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Search: id:A132271
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| A132271 |
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Product{k>=0, 1+floor(n/10^k)}. |
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6
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| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360, 427
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OFFSET
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0,2
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COMMENT
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If n is written in base-10 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product (1+d(m)d(m-1)d(m-2)...d(2)d(1)d(0))*(1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).
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FORMULA
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The following formulas are given for a general parameter p considering the product of terms 1+floor(n/p^k) for 0<=k<=floor(log_p(n)), where p=10 for this sequence.
Recurrence: a(n)=(1+n)*a(floor(n/p)); a(pn)=(1+pn)*a(n); a(n*p^m)=product{1<=k<=m, 1+n*p^k}*a(n).
a(k*p^m-j)=(k*p^m-j+1)*k^m*p^(m(m-1)/2), for 0<k<p, 0<j<p, m>=1, a(p^m)=p^(m(m+1)/2)*product{0<=k<=m, 1+1/p^k}, m>=1.
a(n)=A132272(p*n)=(1+n)*A132272(n).
Asymptotic behavior: a(n)=O(n^((1+log_p(n))/2)); this follows from the inequalities below.
a(n)<=A067080(n)*product{0<=k<=floor(log_p(n)), 1+1/p^k}.
a(n)>=A067080(n)/product{1<=k<=floor(log_p(n)), 1-1/p^k}.
a(n)<c*n^((1+log_p(n))/2)=c*p^A000217(log_p(n)), where c=product{k>=0, 1+1/p^k}=2.2244691382741012... (for p=10 see constant A132325).
a(n)>n^((1+log_p(n))/2)=p^A000217(log_p(n)).
lim sup a(n)/A067080(n)=2*product{k>0, 1+1/p^k}=2.2244691382741012..., for n-->oo (for p=10 see constant A132325).
lim inf a(n)/A067080(n)=1/product{k>0, 1-1/p^k}=1/0.8900100999989990000001000..., for n-->oo (for p=10 see constant A132038).
lim inf a(n)/n^((1+log_p(n))/2)=1, for n-->oo.
lim sup a(n)/n^((1+log_p(n))/2)=2*product{k>0, 1+1/p^k}=2.2244691382741012..., for n-->oo (for p=10 see constant A132325).
lim inf a(n+1)/a(n)=2*product{k>0, 1+1/p^k}=2.2244691382741012... for n-->oo (for p=10 see constant A132325).
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EXAMPLE
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a(12)=(1+floor(12/10^0))*(1+floor(12/10^1))=13*2=26; a(21)=63 since 21=21(base-10) and so
a(21)=(1+21)*(1+2)(base-10)=22*3=66.
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CROSSREFS
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Cf. A132038, A132325, A132269(p=2), A132327(p=3), A132272.
For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.
Sequence in context: A099469 A039112 A160015 this_sequence A116069 A081511 A030283
Adjacent sequences: A132268 A132269 A132270 this_sequence A132272 A132273 A132274
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007
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