This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A283320 #37 Jan 05 2025 19:51:41
%S A283320 4,9,10,12,18,24,42,60,84,90,120,150,180,330,390,420,630,840,1050,
%T A283320 1260,1470,1680,1890,2100,2730,3570,3990,4620,5460,6930,8190,9240,
%U A283320 10920,11550,13650,13860,16170,18480,20790,23100,25410,27720,39270,43890,53130
%N A283320 Composite semisimple numbers.
%C A283320 See A283530 for the definition of semisimple numbers, and A283736 for the full list.
%C A283320 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
%H A283320 M. F. Hasler, <a href="/A283320/b283320.txt">Table of n, a(n) for n = 1..500</a> (Terms up to 3.5*10^18.)
%H A283320 J. Wang, X. Wang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/44-2/quartwang02_2006.pdf">On the set of reduced phi-partitions of a positive integer</a>, Fib. Quart. 44 (2) (2006) 98-102.
%o A283320 (PARI) is_A283320(n)={bittest(n,0)&&return(n==9);(2>n\=2)&&return;my(Q,m);forprime(p=3,,n<p&&return(n-1); Q=factor(n)[,1]; Q[#Q]>p && 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)<p&&return([p, q]), 2)); n%p && return; n\=p)} \\ if n = a*q_1*...*q_k*(p-1)# is semisimple, return either a-1 (if k=0) or else, p and the indices [ i_1 ... i_k ] such that q_m is the ( i_m )-th prime factor of n/(p-1)#. - _M. F. Hasler_, Mar 15 2017
%o A283320 (PARI) 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
%Y A283320 Cf. A283528, A283530, A283736.
%K A283320 nonn
%O A283320 1,1
%A A283320 _N. J. A. Sloane_, Mar 13 2017
%E A283320 a(19)-a(32) from _Alois P. Heinz_, Mar 15 2017
%E A283320 a(33) and beyond from _M. F. Hasler_, Mar 17 2017