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A328920
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let f(N) be the smallest index m such that from the m-th term on, the sequence {k^k mod N: k >= 0} enters into a cycle, then a(n) is the smallest number N such that f(N) = n, or 0 if no such n exists.
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1
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1, 2, 0, 8, 81, 0, 15625, 128, 5764801, 0, 0, 2048, 3138428376721, 1594323, 3937376385699289, 32768, 43046721, 0, 14063084452067724991009, 524288, 37589973457545958193355601, 476837158203125, 31381059609, 8388608, 480250763996501976790165756943041, 847288609443, 0, 134217728, 0, 3219905755813179726837607
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OFFSET
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0,2
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COMMENTS
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By the formula for f(N) shown in A328914, a(n) = 0 if and only if n > 1 and n-1 is powerful (n-1 is in A001694). Note that if f(N) = n, where N = Product_{i=1..t} p_i^e_i, then f(p_i^e_i) <= n, and there exists some i such that f(p_i^e_i) = n.
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LINKS
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FORMULA
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a(0) = 1, a(1) = 2; for n > 1, let p be the smallest prime such that n-1 is divisible by p but not by p^2, then a(n) = p^n, or 0 if no such p exists.
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EXAMPLE
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By the formula shown in A328914, if f(N) = 7, then N must be divisible by either 2^7 = 128 or 3^7 = 729, so the smallest N is a(7) = 128.
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PROG
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(PARI) a(n) = if(n==0, 1, if(n==1, 2, my(v=factor(n-1)); for(i=1, omega(n-1), if(v[i, 2]==1, return(v[i, 1]^n))); return(0)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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