A283736 -id:A283736 - cp's OEIS frontend

4, 9, 10, 12, 18, 24, 42, 60, 84, 90, 120, 150, 180, 330, 390, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2730, 3570, 3990, 4620, 5460, 6930, 8190, 9240, 10920, 11550, 13650, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 39270, 43890, 53130

Offset: 1

N. J. A. Sloane, Mar 13 2017

nonn

See A283530 for the definition of semisimple numbers, and A283736 for the full list.

The term a(94) = A002110(9)/prime(7) is the smallest term larger than some A002110(i+1) without being a multiple of the next smaller primorial A002110(i), here with i=7. For subsequent terms of the form a(n) = A002110(i+2)/prime(i), the ratio a(n)/A002110(i+1) = prime(i+2)/prime(i) is smaller, but one can have a multiple m*a(n) of such a term, provided m*(prime(i+2)-(prime(i))(prime(i+1)-prime(i)) < prime(i). This occurs first at a(419) = 2*a(405) and a(425) = 3*a(405) with a(405) = A002110(15)/prime(13) ~ 1.5e16. No term > 4*p# not a multiple of (p-1)# occurs below 4*79#/71 ~ 1.8e29, and no term > 5*p# not a multiple of (p-1)# occurs below 5*107#/101 ~ 1.3e41. The first term of the form A002110(i+3)/prime(i) also appears for prime(i) = 101. - M. F. Hasler, Mar 16 2017

M. F. Hasler, Table of n, a(n) for n = 1..500 (Terms up to 3.5*10^18.)
J. Wang, X. Wang, On the set of reduced phi-partitions of a positive integer, Fib. Quart. 44 (2) (2006) 98-102.

Cf. A283528, A283530, A283736.

is_A283320(n)={bittest(n,0)&&return(n==9);(2>n\=2)&&return;my(Q,m);forprime(p=3,,np && for(k=1, #Q-m=#select(q->q<=p, Q), forvec(q=vector(k, j, [m+1, #Q]), prod(i=1, k, 1-p/Q[q[i]], n)M. F. Hasler, Mar 15 2017

list_A283320(n,L=4,N=1,s=1,a=List())={forprime(p=2,,L*=nextprime(p+1);until(N>=L,until(is_A283320(N+=s),);listput(a,N);n--||return(Vec(a)));s*=p)} \\ Assumes the gap is a multiple of (p-1)# for N >= (L/2)*p#: With the default L=4, the step is increased to s = 2, 6, 30,... for N >= 12, 60, 420,... For n > 418 one must increase L, since a(419) = 2*A002110(15)/prime(13) ~ 2.3*A002110(14) and a(425) = 3*A002110(15)/prime(13) ~ 3.4*A002110(14) are not multiples of A002110(13). No other such term > 2*p# not a multiple of (p-1)# occurs below 2*67#/59 ~ 2.7e23, and L=8 is sufficient up to 4*73#/71 = 1.8e29. - M. F. Hasler, Mar 16 2017

a(19)-a(32) from Alois P. Heinz, Mar 15 2017

a(33) and beyond from M. F. Hasler, Mar 17 2017

cp's OEIS Frontend

A283320 Composite semisimple numbers.

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