A120628 -id:A120628 - cp's OEIS frontend

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617

Offset: 1

Clark Kimberling, Dec 15 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. 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 A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x        Lower POBA       Upper POBA       POBA
1        A001359/A006512  A006512/A001359  A265759/A265760
3/2      A104163/A158708  A162336/A158709  A265761/A222565
2        A005383/A005382  A005385/A005384  A079149/A120628
3        A091180/A088878  A094525/A023208  A265763/A265764
4        A162857/A062737  A090866/A023212  A265765/A120639
5        A265766/A158318  A265767/A023217  A265768/A265769
6        A227756/A158015  A051644/A007693  A265770/A265771
sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
tau      A265796/A265797  A265798/A265799  A265800/A265801
1/tau    A265799/A265798  A265797/A265796  A265806/A265807
pi       A265808/A265809  A265810/A265811  A265812/A265813
e        A265814/A265815  A265816/A265817  A265818/A265819

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Cf. A000040, A001359, A006512, A265759, A265760.

x = 1; 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, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A087713 Greatest prime factor of the product of the neighbors of the n-th prime.

Original entry on oeis.org

3, 2, 3, 3, 5, 7, 3, 5, 11, 7, 5, 19, 7, 11, 23, 13, 29, 31, 17, 7, 37, 13, 41, 11, 7, 17, 17, 53, 11, 19, 7, 13, 23, 23, 37, 19, 79, 41, 83, 43, 89, 13, 19, 97, 11, 11, 53, 37, 113, 23, 29, 17, 11, 7, 43, 131, 67, 17, 139, 47, 71, 73, 17, 31, 157, 79, 83, 13, 173, 29, 59

Offset: 1

Reinhard Zumkeller, Sep 28 2003

Apparently a(n) = A024710(n) for n>4. - Georg Fischer, Oct 06 2018

Conjecture: The record values are A120628 \ {2}. - Jason Yuen, Jan 19 2025

			a(10) = A006530(prime(10)^2 - 1) = A006530(29*29-1) = A006530(840) = A006530(7*5*3*2^3) = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Davide Rotondo and others, A120628 and record numbers of a087713, Conversation in the SeqFan Google Group, Dec 25 2024.

Cf. A006530, A084920, A120628.

a087713 = a006530 . a084920  -- Reinhard Zumkeller, Aug 27 2013

FactorInteger[#][[-1,1]]&/@((#-1)(#+1)&/@Prime[Range[80]]) (* Harvey P. Dale, Oct 26 2019 *)

a(n) = my(p=prime(n)); vecmax(factor((p-1)*(p+1))[, 1]); \\ Michel Marcus, Jan 20 2025

from sympy import prime, primefactors
def A087713(n): p = prime(n); return max(primefactors(p*p-1))  # Ya-Ping Lu, Mar 07 2025

a(n) = A006530((A000040(n)-1)*(A000040(n)+1)) = A006530(A006093(n)*A008864(n)) = A006530(A084920(n)).

a(n) <= (prime(n)+1)/2, n > 1. - Ya-Ping Lu, Apr 10 2025

Definition clarified by Harvey P. Dale, Oct 26 2019

A333197 Nonprime numbers k such that each nonprime divisor of k is 1 away from a prime number.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 28, 32, 38, 40, 44, 46, 48, 58, 62, 74, 80, 82, 88, 96, 106, 148, 158, 164, 166, 178, 194, 212, 226, 262, 278, 314, 316, 332, 346, 358, 382, 388, 398, 422, 458, 466, 478, 502, 524, 542, 556, 562, 586, 614, 632, 662, 674

Offset: 1

Michel Lagneau, Mar 11 2020

nonn

Let {d(i), i = 1..q} be the set of the q nonprime divisors of a number m. The sequence lists the nonprime numbers such that |d(i) - p(i)| = 1 for all i, where p(i) is prime.
Conjecture: except for a(n) = 4, 8, 16 and 32, a(n) is of the form 2^i*p^j with p = 3, 5, 7, 11, 19, 23, 29, 31, ... ({A120628} minus {2}).
Consequence: 2 * A120628(k) is in the sequence for k >= 1.
Note that all nonprime divisors of a term of the sequence must be 1 or even. Thus a term of the sequence can have at most one odd prime divisor, i.e., it is a power of 2 or 2^i*p where p is an odd prime. In the latter case, since 2*p is a nonprime divisor, p must be in A120628. - Robert Israel, Apr 12 2020

			48 is in the sequence because the nonprime divisors of 48 are {1, 4, 6, 8, 12, 16, 24, 48} and:
|1 - 2| = 1,
|4 - 5| = 1 (or |4 - 3| = 1),
|6 - 7| = 1 (or |6 - 5| = 1),
|8 - 7| = 1,
|12 - 13| = 1 (or |12 - 11| = 1),
|16 - 17| = 1,
|24 - 23| = 1,
|48 - 47| = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002808, A018252, A120628, A334026.

