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A264907
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a(n) is the smallest "cyclic" integer, k, that has exactly n prime factors.
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3
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1, 2, 15, 255, 5865, 146965, 3380195, 125067215, 7378965685, 494390700895, 36090521165335, 2571956263189313, 187752807212819849, 18212022299643525353, 1839414252263996060653, 196817324992247578489871, 21453088424154986055395939, 2981979290957543061700035521
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OFFSET
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0,2
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COMMENTS
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The cyclic numbers are given in A003277.
a(n) = k if k = p_1*p_2*...*p_n where the p_i are distinct primes and no p_j-1 is divisible by any p_i and k is the smallest such integer.
a(n) < a(n+1) for all n. For some n, also a(n) | a(n+1). Note that the divisors of any cyclic number are cyclic. - Jeppe Stig Nielsen, May 22 2021
Are the prime factors of a(n) always a "normal sequence of primes" in the sense of A100564? Equivalently, can you always find a(n) by starting from a suitable smallest prime p_1, and then pick the subsequent prime factors greedily (under the condition that p_j - 1 is not divisible by any p_i) until you have n primes? If yes, then it is easy to calculate a(n), as all one needs to do is find the optimal starting prime. - Jeppe Stig Nielsen, May 23 2021
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LINKS
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EXAMPLE
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The prime factorizations for terms a(1)..a(12) are:
2
3, 5
3, 5, 17
3, 5, 17, 23
5, 7, 13, 17, 19
5, 7, 13, 17, 19, 23
5, 7, 13, 17, 19, 23, 37
5, 7, 13, 17, 19, 23, 37, 59
5, 7, 13, 17, 19, 23, 37, 59, 67
5, 7, 13, 17, 19, 23, 37, 59, 67, 73
7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61
7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61, 73
146965 = 5*7*13*17*19 is cyclic. Since it is the smallest example with 5 primes, 146965 = a(5). It is not a multiple of a(4) = 3*5*17*23. - Jeppe Stig Nielsen, May 22 2021
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PROG
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(PARI) n=0; for(m=1, +oo, if(gcd(m, eulerphi(m))==1&&omega(m)==n, print1(m, ", "); n++)) \\ slow, from Jeppe Stig Nielsen, May 22 2021
(PARI) N=0; for(n=0, +oo, a=+oo; forsubset([N, n], x, m=prod(j=1, n, prime(x[j])); m<a&&gcd(m, eulerphi(m))==1&&(a=m)); print1(a, ", "); a+=(n==1); for(p=1, +oo, m=a*prime(p); if(gcd(m, eulerphi(m))==1, N=p; break()))) \\ Jeppe Stig Nielsen, May 22 2021
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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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Wrong terms a(5), a(6), a(7), a(8), a(10), a(12) corrected, and more terms added, and a(0)=1 preprended by Jeppe Stig Nielsen, May 22 2021
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STATUS
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approved
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