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A007459
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Higgs's primes: a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2.
(Formerly M0660)
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8
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2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 383, 397, 419, 421, 431, 461, 463, 491, 509, 523, 547, 557, 571
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OFFSET
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1,1
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COMMENTS
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Named after the British mathematician Denis A. Higgs (1932-2011). - Amiram Eldar, Jun 05 2021
No prime of the form a*b^k + 1 (those in A089200) with a > 0, b > 1 and k > 2 is a Higgs's prime. - Mauro Fiorentini, Aug 08 2023
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MAPLE
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a:=[2]; P:=1; j:=1;
for n from 2 to 32 do
P:=P*a[n-1]^2;
for i from j+1 to 250 do
if (P mod (ithprime(i)-1)) = 0 then
a:=[op(a), ithprime(i)]; j:=i; break; fi;
od:
od:
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MATHEMATICA
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f[ n_List ] := (a = n; b = Apply[ Times, a^2 ]; d = NextPrime[ a[ [ -1 ] ] ]; While[ ! IntegerQ[ b/(d - 1) ] || d > b, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]
nxt[{p_, a_}]:=Module[{np=NextPrime[a]}, While[PowerMod[p, 2, np-1] != 0, np = NextPrime[np]]; {p*np, np}]; NestList[nxt, {2, 2}, 60][[All, 2]] (* Harvey P. Dale, Jul 09 2021 *)
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PROG
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(Haskell)
a007459 n = a007459_list !! (n-1)
a007459_list = f 1 a000040_list where
f q (p:ps) = if mod q (p - 1) == 0 then p : f (q * p ^ 2) ps else f q ps
(PARI) step(v)=my(N=vecprod(v)^2); forprime(p=v[#v]+1, , if(N%(p-1)==0, return(concat(v, p))))
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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