A062234 -id:A062234 - cp's OEIS frontend

A346477 Dirichlet inverse of A346476.

1, -1, -1, 2, -3, 5, -3, 2, 8, 13, -9, -2, -9, 17, 11, 8, -15, -8, -15, -12, 19, 37, -17, 18, 8, 41, -4, -12, -27, -33, -25, 20, 37, 61, 25, 56, -33, 65, 35, 38, -39, -45, -39, -42, -36, 77, -41, 32, 32, -20, 53, -42, -47, 96, 35, 58, 61, 109, -57, 132, -55, 109, -48, 56, 43, -121, -63, -72, 71, -109, -69, 56

Offset: 1

Antti Karttunen, Jul 29 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346478.

Cf. also A323910, A346248, A346479.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA346476(n/d).
a(n) = A346478(n) - A346476(n).
a(p) = A252748(p) = A346248(p) = -A346476(p) = -A062234(A000720(p)), for any prime p.

A346478 Sum of A346476 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, -3, 1, 6, 0, -11, 0, 6, 6, -5, 0, -23, 0, -29, 6, 18, 0, -3, 9, 18, -15, -37, 0, -60, 0, -9, 18, 30, 18, 23, 0, 30, 18, 1, 0, -84, 0, -83, -61, 34, 0, -13, 9, -67, 30, -91, 0, 45, 54, 5, 30, 54, 0, 75, 0, 50, -77, -5, 54, -184, 0, -137, 34, -176, 0, -13, 0, 66, -55, -145, 54, -188, 0, -37, 49

Offset: 1

Antti Karttunen, Jul 30 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346477.

Cf. also A323911, A346250, A346480.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];
A346478(n) = (A346476(n)+A346477(n));

a(n) = A346476(n) + A346477(n).

a(1) = 2; and for n > 2, a(n) = -Sum_{d|n, 1A346476(n/d) * A346477(d).

A215808 Primes of the form 2*prime(k) - prime(k+1).

Original entry on oeis.org

3, 3, 17, 41, 47, 67, 151, 167, 199, 227, 251, 257, 347, 367, 557, 587, 601, 607, 641, 647, 727, 941, 971, 1051, 1091, 1097, 1117, 1181, 1217, 1277, 1361, 1427, 1447, 1447, 1487, 1487, 1499, 1607, 1697, 1741, 1747, 1741, 1777, 1877, 1901, 2087, 2143, 2281

Offset: 1

Zak Seidov, Sep 06 2012

nonn

Corresponding values of k: 3, 4, 9, 15, 16, 21, 37, 40, 47, 51, 55, 56, 71, 74, 103 (A216075).

			k=3: 2*5-7=3, k=4: 2*7-11=3, k=9: 2*23-29=17.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A062234, A085704, A094104, A094105, A216075.

pr=Prime[Range[1000]]; s=Select[2*Most[pr]-Rest[pr],PrimeQ]
Select[2#[[1]]-#[[2]]&/@Partition[Prime[Range[500]],2,1],PrimeQ] (* Harvey P. Dale, Feb 25 2017 *)

A364411 a(n) = prime(n) + 2*prime(n+1).

Original entry on oeis.org

8, 13, 19, 29, 37, 47, 55, 65, 81, 91, 105, 119, 127, 137, 153, 171, 181, 195, 209, 217, 231, 245, 261, 283, 299, 307, 317, 325, 335, 367, 389, 405, 415, 437, 451, 465, 483, 497, 513, 531, 541, 563, 577, 587, 595, 621, 657, 677, 685, 695, 711, 721, 743, 765, 783

Offset: 1

Paul Curtz, Jul 23 2023

All terms > 8 are odd.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A062234, A094105, A100484, A191472 (first differences), A210497.

ListConvolve[{2,1},Prime[Range[100]]] (* Paolo Xausa, Nov 02 2023 *)

a(n) = a(n-1) + A191472(n-1).
a(n) = A000040(n) + A100484(n+1).
a(n) = A000040(n+1) + A001043(n).

A194581 Primes prime(k) of the form (2*prime(k-1) + prime(k+1))/3.

