A264907 a(n) is the smallest "cyclic" integer, k, that has exactly n prime factors.

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

Offset: 0

Author

Geoffrey Critzer, Nov 28 2015

Keywords

• nonn

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_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

Examples

			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
Corrected by _Jeppe Stig Nielsen_, May 22 2021.
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
		

Crossrefs

Cf. A003277, A100564.

Programs

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]));mJeppe Stig Nielsen, May 22 2021

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

a(17) from Jeppe Stig Nielsen, May 22 2021