A090866 -id:A090866 - cp's OEIS frontend

A158018 Primes p such that (p - 1)/12 is also prime.

37, 61, 157, 229, 277, 349, 373, 709, 733, 853, 877, 997, 1069, 1213, 1237, 1669, 1789, 2293, 2389, 2677, 2749, 2797, 3229, 3253, 3373, 3517, 3733, 4549, 4597, 4813, 4909, 5197, 5557, 5749, 6037, 6277, 6829, 7213, 7573, 7717, 7933, 8293, 8629, 9013, 9133

Offset: 1

Roger L. Bagula, Mar 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005385, A090866, A051644, A055781

Flatten[Table[If[PrimeQ[n] && PrimeQ[(n - 1)/12], n, {}], {n, 1, 10000}]]
Select[Prime[Range[1500]], PrimeQ[(# - 1) / 12]&] (* Vincenzo Librandi, Apr 14 2013 *)

a(n)=12*A075704(n)+1. [From R. J. Mathar, Mar 15 2009]

Definition slightly rephrased - The Assoc. Eds. of the OEIS, Aug 30 2010

A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

Original entry on oeis.org

11, 7, 13, 11, 19, 29, 43, 53, 67, 149, 163, 173, 211, 269, 283, 293, 317, 331, 389, 509, 523, 547, 557, 653, 691, 773, 787, 797, 907, 1051, 1109, 1123, 1171, 1229, 1493, 1531, 1637, 1723, 1733, 1867, 1949, 1997, 2011, 2083, 2251, 2309, 2347, 2371, 2467

Offset: 1

Clark Kimberling, Dec 18 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

			The POBAs for 4 start with  11/2, 7/2, 13/3, 11/3, 19/5, 29/7, 43/11, 53/13, 67/17. For example, if p and q are primes and q > 13, then 53/13 is closer to 3 than p/q is.

Cf. A000040, A023212, A062737, A090866, A120639, A162857.

x = 4; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265765/A120639 *)
Numerator[tL]   (* A162857 *)
Denominator[tL] (* A062737 *)
Numerator[tU]   (* A090866 *)
Denominator[tU] (* A023212 *)
Numerator[y]    (* A265765 *)
Denominator[y]  (* A120639 *)

A180642 Numbers k such that phi(k)/4 is a prime, where phi is the Euler totient function.

Original entry on oeis.org

13, 15, 16, 20, 21, 24, 25, 26, 28, 29, 30, 33, 36, 42, 44, 50, 53, 58, 66, 69, 92, 106, 138, 141, 149, 173, 177, 188, 236, 249, 269, 282, 293, 298, 317, 321, 332, 346, 354, 389, 428, 498, 501, 509, 537, 538, 557, 586, 634, 642, 653, 668, 681, 716, 773, 778, 789

Offset: 1

Carmine Suriano, Sep 14 2010

nonn

Apparently the sequence is infinite, but I have no proof. There are many n-ples of consecutives: (15,16)-(20,21)-(24,25,26)-(537,538)-(1436,1437)-...-(30236-30237)

This sequence is infinite if and only if there are infinitely many primes of the form 2p+1 or 4p+1 with prime p. - Charles R Greathouse IV, Feb 04 2013

			a(5) = 21 since pi(21)/4 = 12/4 = 3 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A065966 (phi(k)/2 is prime), A090866 (subsequence of primes).

Select[Range[800],PrimeQ[EulerPhi[#]/4]&] (* Harvey P. Dale, Feb 11 2015 *)

is(n)=n=eulerphi(n);n%4==0 && isprime(n/4) \\ Charles R Greathouse IV, Feb 04 2013

is(n)=if(n<51,n=eulerphi(n);n%4==0 && isprime(n/4),my(v=[3,4,6]);for(i=1,#v,if(n%(2*v[i])==v[i]&&gcd(n/v[i],v[i])==1&&isprime(n/v[i])&&isprime(eulerphi(n)/4),return(1)));if(n%4==2,n/=2);n%4==1&&isprime(n)&&isprime(n\4)) \\ Charles R Greathouse IV, Feb 04 2013

a(n) >> n log^2 n. - Charles R Greathouse IV, Feb 04 2013

A285016 Primes of the form p*b^b - 1, where p is a prime and b>1.