Original entry on oeis.org

3, 7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 401, 463, 491, 509, 643, 743, 761, 823, 859, 883, 911, 997, 1093, 1237, 1279, 1303, 1429, 1459, 1483, 1489, 1499, 1571, 1609, 1637, 1831, 1873, 1999, 2003, 2069, 2083, 2221, 2239, 2243, 2251, 2269, 2273, 2399

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2011

nonn

Primes prime(k) such that A062234(k) = A062234(k-1). - Thomas Ordowski, Jan 03 2016
Primes prime(k) such that A001223(k) = 2*A001223(k-1). - Robert Israel, Jan 03 2016
Or, primes which are at 1/3 of the distance between the previous and next prime. See A267291 for primes which are at 2/3 between their neighbors. - M. F. Hasler, Jan 12 2016

			a(1)=3 (=(2*2+5)/3), a(2)=7 (=(2*5+11)/3), a(3)=13 (=(2*11+17)/3).

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

Cf. A000040, A001223, A062234.

Primes:= select(isprime, [2,seq(i,i=3..10^4,2)]):
Gaps:= Primes[2..-1]-Primes[1..-2]:
Primes[select(t -> 2*Gaps[t-1] = Gaps[t],[$2..nops(Gaps)])]; # Robert Israel, Jan 03 2016

Table[(2 Prime[k - 1] + Prime[k + 1])/3, {k, 2, 360}] /. {Rational -> Nothing, n /; CompositeQ@ n -> Nothing} (* Michael De Vlieger, Jan 09 2016 *)

for(k=2, 1000, q=2*prime(k-1)+prime(k+1); if(q%3==0 && isprime(q\3), print1(q\3, ", "))) \\ Colin Barker, Jun 27 2014

A194581(n,show=0,o=2,g=0)={forprime(p=o+1,,g*2==(g=-o+o=p)||next; show&&print1(p-g",");n--||return(p-g))} \\ 2nd & 3rd optional args allow printing the whole list and using another starting value. - M. F. Hasler, Jan 12 2016

Entries corrected by R. J. Mathar, Sep 30 2011

A207480 a(n) = (3/2)*(1+prime(n)) - prime(n+1).

Original entry on oeis.org

1, 2, 1, 5, 4, 8, 7, 7, 14, 11, 16, 20, 19, 19, 22, 29, 26, 31, 35, 32, 37, 37, 38, 46, 50, 49, 53, 52, 44, 61, 61, 68, 61, 74, 71, 74, 79, 79, 82, 89, 82, 95, 94, 98, 89, 95, 109, 113, 112, 112, 119, 112, 121, 124, 127, 134, 131, 136, 140, 133, 134, 151

Offset: 2

Zak Seidov, Feb 18 2012

nonn

Conjecture: a(n) > 0 for all n (cf. A062234).
Note that a(1) = 3/2 hence offset is 2.
There are many cases of two successive terms of the same value, the first case is a(8)=a(9)=7: p(8)=19, p(9)=23, p(10)=29, (3/2)*(1+19)-23 = (3/2)*(1+23)-29 = 7.
The first case of 3 equal successive terms is a(691..693)=2588 for corresponding 4 consecutive primes primes p(691..694)= 5189, 5197, 5209, 5227.
The first case of 4 equal successive terms is a(12702874..12702878)=15579672 for corresponding 5 consecutive primes primes p(12702874..12702878)= 231159373,231159389,231159413,231159449,231159503.
Also of interest are cases with a(n)>a(n-1), e.g., a(27..29): 53, 52, 44 (the general tendency is, of course, increasing a(n) with n).

Zak Seidov, Table of n, a(n) for n = 2..1001

Cf. A062234.

a:= n-> 3*(1+ithprime(n))/2-ithprime(n+1):
seq(a(n), n=2..63);  # Alois P. Heinz, Feb 14 2022

(3(#[[1]]+1)/2)-#[[2]]&/@Partition[Prime[Range[2,70]],2,1] (* Harvey P. Dale, Jul 27 2016 *)

a(n) = my(p=prime(n)); (3/2)*(1+p) - nextprime(p+1); \\ Michel Marcus, Feb 14 2022

A360510 a(n) = Product_{i=2..n} p(i) - p(n+1)^2, where p(i) is the i-th prime.

Original entry on oeis.org

-8, -22, -34, -16, 986, 14726, 254894, 4849316, 111545594, 3234845654, 100280243696, 3710369065724, 152125131761756, 6541380665832806, 307444891294242896, 16294579238595018884, 961380175077106315814, 58644190679703485487146, 3929160775540133527934504, 278970415063349480483702366

Offset: 1

N. J. A. Sloane, Feb 24 2023

sign

It is known that a(n) >= 0 for n >= 5.
Remember an empty product equals 1 by convention.
See A064819 for another version.

