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A321288
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a(n) is the unique n-digit odd number (including leading zero(s)) with alternating even and odd digits and divisible by 5^n.
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1
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5, 25, 125, 8125, 78125, 78125, 5078125, 5078125, 705078125, 2705078125, 72705078125, 272705078125, 3272705078125, 23272705078125, 323272705078125, 4323272705078125, 74323272705078125, 474323272705078125, 1474323272705078125, 61474323272705078125, 561474323272705078125
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OFFSET
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1,1
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COMMENTS
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For n > 1, if n is even, then a(n) is the unique number in {a(n-1), a(n-1) + 2*10^(n-1), a(n-1) + 4*10^(n-1), a(n-1) + 6*10^(n-1), a(n-1) + 8*10^(n-1)} that divides 5^n; if n is odd, then a(n) is the unique number in {a(n-1) + 10^(n-1), a(n-1) + 3*10^(n-1), a(n-1) + 5*10^(n-1), a(n-1) + 7*10^(n-1), a(n-1) + 9*10^(n-1)} that divides 5^n.
The unique n-digit even number (including leading zero(s)) with alternating even and odd digits and divisible by 5^n is 0 if n = 1 and 10*a(n-1) if n > 1. - Jianing Song, Feb 24 2021
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n-1} A321289(i)*10^i.
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EXAMPLE
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a(2) is the unique number in {5, 25, 45, 65, 85} that is divisible by 25, which is 25.
a(3) is the unique number in {125, 325, 525, 725, 925} that is divisible by 125, which is 125.
a(4) is the unique number in {125, 2125, 4125, 6125, 8125} that is divisible by 625, which is 8125.
a(5) is the unique number in {18125, 38125, 58125, 78125, 98125} that is divisible by 3125, which is 78125.
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MATHEMATICA
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n = 21; a[1] = 5; For[i=2, i <= n, i++, For[j=0, j <= 4, j++, t = a[i-1] + (2j + Mod[i, 2]) 10^(i-1); If[Mod[t, 5^i] == 0, a[i] = t; Break[]]]]; Array[a, n] (* Jean-François Alcover, Nov 23 2018, from PARI *)
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PROG
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(PARI) seq(n)={my(v=vector(n)); v[1]=5; for(i=2, n, for(j=0, 4, my(t=v[i-1] + (2*j + i%2)*10^(i-1)); if(t%(5^i)==0, v[i]=t; break))); v}
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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