Search: a124331 -id:a124331
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A124330
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a(n)= ((d(n) mod phi(n)) +1)th positive integer which is coprime to n, where phi(n) is number of positive integers which are <= n and are coprime to n and d(n) is the number of positive divisors of n.
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+10
3
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1, 1, 1, 3, 3, 1, 3, 1, 5, 1, 3, 7, 3, 11, 8, 11, 3, 1, 3, 17, 8, 9, 3, 1, 4, 9, 7, 15, 3, 1, 3, 13, 7, 9, 6, 29, 3, 9, 7, 21, 3, 29, 3, 15, 13, 9, 3, 31, 4, 17, 7, 15, 3, 25, 6, 19, 7, 9, 3, 47, 3, 9, 11, 15, 6, 29, 3, 13, 7, 27, 3, 37, 3, 9, 13, 13, 5, 29, 3, 27, 8, 9, 3, 43, 6, 9, 7, 19, 3, 47, 5
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OFFSET
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1,4
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LINKS
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MATHEMATICA
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f[n_] := Block[{k = 0, m = Mod[Length[Divisors[n]], EulerPhi[n]] + 1}, While[m > 0, k++; While[GCD[n, k] > 1, k++ ]; m--; ]; k]; Table[f[n], {n, 100}] (* Ray Chandler, Oct 26 2006 *)
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PROG
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(PARI) A124330(n) = { my(k=1+(numdiv(n)%eulerphi(n))); for(i=1, oo, if(1==gcd(i, n), k--; if(!k, return(i)))); }; \\ Antti Karttunen, Feb 18 2023
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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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A124219
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a(n)= m-th positive divisor of n, where phi(n) is number of positive integers which are <= n and are coprime to n, d(n) is the number of positive divisors of n and m = d(n) if d(n)|phi(n), else m = phi(n) mod d(n).
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+10
2
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1, 1, 3, 2, 5, 2, 7, 8, 9, 10, 11, 4, 13, 2, 15, 4, 17, 18, 19, 2, 21, 2, 23, 24, 5, 26, 3, 28, 29, 30, 31, 8, 33, 34, 35, 3, 37, 2, 39, 40, 41, 6, 43, 2, 45, 2, 47, 8, 49, 2, 51, 52, 53, 2, 55, 56, 57, 58, 59, 4, 61, 2, 63, 8, 65, 6, 67, 2, 69, 70, 71, 72, 73, 74, 15, 76, 77, 78, 79, 2
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OFFSET
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1,3
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LINKS
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MATHEMATICA
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f[n_] := Block[{d = Divisors[n]}, d[[Mod[EulerPhi[n], Length[d], 1]]]]; Table[f[n], {n, 90}] (* Ray Chandler, Oct 26 2006 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A342665
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Numbers k for which phi(k)+1 is a multiple of d(k), where phi is Euler totient function (A000010) and d(n) gives the number of divisors of n (A000005).
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+10
2
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1, 2, 4, 25, 81, 121, 289, 529, 841, 1681, 2209, 2809, 3481, 5041, 6889, 7921, 10201, 11449, 12100, 12769, 17161, 18769, 22201, 27889, 28561, 28900, 29929, 32041, 36481, 38809, 51529, 54289, 57121, 63001, 66049, 69169, 72361, 78961, 84100, 85849, 96721, 100489, 120409, 124609, 128881, 146689, 151321, 160801, 175561
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OFFSET
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1,2
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COMMENTS
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Numbers k such that A124331(k) = k. This is also a subsequence of the records of A124331 (both their values and their positions).
Terms other than 2 are a perfect square. Proof: phi(k) is even for k > 2. So phi(k)+1 is odd for k > 2. d(k) is odd only if k is a perfect square. So for any term k > 2 we need k to be a perfect square. Checking cases <= 2 leaves only 2 as the nonsquare in this sequence. - David A. Corneth, Mar 31 2021
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LINKS
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MATHEMATICA
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Select[Join[{1, 2}, Range[2, 420]^2], Divisible[EulerPhi[#] + 1, DivisorSigma[0, #]] &] (* Amiram Eldar, Mar 31 2021 *)
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PROG
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(PARI) isA342665(n) = !((eulerphi(n)+1) % numdiv(n));
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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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