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Search: id:A135710
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| A135710 |
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Positive integers b such that more than one prime factor p of b attains the maximum of (p-1)*v_p(b) where v_p(b) is the valuation of b at p. |
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| 12, 45, 80, 90, 144, 180, 189, 240, 360, 378, 448, 637, 720, 756, 945, 1274, 1344, 1512, 1625, 1728, 1890, 1911, 2025, 2240, 2548, 2673, 3024, 3185, 3250, 3780, 3822, 4032, 4050, 4875, 5096, 5346, 5733, 6048, 6125, 6370, 6400
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Given b, the number of trailing zeros at the end of the base-b representation of x! is asymptotic to x/M where M is the maximum over p|b of (p-1)*v_p(b).
Usually only one prime p attains the maximum and then the number is v_p(x!)/v_p(b) for all but finitely many x.
But for b=12,45,80,90,..., at least two v_p(x!) must be computed. For example: if b=12 then for x=2006 there are 998 trailing zeros due to v_3 but for x=2007 there are 999 due to v_2.
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EXAMPLE
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Example: for b=90 we have (p-1)*v_p(b) = 1, 4, 4 for p = 2, 3, 5 respectively so the maximum of 4 is attained twice (p=3 and p=5).
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PROGRAM
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(PARI) { F(n, f, p, v, vmax)= f=factor(n); p=f[, 1]; v=vector(length(p), i, f[i, 2]*(p[i]-1)); vmax=vecmax(v); sum(i=1, length(v), v[i]==vmax) } for(n=1, 6400, if(F(n)>1, print(n)))
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CROSSREFS
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Cf. A027868, A011371, A054861.
Sequence in context: A007899 A100156 A009785 this_sequence A070996 A015237 A024223
Adjacent sequences: A135707 A135708 A135709 this_sequence A135711 A135712 A135713
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KEYWORD
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easy,nonn
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AUTHOR
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Noam D. Elkies (elkies(AT)math.harvard.edu), Nov 25 2007
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