with(numtheory):
for n from 1 to 50 do:
if type(n,prime)=false
  then
d:=divisors(n):n0:=nops(d):it:=0:
  for k from 1 to n0 do :
   if nextprime(d[k])- d[k]= 1
      or
      d[k] - prevprime(d[k])= 1
      or
      isprime(d[k])
       then
       it:=it+1:
       eles
    fi:
   od:
   if it=n0
   then
   printf(`%d, `,n):
     else fi:
  fi:
od:
# Alternative:
N:= 1000: # for terms <= N
P,NP:= selectremove(isprime, [$1..N]):
P:= convert(P,set):
P1:= P union map(`+`,P,1) union map(`-`,P,1):
filter:= proc(n) numtheory:-divisors(n) subset P1 end proc:
select(filter, NP); # Robert Israel, Apr 12 2020

seqQ[n_] := !PrimeQ[n] &&  AllTrue[Divisors[n], AnyTrue[# + {-1,0,1}, PrimeQ] &]; Select[Range[700], seqQ] (* Amiram Eldar, Mar 11 2020 *)

isok(m) = !isprime(m) && (sumdiv(m, d, !isprime(d) && (isprime(d+1) || ((d>1) && isprime(d-1)))) == sumdiv(m, d, !isprime(d))); \\ Michel Marcus, Mar 11 2020

A334026 Primes p such that 2p and 4p are 1 away from a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 37, 41, 53, 79, 83, 97, 131, 139, 173, 199, 281, 293, 307, 431, 499, 577, 593, 619, 683, 727, 743, 911, 997, 1013, 1297, 1429, 1481, 1511, 1811, 1901, 1931, 2003, 2029, 2141, 2273, 2351, 2693, 3037, 3067, 3109, 3491, 3499, 3739, 3769, 3863, 3911, 4211, 4373, 4447, 4481, 4567, 4871

Offset: 1

Robert Israel, Apr 12 2020

nonn

Primes p such that at least one of 2*p-1 and 2*p+1 is prime, and at least one of 4*p-1 and 4*p+1 is prime.
Primes p such that either 2*p-1 and 4*p+1 are prime, or 2*p+1 and 4*p-1 are prime.
Primes p such that 4*p is in A333197.

			a(3) = 5 is a member because 5, 2*5+1=11 and 4*5-1=19 are primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A120628, A333197.

filter:= proc(t) isprime(t) and (isprime(2*t+1) or isprime(2*t-1)) and (isprime(4*t+1) or isprime(4*t-1)) end proc:
select(filter, [2,seq(i,i=3..10000,2)]);

Select[Prime[Range[700]],AnyTrue[2#+{1,-1},PrimeQ]&&AnyTrue[4#+{1,-1},PrimeQ] &] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 17 2021 *)

A276983 Semiprimes n such that n-1 or n+1 is prime.

Original entry on oeis.org

4, 6, 10, 14, 22, 38, 46, 58, 62, 74, 82, 106, 158, 166, 178, 194, 226, 262, 278, 314, 346, 358, 382, 398, 422, 458, 466, 478, 502, 542, 562, 586, 614, 662, 674, 718, 734, 758, 838, 862, 878, 886, 982, 998, 1018, 1094, 1154, 1186, 1202, 1214, 1238, 1282, 1306, 1318, 1322

Offset: 1

Gary E. Davis, Sep 24 2016

nonn

Union of A077065 and A077068.

			a(3) = 10 = 2*5 is a product of 2 primes and 10+1 = 11 is prime.
a(4) = 14 = 2*7 is a product of 2 primes and 14-1 = 13 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A077065, A077068, A120628.

select(t -> isprime(t/2) and (isprime(t-1) or isprime(t+1)), [seq(i,i=2..10000,2)]); # Robert Israel, Sep 30 2016

func[n_] := PrimeOmega[n] == 2 && (PrimeQ[n + 1] || PrimeQ[n - 1])
Select[Range[1000], func[#] &]

isok(n) = (bigomega(n)==2) && (isprime(n-1) || isprime(n+1)); \\ Michel Marcus, Sep 24 2016

lista(nn) = forprime(p=2, nn, if(isprime(2*p+1) || isprime(2*p-1), print1(2*p, ", "))); \\ Altug Alkan, Sep 30 2016

from sympy import isprime, primerange
def aupto(N): return [t for t in (2*p for p in primerange(2, N//2+1)) if isprime(t-1) or isprime(t+1)]
print(aupto(1322)) # Michael S. Branicky, Aug 21 2022

a(n) = 2*A120628(n).

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A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

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A087713 Greatest prime factor of the product of the neighbors of the n-th prime.

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A333197 Nonprime numbers k such that each nonprime divisor of k is 1 away from a prime number.

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A334026 Primes p such that 2*p and 4*p are 1 away from a prime.

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Maple

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A276983 Semiprimes n such that n-1 or n+1 is prime.

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Author

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Programs

Maple

Mathematica

PARI

PARI

Python

Formula

A334026 Primes p such that 2p and 4p are 1 away from a prime.