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A323185 Denominators of rationals whose continued fraction representations show the prime factors of n (for n > 1) in nondecreasing order.

2, 3, 5, 5, 7, 7, 12, 10, 11, 11, 17, 13, 15, 16, 29, 17, 23, 19, 27, 22, 23, 23, 41, 26, 27, 33, 37, 29, 37, 31, 70, 34, 35, 36, 56, 37, 39, 40, 65, 41, 51, 43, 57, 53, 47, 47, 99, 50, 57, 52, 67, 53, 76, 56, 89, 58, 59, 59, 90, 61, 63, 73, 169, 66, 79, 67, 87

Offset: 2

Chris Boyd, Jan 06 2019

a(n) is the denominator of the generating rational of n (see comments and numerators in A323184).

If n is prime, a(n) is n.

			a(28) = 37 because 15/37 = [0; 2, 2, 7] and 2*2*7 = 28.
a(29) = 29 because 1/29 = [0; 29] = 29.

Georg Fischer, Table of n, a(n) for n = 2..1000

Cf. A323184.

Array[Denominator@ FromContinuedFraction@ Prepend[Flatten@ Map[ConstantArray[ #1, #2] &  @@ # &, FactorInteger@ #], 0] &, 67, 2] (* Michael De Vlieger, Jan 07 2019 *)

vectorise_factors(m)={v=[0];F=factor(m);for(i=1,matsize(F)[1],for(j=1,F[i,2],v=concat(v,F[i,1])));}
A323185(n)={vectorise_factors(n); contfracpnqn(v)[2,1];}
for(k=2,75,print1(A323185(k)", "))

A323184 Numerators of rationals whose continued fraction representations show the prime factors of n (for n>1) in nondecreasing order.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 3, 5, 1, 7, 1, 7, 5, 12, 1, 10, 1, 11, 7, 11, 1, 17, 5, 13, 10, 15, 1, 16, 1, 29, 11, 17, 7, 23, 1, 19, 13, 27, 1, 22, 1, 23, 16, 23, 1, 41, 7, 26, 17, 27, 1, 33, 11, 37, 19, 29, 1, 37, 1, 31, 22, 70, 13, 34, 1, 35, 23, 36, 1, 56, 1, 37, 26

Offset: 2

Chris Boyd, Jan 06 2019

Denominators are given in A323185.
From its prime factorization, each natural number N>1 can be uniquely represented as a tuple of nondecreasing first powers (e.g., 60 = 2*2*3*5 -> (2, 2, 3, 5)).
There is a unique positive finite continued fraction associated with N whose coefficients in the standard abbreviated notation (except for the first coefficient, which is arbitrarily set to zero) map 1-to-1 to the elements of the tuple, from which the corresponding generating rational can be calculated (e.g. 60 -> [0; 2, 2, 3, 5] = 37/90).
The first few generating rationals are:
N ... GR .... continued fraction
2 ... 1/2 ... [0; 2]
3 ... 1/3 ... [0; 3]
4 ... 2/5 ... [0; 2, 2]
5 ... 1/5 ... [0; 5]
6 ... 3/7 ... [0; 2, 3]
7 ... 1/7 ... [0; 7]
8 ... 5/12 .. [0; 2, 2, 2]
9 ... 3/10 .. [0; 3, 3]
10 .. 5/11 .. [0; 2, 5]
a(n) is the numerator of the generating rational of n.
Iff n is prime, a(n) is 1.

			a(28) = 15 because 15/37 = [0; 2, 2, 7] and 2*2*7 = 28.
a(29) = 1 because 1/29 = [0; 29] = 29.

Georg Fischer, Table of n, a(n) for n = 2..1000

Cf. A323185.

Array[Numerator@ FromContinuedFraction@ Prepend[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #], 0] &, 74, 2] (* Michael De Vlieger, Jan 07 2019 *)

vectorise_factors(m)={v=[0];F=factor(m);for(i=1,matsize(F)[1],for(j=1,F[i,2],v=concat(v,F[i,1])));}
A323184(n)={vectorise_factors(n); contfracpnqn(v)[1,1];}
for(k=2,75,print1(A323184(k)", "))

A266214 Numbers n that are not coprime to the numerator of zeta(2n)/(Pi^(2n)).

