

A264907


a(n) is the smallest "cyclic" integer, k, that has exactly n prime factors.


3



1, 2, 15, 255, 5865, 146965, 3380195, 125067215, 7378965685, 494390700895, 36090521165335, 2571956263189313, 187752807212819849, 18212022299643525353, 1839414252263996060653, 196817324992247578489871, 21453088424154986055395939, 2981979290957543061700035521
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OFFSET

0,2


COMMENTS

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_j1 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


LINKS



EXAMPLE

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


PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