Original entry on oeis.org

7, 11, 19, 43, 53, 67, 163, 211, 283, 331, 523, 547, 691, 787, 907, 1051, 1123, 1171, 1279, 1531, 1723, 1867, 2011, 2083, 2251, 2347, 2371, 2467, 2707, 2731, 2803, 2971, 3187, 3307, 3547, 3643, 3907, 3931, 4051, 4243, 4363, 4603, 4651, 4723, 5107, 5227

Offset: 1

Vincenzo Librandi, May 12 2017

			a(1) = 2*(2^2)-1 = 7.
a(2) = 3*(2^2)-1 = 11.
a(3) = 5*(2^2)-1 = 19.
a(4) = 11*(2^2)-1 = 43.

Vincenzo Librandi, Table of n, a(n) for n = 1..2300

Cf. A000312, A090866, A285015.

nmax=10^4; pimax=PrimePi[nmax]; bmax=1;While[(bmax+1)^(bmax+1)<=nmax,bmax++]; Select[Union@Flatten@Table[Prime[pi] b^b-1,{b,2,bmax},{pi,pimax}],PrimeQ[#]&&#<=nmax&]

is(n)=for(b=2,oo, my(B=b^b); if((n+1)%B==0 && isprime((n+1)/B), return(isprime(n))); if(2*B+1>n, return(0))) \\ Charles R Greathouse IV, Jun 16 2022

list(lim)=my(v=List()); lim\=1; for(b=2,oo, my(p=2*b^b-1); if(p>lim, break); if(isprime(p), listput(v,p))); forstep(b=2,oo,2, my(B=b^b); if(3*B-1>lim, break); forprime(q=3,(lim+1)\B, my(p=q*B-1); if(isprime(p), listput(v,p)))); Set(v) \\ Charles R Greathouse IV, Jun 16 2022

A318251 Lesser of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

Original entry on oeis.org

6144, 19329024, 939524096, 4026531840, 309237645312, 6146186280960, 52158082842624, 29273397577908224

Offset: 1

Amiram Eldar, Aug 22 2018

The larger numbers in each pair are in A318252.
Analogous to A002025 as A163272 is analogous to A000396.
If p and 4p+1 are primes then 2^(4p-1)*p is in this sequence, therefore if A023212 is infinite then also this sequence is.
The terms were calculated using an extended list of terms of A025487.

			6144 is in the sequence since A074206(6144) = 13312 and A074206(13312) = 6144.

Cf. A023212, A074206, A090866, A163272, A318252.

f(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, f(A[k])));
isok(n)={my(a=f(n)); a>n && f(a)==n;} \\ Michel Marcus, Sep 26 2018

A318252 Larger of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

Original entry on oeis.org

13312, 81551360, 1946157056, 128580583424, 12695923326976, 33590071001088, 2182874178519040, 59672695062659072

Offset: 1

Amiram Eldar, Aug 22 2018

The lesser numbers in each pair are in A318251.
Analogous to A002046 as A163272 is analogous to A000396.
If p and 4p+1 are primes then 2^(4p-2)*(4p+1) is in this sequence.

			13312 is in the sequence since A074206(13312) = 6144 and A074206(6144) = 13312.

Cf. A023212, A074206, A090866, A163272, A318251.

f(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, f(A[k])));
isok(n)={my(a=f(n)); aMichel Marcus, Sep 26 2018

cp's OEIS Frontend

A158018 Primes p such that (p - 1)/12 is also prime.

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A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

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A180642 Numbers k such that phi(k)/4 is a prime, where phi is the Euler totient function.

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A285016 Primes of the form p*b^b - 1, where p is a prime and b>1.

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A318251 Lesser of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

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A318252 Larger of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

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