A076274 -id:A076274 - cp's OEIS frontend

A083251 Numbers n such that abs(A045763(n) - A073757(n)) = 2, i.e., signed difference of size of related and unrelated sets to n equals either 2 or -2.

2, 48, 72, 80, 112, 176, 208, 272, 304, 368, 464, 496, 592, 656, 688, 752, 848, 944, 976, 1072, 1136, 1168, 1264, 1328, 1424, 1552, 1616, 1648, 1712, 1744, 1808, 2032, 2096, 2192, 2224, 2384, 2416, 2512, 2608, 2672, 2768, 2864, 2896

Offset: 1

Labos Elemer, May 07 2003

nonn

			For n=2896: d=10 divisors, r=1440 coprimes, u=1447 unrelated or n - u = r + d - 1 = 1449 related numbers to n; thus abs(1449 - 1447) = 2.

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

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 2], Print[n(*, {d, r, u}*)]], {n, 1, 3000}]

a(n) = 8 * (A076274(n-1) + 1) for n > 3, as proved by Lawrence Sze. - Ralf Stephan, Nov 16 2004

A023582 Number of distinct prime divisors of 2*prime(n)-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 3, 2, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 3, 2, 1, 2, 3, 2, 3, 1, 2, 1, 2, 1, 1, 3, 2, 3, 2, 1, 2, 1, 3, 3, 2, 2, 2, 2, 1, 3, 2, 1, 3, 3, 2, 2, 2, 2, 3, 2, 2, 1, 3, 2, 2, 3

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

seq(nops(numtheory:-factorset(2*ithprime(n)-1)),n=1..120); # Muniru A Asiru, Apr 29 2019

PrimeNu[2*Prime[Range[100]]-1] (* Harvey P. Dale, Jan 25 2015 *)

a(n) = omega(2*prime(n)-1); \\ Michel Marcus, Oct 01 2013

a(n) = A001221(A076274(n)). - Michel Marcus, Oct 01 2013

A023583 Greatest prime divisor of 2*prime(n)-1.

Original entry on oeis.org

3, 5, 3, 13, 7, 5, 11, 37, 5, 19, 61, 73, 3, 17, 31, 7, 13, 11, 19, 47, 29, 157, 11, 59, 193, 67, 41, 71, 31, 5, 23, 29, 13, 277, 11, 43, 313, 13, 37, 23, 17, 19, 127, 11, 131, 397, 421, 89, 151, 457, 31, 53, 37, 167, 19, 7, 179, 541, 79, 17, 113, 13, 613, 23, 5

Offset: 1

Clark Kimberling

a(n) = 2*prime(n)-1 if n is in A137288. - Robert Israel, May 19 2020

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

Cf. A006530, A076274, A137288.

f:= n -> max(numtheory:-factorset(2*ithprime(n)-1)):
map(f, [$1..100]); # Robert Israel, May 19 2020

a[n_] := FactorInteger[2*Prime[n]-1][[-1, 1]]; Array[a, 100] (* Amiram Eldar, Oct 27 2024 *)

a(n) = {my(f = factor(2*prime(n)-1)); f[#f~, 1];} \\ Amiram Eldar, Oct 27 2024

a(n) = A006530(A076274(n+1)). - Bernard Schott, May 20 2020

Name edited by Robert Israel, May 19 2020

A141295 Largest m<=n such that all k with 1<=k<=m are divisors of n or coprime to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 5, 5, 3, 11, 7, 13, 3, 5, 5, 17, 3, 19, 5, 5, 3, 23, 8, 9, 3, 5, 5, 29, 3, 31, 5, 5, 3, 9, 7, 37, 3, 5, 5, 41, 3, 43, 5, 5, 3, 47, 8, 13, 3, 5, 5, 53, 3, 9, 5, 5, 3, 59, 7, 61, 3, 5, 5, 9, 3, 67, 5, 5, 3, 71, 9, 73, 3, 5, 5, 13, 3, 79, 5, 5, 3, 83, 7, 9, 3, 5, 5, 89, 3, 13, 5, 5, 3

Offset: 1

Reinhard Zumkeller, Jun 23 2008

nonn

n mod a(n) = 0 or GCD(n,a(n)) = 1;
a(n) = n iff n=1 or n=4 or n is prime; a(A046022(n))=A046022(n);
a(p^2) = 2*p - 1 for odd primes p.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027750, A038566, A076274.

