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A274212
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The factorization of n contains only lesser of twin primes.
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2
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1, 3, 5, 9, 11, 15, 17, 25, 27, 29, 33, 41, 45, 51, 55, 59, 71, 75, 81, 85, 87, 99, 101, 107, 121, 123, 125, 135, 137, 145, 149, 153, 165, 177, 179, 187, 191, 197, 205, 213, 225, 227, 239, 243, 255, 261, 269, 275, 281, 289, 295, 297, 303, 311, 319, 321, 347, 355, 363, 369, 375, 405, 411, 419, 425, 431, 435, 447, 451, 459, 461, 493, 495, 505, 521, 531, 535, 537, 561, 569
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OFFSET
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1,2
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COMMENTS
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1 is in the sequence by convention.
The longest chain of consecutive odd numbers in this sequence has length 6 (otherwise one of the terms is divisible by 7). - Zak Seidov, Jun 20 2016
The first occurrence of five consecutive odd numbers is a(1473448)..a(1473452) = {80008203 = 3 * 11 * 2424491, 80008205 = 5 * 17^3 * 3257, 80008207 = 2081 * 38447, 80008209 = 3^3 * 2963267, 80008211}. - Charles R Greathouse IV, Jun 30 2016
Smallest n (if any) such that n+{0,2,4,6,8,10} all are terms is n > 10^12 according to Giovanni Resta. - Zak Seidov, Jul 01 2016
For any two terms, the sequence also contains their product. Reciprocally, this allows us to generate the whole sequence which is the closure, with respect to multiplication, of the set A001359 of lesser of twin primes. - M. F. Hasler, Jun 23 2016
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LINKS
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FORMULA
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Arithmetic conjecture: the equation a(n+1) - a(n) = 2r has infinitely many solutions for any fixed integer value r >= 1.
Analytic conjecture: a(n) is asymptotic to C*n*log(n)^2 for a constant C > 0.2.
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PROG
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(PARI) for(n=1, 1000, if(prod(i=1, omega(n), isprime(factor(n)[i, 1]+2))==1, print1(n, ", ")))
(PARI) is(n)=!for(i=1, #n=factor(n)~, isprime(n[1, i]+2)||return) \\ prefix "bittest(n, 0) &&" for efficiency, if the selection is to be applied to numbers of unknown parity. - M. F. Hasler, Jun 23 2016
(PARI) list(lim, mx=lim)=my(u, v=List([1]), P=List(), p=2); forprime(q=3, min(mx, lim)+2, if(q-p==2, listput(P, p)); p=q); for(i=1, #P, p=P[i]; if(3*p>lim, for(j=i, #P, listput(v, P[j])); break); u=list(lim\p, p); for(j=1, #u, listput(v, p*u[j]))); Set(v) \\ Charles R Greathouse IV, Jun 30 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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