Original entry on oeis.org

14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, 62, 70, 74, 76, 78, 82, 84, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 134, 138, 140, 142, 146, 148, 150, 152, 154, 156, 158, 162, 164, 166, 168, 170

Offset: 1

Chris Boyd, Robert Israel, Dec 24 2015

Equivalently, n not coprime to the numerator of 2^(2n-1)*Bernoulli(2n)/(2n)! (see Lekraj Beedassy comment in A046988).
Conjecture 1: for n>=1, a(n) is identical to 2*A072823(n+1).
Conjecture 2: The corresponding GCDs are powers of 2.
Verified for n <= 10000, e.g.,
GCD = 2 for 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, ...
GCD = 4 for 60, 92, 108, 116, 120, 124, 156, 172, 180, 184, 188, ...
GCD = 8 for 248, 376, 440, 472, 488, 496, 504, 632, 696, 728, 744, ...
GCD = 16 for 1008, 1520, 1776, 1904, 1968, 2000, 2016, 2032, 2544, ...
GCD = 32 for 4064, 6112, 7136, 7648, 7904, 8032, 8096, 8128, 8160
Taking GCDs vertically, column 1 = "14, 60, 248, 1008, 4064, ..." appears to be essentially the same as A171499 and A131262; (ii) column 2 = "22, 92, 376, 1520, 6112, ..." appears to be essentially the same as A010036.
From Chris Boyd, Jan 25 2016: (Start)
Determining whether n is a term of this sequence can be approached by considering odd and even factors separately, and exploiting the fact that numerator(zeta(2n)/(Pi^(2n))) = numerator(2^(2n-2)*N_2n/(D_2n*(2n)!)), where N_2n and D_2n are odd coprime integers such that Bernoulli(2n) = N_2n/2D_2n.
Case 1: odd factors. n is a term if it has an odd prime factor p (necessarily irregular) that divides N_2n at a higher multiplicity than it divides (2n)!. No such factor p of N_2n up to 2n = 10000 is of sufficient multiplicity, and the apparent scarcity of squared and higher power factors of N_2n values (see A090997) suggests that no such p is likely to exist.
Case 2: even factors. An even n is a term if 2 divides 2^(2n-2) at a higher multiplicity than it divides (2n)!. The multiplicity of 2 in 2^(2n-2) is 2n-2, and in (2n)! is 2n minus the number of 1's in the binary expansion of 2n (see A005187). Qualifying n values are therefore those where the number of 1's in the binary expansion of 2n is greater than 2. Except for its first term, A072823 comprises integers with three or more 1-bits in their binary expansion. It follows that for m > 1, 2*A072823(m) values belong to this sequence.
In summary, this sequence is essentially a supersequence of 2*A072823. Conjectures 1 and 2 are true if there are no irregular odd primes p that divide n and the numerator of Bernoulli(2n)/(2n)!. (End)

Chris Boyd, Table of n, a(n) for n = 1..10000
C. Boyd in reply to R. Israel, A046988 query, SeqFan list, Dec 22 2015.

Cf. A010036, A072823, A131262, A171499.

select(n -> igcd(n,numer(2^(2*n-1)*bernoulli(2*n)/(2*n)!)) > 1), [$1..1000]);

Select[Range@ 170, ! CoprimeQ[#, Numerator[Zeta[2 #]/Pi^(2 #)]] &] (* Michael De Vlieger, Dec 24 2015 *)

test(n) = if(gcd(numerator(2^(2*n-1)*bernfrac(2*n)/(2*n)!),n)!=1,1,0)
for(i=1,200,if(test(i),print1(i,", ")))

A264722 Composite numbers that are less than the average of their closest flanking primes.

Original entry on oeis.org

8, 14, 20, 24, 25, 32, 33, 38, 44, 48, 49, 54, 55, 62, 63, 68, 74, 75, 80, 84, 85, 90, 91, 92, 98, 104, 110, 114, 115, 116, 117, 118, 119, 128, 132, 133, 140, 141, 142, 143, 152, 153, 158, 159, 164, 168, 169, 174, 175, 182, 183, 184, 185, 194, 200, 201, 202, 203

Offset: 1

Chris Boyd, Nov 21 2015

Composite numbers that are nearer to the immediately previous prime than to the immediately next prime.
Members of this sequence are the numbers C, necessarily composite, such that P_n < C < I_n, where P_n is the n-th odd prime and I_n the interprime (A024675) between P_n and P_n+1.
Prime-free subsequence of A264720.

			a(7) = 33 because 33 < (31 + 37)/2 = 34.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A264719, A264720, A264721.

Select[Range@ 204, And[CompositeQ@ #, # < (NextPrime[#, -1] + NextPrime@ #)/2] &] (* Michael De Vlieger, Nov 22 2015 *)
Range[#[[1]]+1,Total[#]/2 -1]&/@Select[Partition[Prime[Range[50]],2,1], #[[2]]- #[[1]]>2&]//Flatten  (* Harvey P. Dale, Jul 28 2020 *)

test(n)= {if(n-precprime(n-1)2&&!isprime(n),return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A264721 Composite numbers that are greater than the average of their closest flanking primes.

