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A091137
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Largest number m such that number of times m divides k! is almost k/n for large k, i.e., largest m with A090624(m)=n.
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36
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1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 479001600, 958003200, 2615348736000, 5230697472000, 31384184832000, 62768369664000, 32011868528640000, 64023737057280000, 51090942171709440000
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OFFSET
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0,2
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COMMENTS
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This is always a relatively small multiple of n!, since the multiplicity with which a prime p divides n! is always <= n/(p-1); it is equal to floor(n/(p-1)) at least when n is a power of p. - Franklin T. Adams-Watters, May 31 2010
At least for most small n, a(n) = A002790(n) * n!; the first difference is n=15. It appears that A002790(n) * n! always divides a(n).
Conjecture: The denominators of the series reversion of the sequence with e.g.f. Polylog(2,x). - Benedict W. J. Irwin, Jan 05 2017
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REFERENCES
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P. Curtz, Integration numérique ..., Note 12, C.C.S.A., Arcueil, 1969; see pp. 36, 56.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 3rd. ed., 1966; Lemma 1.7.3, p. 14. [From N. J. A. Sloane, Sep 06 2010]
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LINKS
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Table of n, a(n) for n=0..18.
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See sigma_n p. 3.
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FORMULA
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a(n) = Product_p {p prime} p^floor(n/(p-1)).
a(2n+1) = 2*a(2n).
a(n+1) = A027760(n+1)*a(n). - Paul Curtz, Aug 01 2008
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EXAMPLE
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a(2) = 12 noting that 100! is a multiple of 12^48 and 48 is almost 100/2.
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MAPLE
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A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end: for n from 0 to 20 do printf("%a, ", A091137(n) ) ; od: # R. J. Mathar, Feb 23 2009
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MATHEMATICA
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A027760[n_] := Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]; a[n_] := a[n] = A027760[n]*a[n-1]; a[0] = 1; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 04 2011 *)
(* Conjecture - gives first 19 terms *) Rest@Denominator@Table[CoefficientList[InverseSeries[Series[PolyLog[2, x], {x, 0, 20}]], x][[n]] (n - 1)!, {n, 1, 20}] (* Benedict W. J. Irwin, Jan 05 2017 *)
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PROG
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(PARI) a(n)=local(r); r=1; forprime(p=2, n+1, r*=p^(n\(p-1))); r \\ Franklin T. Adams-Watters, May 31 2010
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CROSSREFS
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Starts similarly to A002207 especially for even n and all values of A002207 seen so far seem to divide a(n). Cf. A090622, A090624, A091136.
Cf. A171080.
Cf. A002790, A000142. - Franklin T. Adams-Watters, May 31 2010
Sequence in context: A002207 A181814 A232248 * A347284 A092825 A135396
Adjacent sequences: A091134 A091135 A091136 * A091138 A091139 A091140
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, Dec 19 2003
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STATUS
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approved
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