| Displaying 1-6 of 6 results found.
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1, 2, 4, 8, 10, 16, 20, 24, 29, 33, 36, 46, 76, 99, 108, 132, 179, 213, 217, 251, 286, 397, 431, 439, 445, 471, 535, 658, 677, 702, 780, 889, 1227, 1296, 1388, 1395, 1430, 1438, 1624, 1817, 2082, 2396, 2423, 2978, 3133, 3138, 3432, 3511, 3699, 3838, 4024, 4104, 4589, 4930
COMMENTS
Or: positions m for which A177929(m)-1 and A177929(m)+1 are twin primes.
MAPLE
A020639 := proc(n) numtheory[factorset](n) ; min(op(%)) ; end proc:
A177929 := proc(n) option remember; if n = 1 then 4; else d1 := A020639(procname(n-1)-1) ; d2 := A020639(procname(n-1)+1) ; procname(n-1)+min(d1, d2) -1; end if; end proc:
read("transforms") ; L := [seq( A177930(n), n=1..1300)] ; RECORDS(L)[2] ; # R. J. Mathar, May 31 2010
MATHEMATICA
lpf[n_] := FactorInteger[n][[1, 1]];
b[n_] := b[n] = If[n == 1, 4, b[n-1] + lpf[b[n-1]^2-1]-1];
Position[Table[b[n], {n, 1, 1000}], k_ /; PrimeQ[k-1] && PrimeQ[k+1]] // Flatten (* Jean-François Alcover, Feb 24 2024 *)
3, 5, 11, 29, 59, 137, 281, 569, 1151, 2309, 4649, 9341, 19139, 38711, 77489, 155381, 311681, 624047, 1248101, 2497421, 4998941, 10002437, 20005289, 40010609, 80021309, 160043909, 320090921, 640196267, 1280392739, 2560793201, 5121618767
a(1)=2. Otherwise the average of the smallest prime divisors of 2n-1 and 2n+1.
+10
7
2, 4, 6, 5, 7, 12, 8, 10, 18, 11, 13, 14, 4, 16, 30, 17, 4, 21, 20, 22, 42, 23, 25, 27, 5, 28, 29, 4, 31, 60, 32, 4, 36, 35, 37, 72, 38, 5, 43, 41, 43, 44, 4, 46, 48, 5, 4, 51, 50, 52, 102, 53, 55, 108, 56, 58, 59, 4, 5, 9, 7, 4, 66, 65, 67, 69, 5, 70, 138, 71, 7, 8, 4, 76, 150, 77, 4, 81
COMMENTS
As n tends to infinity, we have 1) lim inf (a(n)/n)=0; 2) if there exist infinitely many twin primes, then lim sup (a(n)/n)=2, otherwise, lim sup (a(n)/n)=1.
MAPLE
N:= 100: # to get a(1) to a(N)
S:= [1, seq(min(numtheory:-factorset(2*i-1)), i=2..N+1)]:
MATHEMATICA
Table[If[n == 1, 2, Mean[{FactorInteger[2 n - 1][[1, 1]], FactorInteger[2 n + 1][[1, 1]]}]], {n, 78}] (* Michael De Vlieger, Aug 02 2015 *)
PROG
(PARI) a(n) = if (n==1, 2, (vecmin(factor(2*n-1)[, 1]) + vecmin(factor(2*n+1)[, 1]))/2); \\ Michel Marcus, Feb 07 2016 (Magma) [2] cat [1/2*(Min(PrimeFactors(2*n-1))+ Min(PrimeFactors(2*n+1))):n in [2..80]]; // Vincenzo Librandi, Feb 07 2016
Smallest prime divisor of ( A177941(n))^2-4.
+10
2
3, 3, 7, 13, 5, 3, 5, 3, 37, 7, 79, 7, 163, 5, 3, 5, 3, 11, 3, 349, 17, 3, 5, 3, 5, 3, 17, 3, 5, 3, 5, 3, 757, 17, 3, 5, 3, 23, 3, 5, 3, 1567, 13, 5, 3, 5, 3, 7, 3163, 5, 3, 5, 3, 17, 3, 5, 3, 5, 3, 23, 3, 5, 3, 37, 7, 17, 3, 7, 23, 3, 13, 7, 5, 3, 5, 3, 7, 11, 3, 5, 3, 5, 3, 6547, 7, 13099, 7, 73
PROG
(PARI) listap(nn) = {my(va = vector(nn), p); va[1] = 5; for (n=2, nn, p = factor(va[n-1]^2-4)[1, 1]; print1(p, ", "); va[n] = va[n-1] + p - 1; ); } \\ Michel Marcus, Dec 14 2018
1, 3, 4, 9, 11, 13, 20, 33, 42, 49, 84, 86, 107, 109, 123, 128, 191, 295, 296, 318, 330, 337, 396, 453, 481, 616, 663, 771, 882, 1105, 1180, 1257, 1431, 1659, 1856, 1936, 2130, 2370, 2584, 2651, 2790, 2959, 3009, 3080, 3121, 3189, 3503, 3639, 3879, 3902, 3961
PROG
(PARI) listai(nn) = {my(va = vector(nn), rec = 0, ind = 1); va[1] = 5; for (n=2, nn, p = factor(va[n-1]^2-4)[1, 1]; if (p > rec, print1(n-1, ", "); rec = p); va[n] = va[n-1] + p - 1; ); } \\ Michel Marcus, Dec 14 2018
EXTENSIONS
1 inserted, a(10) corrected and sequence extended by R. J. Mathar, Jun 30 2010
3, 7, 13, 37, 79, 163, 349, 757, 1567, 3163, 6547, 13099, 26497, 52999, 106273, 212557, 426889, 855427, 1710853, 3421903, 6845869, 13691767, 27385087, 54771007, 109542907, 219096259, 438203677, 876417229, 1752875893, 3505814527, 7011656629, 14023322167, 28046754727, 56093637367
PROG
(PARI) listar(nn) = {my(va = vector(nn), rec = 0); va[1] = 5; for (n=2, nn, p = factor(va[n-1]^2-4)[1, 1]; if (p > rec, print1(p, ", "); rec = p); va[n] = va[n-1] + p - 1; ); } \\ Michel Marcus, Dec 14 2018
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