Original entry on oeis.org

10, 16, 22, 27, 28, 35, 36, 40, 46, 51, 52, 57, 58, 65, 66, 70, 77, 78, 82, 87, 88, 94, 95, 96, 100, 106, 112, 121, 122, 123, 124, 125, 126, 130, 135, 136, 145, 146, 147, 148, 155, 156, 161, 162, 166, 171, 172, 177, 178, 187, 188, 189, 190, 196, 206, 207, 208

Offset: 1

Chris Boyd, Nov 21 2015

Composite numbers that are nearer to the immediately next prime than to the immediately previous prime.
Members of this sequence are the numbers C, necessarily composite, such that I_n < C < P_n+1, where P_n is the n-th odd prime and I_n the interprime (A024675) between P_n and P_n+1.
Prime-free subsequence of A264719.

			a(7) = 36 because 36 > (31 + 37)/2 = 34.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A264719, A264720, A264722.

Select[Range@ 208, And[CompositeQ@ #, # > (Abs@ NextPrime[#, -1] + NextPrime@ #)/2] &] (* Michael De Vlieger, Nov 22 2015 *)

test(n)= {if(n-precprime(n-1)>nextprime(n+1)-n&&n>2&&!isprime(n),return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A264720 Numbers that are less than the average of their closest flanking primes.

Original entry on oeis.org

3, 7, 8, 13, 14, 19, 20, 23, 24, 25, 31, 32, 33, 38, 43, 44, 47, 48, 49, 54, 55, 61, 62, 63, 68, 73, 74, 75, 80, 83, 84, 85, 89, 90, 91, 92, 98, 103, 104, 109, 110, 113, 114, 115, 116, 117, 118, 119, 128, 131, 132, 133, 139, 140, 141, 142, 143, 151, 152, 153, 158

Offset: 1

Chris Boyd, Nov 21 2015

Numbers that are nearer to the immediately previous prime than to the immediately next prime.
This sequence may be viewed as a generalization of A051635 (the weak primes) that includes qualifying composite numbers.
The union of this sequence with A264719 & A145025 is A000027 (omitting 1 & 2).

			a(11) = 31 because 31 < (29 + 37)/2 = 33.
a(12) = 32 because 32 < (31 + 37)/2 = 34.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A051634, A145025, A264719, A264721, A264722.

Select[Range@ 162, # < (NextPrime[#, -1] + NextPrime@ #)/2 &] (* Michael De Vlieger, Nov 22 2015 *)

test(n)= {if(n-precprime(n-1)2,return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A264719 Numbers that are greater than the average of their closest flanking primes.

Original entry on oeis.org

10, 11, 16, 17, 22, 27, 28, 29, 35, 36, 37, 40, 41, 46, 51, 52, 57, 58, 59, 65, 66, 67, 70, 71, 77, 78, 79, 82, 87, 88, 94, 95, 96, 97, 100, 101, 106, 107, 112, 121, 122, 123, 124, 125, 126, 127, 130, 135, 136, 137, 145, 146, 147, 148, 149, 155, 156, 161, 162

Offset: 1

Chris Boyd, Nov 21 2015

Numbers that are nearer to the immediately next prime than to the immediately previous prime.
This sequence may be viewed as a generalization of A051634 (the strong primes) that includes qualifying composite numbers.
The union of this sequence with A264720 & A145025 is A000027 (omitting 1 & 2).

			a(11) = 37 because 37 > (31 + 41)/2 = 36.
a(12) = 40 because 40 > (37 + 41)/2 = 37.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A051634, A145025, A264720, A264721, A264722.

Select[Range@ 162, # > (Abs@ NextPrime[#, -1] + NextPrime@ #)/2 &] (* Michael De Vlieger, Nov 22 2015 *)

test(n)= {if(n-precprime(n-1)>nextprime(n+1)-n&&n>2,return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A257231 a(n) = n^2 mod p where p is the least prime greater than n.

Original entry on oeis.org

1, 1, 4, 1, 4, 1, 5, 9, 4, 1, 4, 1, 16, 9, 4, 1, 4, 1, 16, 9, 4, 1, 7, 25, 16, 9, 4, 1, 4, 1, 36, 25, 16, 9, 4, 1, 16, 9, 4, 1, 4, 1, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 4, 1, 36, 25, 16, 9, 4, 1, 16, 9, 4, 1, 4, 1, 36, 25, 16, 9, 4, 1, 16, 9, 4, 1, 36, 25, 16, 9, 4

Offset: 1

Chris Boyd, Apr 19 2015

Conjecture: a(n) is always a positive square, except for the terms 5, 7, 69, 42 and 17 given by n = 7, 23, 113, 114, and 115 respectively. It is easy to show that nonsquare terms are in [p, q) iff p and q are consecutive primes and q-p > sqrt(q). There are no gaps between consecutive primes greater than sqrt(q) for 127 < q < 4*10^18 (see Nicely's table of maximal prime gaps).

			a(23) = 7 because 23^2 mod 29 = 7.
a(24) = 25 because 24^2 mod 29 = 25.

