A134854 -id:A134854 - cp's OEIS frontend

A098006 (p-1)/2 - phi(p-1) as p runs through the odd primes.

0, 0, 1, 1, 2, 0, 3, 1, 2, 7, 6, 4, 9, 1, 2, 1, 14, 13, 11, 12, 15, 1, 4, 16, 10, 19, 1, 18, 8, 27, 17, 4, 25, 2, 35, 30, 27, 1, 2, 1, 42, 23, 32, 14, 39, 57, 39, 1, 42, 4, 23, 56, 25, 0, 1, 2, 63, 50, 44, 49, 2, 57, 35, 60, 2, 85, 72, 1, 62, 16, 1, 63, 66, 81, 1, 2, 78, 40, 76, 29, 114, 47

Offset: 2

N. J. A. Sloane, Sep 08 2004

In the Luca-Walsh paper it is shown that there are infinitely many numbers not in this sequence. See A098047.

a(n)=0 for Fermat primes (A019434). a(n)=1 for safe primes (A005385). a(n)=2 for A090866. The least prime p for which (p-1)/2-phi(p-1)=n or 0 if there is no such prime is given by A134765(n). Sequence A134854(k) gives the least prime for which a(n)=2^(k-1). For k not a power of 2, it can be shown that if k is in this sequence, then it appears for a prime p <= 1+k^2. - T. D. Noe, Nov 13 2007

J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.
F. Luca and P. G. Walsh, On the number of nonquadratic residues which are not primitive roots, Colloq. Math., 100 (2004), 91-93.

T. D. Noe, Table of n, a(n) for n = 2..10000
T. D. Noe, Finding primes p for which (p-1)/2 - phi(p-1) = k

Cf. A000010, A051953, A098047, A176095 (p runs through the odd numbers).

a098006 n = a005097 (n-1) - a000010 (a006093 n)
-- Reinhard Zumkeller, Mar 26 2013

[(NthPrime(n)-1)/2 - EulerPhi(NthPrime(n)-1): n in [2..100]]; // Vincenzo Librandi, Jan 10 2017

A098006 := proc(n)
    local p;
    p := ithprime(n+1) ;
    (p-1)/2-numtheory[phi](p-1) ;
end proc:
seq(A098006(n),n=1..30) ; # R. J. Mathar, Jan 09 2017

Table[(Prime[n] - 1)/2 - EulerPhi[Prime[n] - 1], {n, 2, 85}] (* Robert G. Wilson v, Sep 09 2004 *)
Table[(n-1)/2-EulerPhi[n-1],{n,Prime[Range[2,100]]}] (* Harvey P. Dale, Oct 23 2016 *)

forprime(p=3,1e3,print1(p\2-eulerphi(p-1)", ")) \\ Charles R Greathouse IV, Feb 04 2013

a(n) = A005097(n-1) - A000010(A006093(n)); a(A159611(n)) = 0. - Reinhard Zumkeller, Mar 26 2013

A134855 Least odd prime p such that 1 + p*2^n is also prime.

Original entry on oeis.org

3, 3, 5, 7, 3, 3, 5, 3, 23, 13, 29, 3, 5, 7, 5, 37, 53, 3, 11, 7, 11, 37, 71, 73, 5, 7, 17, 13, 23, 3, 239, 43, 113, 163, 59, 3, 89, 349, 5, 97, 3, 73, 11, 67, 101, 19, 101, 61, 23, 7, 17, 7, 233, 127, 5, 541, 29, 103, 71, 31, 53, 109, 179, 163, 71, 3, 929, 31, 23, 193, 101, 127

Offset: 1

T. D. Noe, Nov 13 2007

nonn

Let q = 1 + a(n)*2^n. Then q is least prime such that A098006(pi(q)) = 2^(n-1). See A134854 for the values of q.

a(n) = prime(k) for some k < 5*n for n <= 10000 . - Pierre CAMI, Jul 20 2014

Pierre CAMI, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe)

Cf. A098006, A134854.

Table[Select[Prime[Range[2,10000]], PrimeQ[1+2^k # ]&, 1][[1]], {k,100}]
lop[n_]:=Module[{k=3,c=2^n},While[!PrimeQ[1+k*c],k=NextPrime[k]];k]; Array[ lop,80] (* Harvey P. Dale, Sep 01 2022 *)

a(n) = p=3; t=2^n; while(!isprime(1+p*t), p=nextprime(p+1)); p \\ Colin Barker, Jul 22 2014

A134765 Least prime p for which (p-1)/2 - phi(p-1) = n, or 0 if there is no such prime.

Original entry on oeis.org

3, 7, 13, 19, 41, 0, 37, 31, 113, 43, 101, 71, 73, 67, 61, 79, 97, 131, 109, 103, 0, 0, 0, 191, 0, 139, 677, 127, 0, 419, 157, 0, 193, 0, 0, 151, 0, 0, 0, 199, 401, 683, 181, 0, 281, 0, 0, 431, 0, 283, 277, 0, 0, 659, 461, 0, 241, 211, 0, 743, 313, 0, 349, 271, 641, 827

Offset: 0

T. D. Noe, Nov 13 2007, Nov 19 2007

nonn

The graph of this sequence shows that for n>8 either a(n)=0 or a(n)<=1+n^2. See A098006 for the values of (p-1)/2 - phi(p-1) for odd primes p. Sequence A098047 lists the n for which a(n)=0. A134854(n)=a(2^(n-1)).

T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Finding primes p for which (p-1)/2 - phi(p-1) = k
T. D. Noe, Graph for n <= 50000

nn=1000; lc=Table[0,{nn}]; Do[p=Prime[n]; r=(p-1)/2-EulerPhi[p-1]; If[0

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A098006 (p-1)/2 - phi(p-1) as p runs through the odd primes.

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A134855 Least odd prime p such that 1 + p*2^n is also prime.

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A134765 Least prime p for which (p-1)/2 - phi(p-1) = n, or 0 if there is no such prime.

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