A196127 Union of p-1, 2p-1 and 3p-1 where p is a prime (without repetition).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 18, 20, 21, 22, 25, 28, 30, 32, 33, 36, 37, 38, 40, 42, 45, 46, 50, 52, 56, 57, 58, 60, 61, 66, 68, 70, 72, 73, 78, 81, 82, 85, 86, 88, 92, 93, 96, 100, 102, 105, 106, 108, 110, 112, 117, 121, 122, 126, 128, 130

Offset: 1

Juri-Stepan Gerasimov, Sep 28 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A006093, A076274.

lim:=70: S:={}: p:=1: for n from 1 to lim do p:=nextprime(p): S := S union {p-1,2*p-1,3*p-1}: od: op(1..lim,S); # Nathaniel Johnston, Sep 28 2011

With[{upto=130},Select[{#-1,2#-1,3#-1}&/@Prime[Range[PrimePi[upto]+1]]//Flatten//Union,#<=upto&]] (* Harvey P. Dale, Apr 22 2016 *) (* or *)
Select[Range@130, Or @@ PrimeQ[(#+1)/{1,2,3}] &] (* Giovanni Resta, Apr 30 2019 *)

Entries checked by R. J. Mathar, Sep 28 2011

A229989 Number of primes in the interval [floor(n/2), floor(3n/2)].

Original entry on oeis.org

0, 2, 2, 3, 4, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 13, 14, 14, 14, 15, 16, 16, 16, 17, 18, 18, 18

Offset: 1

Clark Kimberling, Oct 09 2013

Conjectures:
(1) a(n+1) - a(n) = 1 for infinitely many n;
(2) a(n+1) - a(n) = -1 for infinitely many n;
(3) a(n+1) - a(n) = -1 if and only if n = 2*prime(m+1) - 1.

			a(5) = 4 counts the primes in the interval [2,7].

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A076274, A229990, A056172.

with(numtheory): A229989 := proc(n) return pi(floor((3/2)*n))-pi(floor(n/2)-1): end proc: seq(A229989(n), n=1..75); # Nathaniel Johnston, Oct 11 2013

z = 1000; c[n_] := PrimePi[Floor[3 n/2]] - PrimePi[Floor[n/2]-1];
t = Table[c[n], {n, 1, z}];            (* A229989 *)
Flatten[Position[Differences[t], -1]]  (* A076274? *)
Flatten[Position[Differences[t], 1]]   (* A229990 *)

A229990 Numbers k such that the interval [floor((k+1)/2), floor(3(k+1)/2)] contains more primes than the interval [floor(k/2), floor(3k/2)] does.

Original entry on oeis.org

1, 3, 4, 8, 12, 19, 20, 24, 28, 31, 40, 44, 48, 52, 55, 64, 67, 68, 71, 72, 84, 91, 92, 99, 100, 104, 108, 111, 115, 120, 127, 128, 131, 132, 140, 148, 151, 152, 155, 160, 171, 175, 180, 184, 187, 188, 204, 208, 211, 220, 224, 231, 232, 235, 239, 244, 248, 252

Offset: 1

Clark Kimberling, Oct 09 2013

nonn

			4 is in this sequence because [[5/2], [15/2]] contains the primes 2,3,5,7, while [[4/2], [12/2]] contains the primes 2,3,5.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A076274, A229989, A056172.

with(numtheory): isA229990 := proc(n) return pi(floor(3*(n+1)/2))-pi(floor((n+1)/2)-1) > pi(floor(3*n/2))-pi(floor(n/2)-1): end proc: seq(`if`(isA229990(n),n,NULL), n=1..252); # Nathaniel Johnston, Oct 11 2013

z = 1000; c[n_] := PrimePi[Floor[3 n/2]] - PrimePi[Floor[n/2]-1];
t = Table[c[n], {n, 1, z}];            (* A229989 *)
Flatten[Position[Differences[t], -1]]  (* A076274? *)
Flatten[Position[Differences[t], 1]]   (* A229990 *)

A247787 Sum of divisors of 2*prime(n)-1.

