions)
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CROSSREFS
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Cf. A276078 (same terms in ascending order).
Cf. also A000142, A001221, A001222, A002110, A007489, A019565, A033312, A034968, A048675, A051903, A059590, A060130, A076954, A246359, A248663, A276073, A276074, A351576, A351577, A351950, A351951, A351952, A351954.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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