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A345531
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Smallest prime power greater than the n-th prime.
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4
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3, 4, 7, 8, 13, 16, 19, 23, 25, 31, 32, 41, 43, 47, 49, 59, 61, 64, 71, 73, 79, 81, 89, 97, 101, 103, 107, 109, 113, 121, 128, 137, 139, 149, 151, 157, 163, 167, 169, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 256, 263, 269, 271, 277
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OFFSET
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1,1
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COMMENTS
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Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.
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LINKS
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FORMULA
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EXAMPLE
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a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.
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MATHEMATICA
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a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
tabseq = Table[a[i], {i, 1, 100}];
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PROG
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(PARI)
A000015(n) = for(k=n, oo, if((1==k)||isprimepower(k), return(k)));
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CROSSREFS
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Cf. A000015, A000040, A007814, A007949, A008864, A343249, A343250, A112765, A343251, A214411, A343252, A343253.
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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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