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A283320
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Composite semisimple numbers.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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PROG
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(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
(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
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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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STATUS
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approved
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