Original entry on oeis.org

4, 6, 13, 14, 32, 31, 48, 38, 78, 80, 62, 74, 121, 108, 128, 192, 182, 133, 160, 192, 180, 158, 288, 240, 194, 272, 252, 288, 256, 403, 288, 390, 448, 278, 480, 352, 314, 434, 494, 576, 576, 381, 512, 576, 528, 398, 422, 540, 608, 458, 768, 702, 532, 672

Offset: 1

Jaroslav Krizek, Sep 24 2014

See A005382 (primes p such that sigma(2p-1) = 2p).

			For n = 3; prime(3) = 5, a(3) = sigma(2*5-1) = sigma(9) = 13.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000203, A072057, A005382, A247788.

[SumOfDivisors(2*p-1): p in PrimesUpTo(2000)]

vector(100,n,sigma(2*prime(n)-1)) \\ Derek Orr, Sep 25 2014

a(n) = sigma(A076274(n)) = A000203(A076274(n)).

A247788 Primes p such that sigma(2p-1) = 3*(p-1).

Original entry on oeis.org

3, 17, 131, 193, 449, 13469, 23297, 581150417

Offset: 1

Jaroslav Krizek, Sep 24 2014

Primes p such that A247787(p) = A000203(A076274(p)) = 3*(p-1).

If a(9) exists it must be bigger than 10^10.

			Prime 17 is in sequence because sigma(2*17-1) = sigma(33) = 48 = 3*(17-1).

Cf. A000203, A076274, A247787.

[p: p in PrimesUpTo(20000)| SumOfDivisors(2*p-1) eq 3*p-3]

Select[Prime@ Range[10^5], DivisorSigma[1, 2 # - 1] == 3 (# - 1) &] (* Michael De Vlieger, Jan 03 2017 *)

forprime(p=1,10^7,if(sigma(2*p-1)==3*(p-1),print1(p,", "))) \\ Derek Orr, Sep 25 2014

a(8) from Matthew Campbell, Jan 03 2017

A023585 Least prime divisor of 2*prime(n)-1.

Original entry on oeis.org

3, 5, 3, 13, 3, 5, 3, 37, 3, 3, 61, 73, 3, 5, 3, 3, 3, 11, 7, 3, 5, 157, 3, 3, 193, 3, 5, 3, 7, 3, 11, 3, 3, 277, 3, 7, 313, 5, 3, 3, 3, 19, 3, 5, 3, 397, 421, 5, 3, 457, 3, 3, 13, 3, 3, 3, 3, 541, 7, 3, 5, 3, 613, 3, 5, 3, 661, 673, 3, 17, 3, 3, 733, 5, 757, 3, 3, 13, 3, 19, 3

Offset: 1

Clark Kimberling

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A020639, A076274.

FactorInteger[#][[1,1]]&/@(2Prime[Range[90]]-1) (* Harvey P. Dale, Oct 08 2017 *)

a(n) = factor(2*prime(n)-1)[1, 1]; \\ Michel Marcus, Oct 01 2013

Definition edited by Michel Marcus, Oct 01 2013

cp's OEIS Frontend

A083251 Numbers n such that abs(A045763(n) - A073757(n)) = 2, i.e., signed difference of size of related and unrelated sets to n equals either 2 or -2.

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A023582 Number of distinct prime divisors of 2*prime(n)-1.

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Mathematica

PARI

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A023583 Greatest prime divisor of 2*prime(n)-1.

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A141295 Largest m<=n such that all k with 1<=k<=m are divisors of n or coprime to n.

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A196127 Union of p-1, 2p-1 and 3p-1 where p is a prime (without repetition).

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Maple

Mathematica

Extensions

A229989 Number of primes in the interval [floor(n/2), floor(3n/2)].

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A229990 Numbers k such that the interval [floor((k+1)/2), floor(3*(k+1)/2)] contains more primes than the interval [floor(k/2), floor(3*k/2)] does.

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Maple

Mathematica

A247787 Sum of divisors of 2*prime(n)-1.

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A229990 Numbers k such that the interval [floor((k+1)/2), floor(3(k+1)/2)] contains more primes than the interval [floor(k/2), floor(3k/2)] does.