

A209236


List of integers m>0 with m1 and m+1 both prime, and m2, m, m+2 all practical.


14



4, 6, 18, 30, 198, 462, 1482, 2550, 3330, 4422, 9042, 11778, 26862, 38610, 47058, 60258, 62130, 65538, 69498, 79902, 96222, 106782, 124542, 143262, 149058, 151902, 184830, 200382, 208962, 225342, 237690, 249858, 251262, 295038, 301182, 312702, 345462, 348462
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OFFSET

1,1


COMMENTS

Conjecture: a(n) always exists. In other words, there are infinitely many quintuples (m2, m1, m, m+1, m+2) with m1 and m+1 both prime and m2, m, m+2 all practical.
Note that this sequence is a subsequence of A014574.
ZhiWei Sun observed that if m2, m, m+2 are all practical with m>4 then m is congruent to 2 modulo 4. His PhD student ShanShan Du gave the following explanation: If m>4 is a multiple of 4, then m2 and m+2 are congruent to 2 modulo 4, and one of them is not divisible by 3 and hence not practical (since 4=1+3).


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205210 [MR96i:11106].
ZhiWei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 20122017.


MAPLE

a(3)=18 since {17, 19} is a twin prime pair and 16, 18, 20 are practical numbers.


MATHEMATICA

f[n_] := f[n] = FactorInteger[n]; Pow[n_, i_] := Pow[n, i] = Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]); Con[n_] := Con[n] = Sum[If[Part[Part[f[n], s+1], 1] <= DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]] +1, 0, 1], {s, 1, Length[f[n]]1}]; pr[n_] := pr[n] = n>0 && (n<3  Mod[n, 2] + Con[n]==0); n=0; t = {}; Do[If[PrimeQ[Prime[k]+2] == True && pr[Prime[k]1] == True && pr[Prime[k]+1] == True && pr[Prime[k]+3] == True, n = n+1; AppendTo[t, Prime[k]+1]], {k, 100}]; t


PROG

(PARI) o=3; forprime(p=5, , (2+o==o=p)next; is_A005153(p3) & is_A005153(p1) & is_A005153(p+1) & print1(p1, ", ")) \\  M. F. Hasler, Jan 13 2013


CROSSREFS

Cf. A005153, A014574, A208243, A208244, A208246, A208249.
Sequence in context: A026689 A138276 A287682 * A182643 A156096 A281861
Adjacent sequences: A209233 A209234 A209235 * A209237 A209238 A209239


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 13 2013


STATUS

approved