S. W. Golomb, Elementary Problem E3137, Amer. Math. Monthly, Proposed 93 (1986), p. 215; Solution and Editorial Comments, 94 (1987), 883-884.

Paolo Xausa, Table of n, a(n) for n = 1..300

Cf. A062234, A064819, A360511.

FoldList[Times, 1, Most[#]] - #^2 & [Prime[Range[2, 25]]] (* Paolo Xausa, Nov 06 2024 *)

from sympy import prime, primorial
def A360510(n): return (primorial(n)>>1)-prime(n+1)**2 # Chai Wah Wu, Feb 24 2023

A360511 a(n) = Product_{i=1..n} p(i) - p(n+1)^3, where p(i) is the i-th prime.

Original entry on oeis.org

-25, -119, -313, -1121, 113, 25117, 503651, 9687523, 223068481, 6469663439, 200560439477, 7420738065889, 304250263447703, 13082761331566207, 614889782588342533, 32589158477189839351, 1922760350154212412089, 117288381359406970682507, 7858321551080267055521179, 557940830126698960967026373

Offset: 1

N. J. A. Sloane, Feb 24 2023

sign

It is known that a(n) >= 0 for n >= 5.

S. W. Golomb, Elementary Problem E3137, Amer. Math. Monthly, Proposed 93 (1986), p. 215; Solution and Editorial Comments, 94 (1987), 883-884.

Paolo Xausa, Table of n, a(n) for n = 1..300

Cf. A062234, A064819, A360510.

FoldList[Times, Most[#]] - Rest[#]^3 & [Prime[Range[25]]] (* Paolo Xausa, Nov 06 2024 *)

from sympy import prime, primorial
def A360511(n): return primorial(n)-prime(n+1)**3 # Chai Wah Wu, Feb 24 2023

A110970 Squares of the form 2*prime(n) - prime(n+1).

Original entry on oeis.org

1, 9, 25, 81, 225, 361, 441, 1089, 1225, 2025, 2601, 3249, 3721, 5041, 7569, 7921, 12321, 13689, 15129, 18225, 21609, 30625, 31329, 38809, 42025, 47961, 53361, 59049, 65025, 77841, 88209, 91809, 94249, 99225, 110889, 123201, 126025, 131769

Offset: 1

Giovanni Teofilatto, Sep 27 2005

nonn

How is the upper limit for the search determined, which ensures that a square difference not previously made does not occur with much larger values? - Hugo Pfoertner, Mar 02 2020

Cf. A110975, A062234.

Contains n^2 for n in A086381.

Union[Select[(2Prime[ # ] - Prime[ # + 1]) & /@ Range[13000], IntegerQ[Sqrt[ # ]] &]] (* Ray Chandler, Oct 07 2005 *)

Corrected and extended by Ray Chandler, Oct 07 2005

A120633 Number of composite numbers bounded inclusively between p(n+1) and 2*p(n) where p(x) is prime(x).

Original entry on oeis.org

1, 1, 3, 2, 7, 7, 12, 12, 13, 22, 19, 25, 31, 31, 33, 37, 45, 44, 51, 56, 55, 61, 63, 66, 75, 80, 81, 87, 88, 82, 101, 103, 111, 105, 121, 120, 124, 132, 134, 140, 148, 142, 158, 158, 164, 156, 165, 182, 188, 188, 189, 199, 193, 205, 210

Offset: 1

Lekraj Beedassy, Jun 21 2006

nonn

Table[Count[Range[Prime[n+1],2Prime[n]],?CompositeQ],{n,60}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Oct 01 2019 *)

a(n) = A062234(n) - A070046(n) + 1 = A120632(n) - A040976(n+1).

Definition clarified by Harvey P. Dale, Oct 01 2019

cp's OEIS Frontend

A346477 Dirichlet inverse of A346476.

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A346478 Sum of A346476 and its Dirichlet inverse.

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A215808 Primes of the form 2*prime(k) - prime(k+1).

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A364411 a(n) = prime(n) + 2*prime(n+1).

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A194581 Primes prime(k) of the form (2*prime(k-1) + prime(k+1))/3.

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A207480 a(n) = (3/2)*(1+prime(n)) - prime(n+1).

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A360510 a(n) = Product_{i=2..n} p(i) - p(n+1)^2, where p(i) is the i-th prime.

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A360511 a(n) = Product_{i=1..n} p(i) - p(n+1)^3, where p(i) is the i-th prime.

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