Chris Boyd, Table of n, a(n) for n = 1..10000
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Tomás Oliveira e Silva, Gaps between consecutive primes

Cf. A257230.

[n^2 mod NextPrime(n): n in [1..80]]; // Vincenzo Librandi, Apr 19 2015

Table[Mod[n^2, NextPrime@ n], {n, 87}] (* Michael De Vlieger, Apr 19 2015 *)
Table[PowerMod[n,2,NextPrime[n]],{n,90}] (* Harvey P. Dale, May 24 2015 *)

PARI
```
a(n)=n^2%nextprime(n+1)
```

A257230 Floor(sqrt(q)-(q-p)), where p and q are consecutive primes.

Original entry on oeis.org

0, 0, 0, -1, 1, 0, 2, 0, -1, 3, 0, 2, 4, 2, 1, 1, 5, 2, 4, 6, 2, 5, 3, 1, 6, 8, 6, 8, 6, -3, 7, 5, 9, 2, 10, 6, 6, 8, 7, 7, 11, 3, 11, 10, 12, 2, 2, 11, 13, 11, 9, 13, 5, 10, 10, 10, 14, 10, 12, 14, 7, 3, 13, 15, 13, 4, 12, 8, 16, 14, 12, 11, 13, 13, 15, 13, 11, 16, 12, 10, 18

Offset: 1

Chris Boyd, Apr 19 2015

Conjecture: a(n) is always positive for n > 30, and is negative only for n = 4, 9 and 30, corresponding to prime pairs (7, 11), (23, 29) and (113, 127).

Related to prime gap conjectures by (e.g.) Legendre, Oppermann, Andrica and Brocard.

			a(30) = -3 because sqrt(127)-(127-113) = -2.73057...
a(31) = 7 because sqrt(131)-(131-127) = 7.44552...

Chris Boyd, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Legendre's conjecture

Cf. A257231.

[Floor(Sqrt(NthPrime(n+1))-(NthPrime(n+1)-NthPrime(n))): n in [1..100]]; // Vincenzo Librandi, Apr 19 2015

Table[Floor[Sqrt[NextPrime[Prime@ p]] - (NextPrime[Prime@ p] - Prime@ p)], {p, 81}] (* Michael De Vlieger, Apr 19 2015 *)

a(n)=floor(sqrt(prime(n+1))-(prime(n+1)-prime(n)))

A240475 Primes that are midway between the closest flanking squarefree numbers.

Original entry on oeis.org

2, 17, 19, 53, 89, 163, 197, 199, 233, 251, 269, 271, 293, 307, 337, 379, 449, 487, 491, 521, 557, 593, 631, 701, 739, 751, 809, 811, 881, 883, 919, 953, 991, 1013, 1049, 1061, 1063, 1097, 1151, 1171, 1279, 1459, 1471, 1493, 1531, 1549, 1567, 1601, 1637

Offset: 1

Chris Boyd, Apr 06 2014

nonn

Primes for which the corresponding A240473(m) is equal to A240474(m).
Primes equal to the average of the closest flanking squarefree numbers.
Primes equal to the average of three consecutive squarefree numbers.
Most terms are such that a(n)+2 and a(n)-2 are the closest squarefree numbers. The first term > 2 for which this is not the case is a(880) = 47527.
494501773, 765921647, 930996623 are the terms < 10^9 that also belong to A176141.

			19 is a term because it is midway between the closest flanking squarefree numbers 17 and 21.
On the other hand, 29 is not a term because it is not midway between the closest flanking squarefree numbers 26 and 30.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A075430, A075432, A166003, A176141, A240473, A240474, A240476.

Select[Mean/@Partition[Select[Range[2000],SquareFreeQ],3,1],PrimeQ] (* Harvey P. Dale, Jul 27 2024 *)

forprime(p=1,1650,forstep(j=p-1,1,-1,if(issquarefree(j),L=j;break));for(j=p+1,2*p,if(issquarefree(j),G=j;break));if(G-p==p-L,print1(p", ")))

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User: Chris Boyd

A323185 Denominators of rationals whose continued fraction representations show the prime factors of n (for n > 1) in nondecreasing order.

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A323184 Numerators of rationals whose continued fraction representations show the prime factors of n (for n>1) in nondecreasing order.

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A266214 Numbers n that are not coprime to the numerator of zeta(2*n)/(Pi^(2*n)).

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A264722 Composite numbers that are less than the average of their closest flanking primes.

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A264721 Composite numbers that are greater than the average of their closest flanking primes.

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Mathematica

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A264720 Numbers that are less than the average of their closest flanking primes.

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Mathematica

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A264719 Numbers that are greater than the average of their closest flanking primes.

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A257231 a(n) = n^2 mod p where p is the least prime greater than n.

A266214 Numbers n that are not coprime to the numerator of zeta(2n)/(Pi^(2n)).