A002235 -id:A002235 - cp's OEIS frontend

A230537 Numbers n such that 3^7*2^n - 1 is prime.

1, 2, 6, 13, 22, 29, 30, 33, 36, 50, 61, 118, 180, 226, 405, 433, 522, 789, 929, 960, 1026, 1030, 1118, 1266, 1521, 1718, 2536, 3029, 3366, 4253, 9157, 10165, 23641, 29877, 30648, 47265, 56097, 90501, 101981, 103021, 108370, 117909, 157237, 169156, 174168

Offset: 1

Lei Zhou, Oct 22 2013

Riesel Primes with k = 3^7 = 2187.

Checked up to n = 1000000.

			2187*2^1-1=4373 is a prime number.

Lei Zhou, Table of n, a(n) for n = 1..52
K. Bonath, Riesel Prime Database
C. K. Caldwell, a(51) = 2187*2^449776-1
C. K. Caldwell, a(52) = 2187*2^916686-1
C. K. Caldwell, a(53) = 2187*2^992632-1

Cf. A002235, A002236, A050539, A050566, A050880, A230527.

b=3^7;i=0; Table[While[i++; cp=b*2^i-1; !PrimeQ[cp]]; i, {j, 1, 30}]

is(n)=ispseudoprime(3^7*2^n-1) \\ Charles R Greathouse IV, May 22 2017

Lei Zhou, Nov 08 2013, added a Mathematica program for small elements.

A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.

Original entry on oeis.org

0, 3, 4, 0, 0, 0, 0, 5, 6, 5, 7, 6, 9, 5, 0, 7, 6, 6, 0, 0, 10, 0, 6, 0, 7, 9, 6, 7, 8, 0, 17, 8, 0, 0, 7, 0, 0, 18, 0, 0, 0, 8, 0, 10, 8, 9, 18, 0, 0, 7, 0, 0, 8, 12, 0, 7, 0, 11, 16, 0, 21, 0, 0, 0, 8, 14, 0, 0, 18, 9, 10, 8, 77, 0, 0, 0, 12, 8, 0, 11, 18, 0

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 05 2014

nonn

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:
For k = 0: 2, 3, 5, 7, 13, 17, ...   Intersection of A000043 and A000043
for k = 1: 3, 4, 6, 94, ...          Intersection of A050414 and A002235
for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770
for k = 3:                           Intersection of A059609 and A001771
for k = 4: 21, ...                   Intersection of A059610 and A002236
for k = 5:                           Intersection of A096817 and A001772
for k = 6:                           Intersection of A096818 and A001773
for k = 7: 5, 10, 14, ...            Intersection of A059612 and A002237
for k = 8: 6, 16, 20, 36, ...        Intersection of A059611 and A001774
for k = 9: 5, 21, ...                Intersection of A096819 and A001775
for k = 10: 7, 13, ...               Intersection of A096820 and A002238
for k = 11: 6, 8, 12, ...
for k = 12: 9, ...
for k = 13: 5, 8, 10, ...

			a(1) = 3 because 2^3 - (2*1+1) = 5 and (2*1+1)*2^3 - 1 = 23 are both prime, 3 = 2*1+1,
a(2) = 4 because 2^4 - (2*2+1) = 11 and (2*2+1)*2^4 - 1 = 79 are both prime, 4 < 2*2+1 = 5.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A238748, A238904 (smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists).

a[n_] := Catch@ Block[{k = 1}, While[k <= 2*n+1, If[2^k - (2*n + 1) > 0 && PrimeQ[2^k - (2*n+1)] && PrimeQ[(2*n + 1)*2^k-1], Throw@k]; k++]; 0]; a/@ Range[0, 80] (* Giovanni Resta, Mar 15 2014 *)

a(0), a(19), a(20) corrected by Giovanni Resta, Mar 13 2014

A245241 Integers n such that 6 * 7^n + 1 is prime.

Original entry on oeis.org

0, 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572

Offset: 1

Robert Price, Nov 14 2014

All terms correspond to verified primes, that is, not merely probable primes.

a(14) > 2*10^5.

			4 is in this sequence because 6 * 7^4 + 1 = 14407, which is prime.

Cf. A003307, A002235, A046865, A079906, A001771, A005541, A056725, A046867, A079907.

Cf. A005537, A005538, A005539, A216889, A216890, A247260.

Select[Range[0,200000], PrimeQ[6 * 7^# + 1] &]

A247202 Smallest odd k > 1 such that k*2^n - 1 is a prime number.

Original entry on oeis.org

3, 3, 3, 3, 7, 3, 3, 5, 7, 5, 3, 5, 9, 5, 9, 17, 7, 3, 51, 17, 7, 33, 13, 39, 57, 11, 21, 27, 7, 213, 15, 5, 31, 3, 25, 17, 21, 3, 25, 107, 15, 33, 3, 35, 7, 23, 31, 5, 19, 11, 21, 65, 147, 5, 3, 33, 51, 77, 45, 17, 69, 53, 9, 3, 67, 63, 43, 63, 51, 27, 73, 5

Offset: 1

Pierre CAMI, Nov 25 2014

nonn

Limit_{N->oo} (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) = log(2). [[Is there a proof or is this a conjecture? - Peter Luschny, Feb 06 2015]]

Records: 3, 7, 9, 17, 51, 57, 213, 255, 267, 321, 615, 651, 867, 901, 909, 1001, 1255, 1729, 1905, 2163, 3003, 3007, 3515, 3797, 3825, 4261, 4335, 5425, 5717, 6233, 6525, 6763, 11413, 11919, 12935, 20475, 20869, 25845, 30695, 31039, 31309, 42991, 55999, ... . - Robert G. Wilson v, Feb 08 2015

Robert G. Wilson v, Table of n, a(n) for n = 1..10031 (first 5000 terms from Pierre CAMI)

Cf. A126715, A247479.

f:= proc(n)
local k,p;
  p:= 2^n;
for k from 3 by 2 do if isprime(k*p-1) then return k fi od;
end proc:
seq(f(n), n=1 .. 100); # Robert Israel, Feb 05 2015

f[n_] := Block[{k = 3, p = 2^n}, While[ !PrimeQ[k*p - 1], k += 2]; k]; Array[f, 70]

a(n) = {k=3; while (!isprime(k*2^n-1), k+=2); k;} \\ Michel Marcus, Nov 25 2014

a(A002235(n)) = 3.

A066679 Numbers n such that sigma(n) is congruent to n mod phi(n).

Original entry on oeis.org

1, 2, 6, 10, 12, 44, 90, 184, 440, 528, 588, 672, 752, 3796, 8928, 9888, 12224, 35640, 37680, 49024, 50976, 89152, 94200, 108192, 146412, 159840, 279864, 1734720, 2554368, 2977920, 12580864, 14239872, 16544880, 28321920, 41362200, 56976480, 60610624

Offset: 1

Joseph L. Pe, Jan 11 2002

nonn

Up to 1.5*10^8 there exist 43 terms of the sequence. - Farideh Firoozbakht, Apr 15 2006

If p=3*2^n-1 is an odd prime then m=2^n*p is in the sequence. Proof: sigma(m)-m=(2^(n+1)-1)*(p+1)-2^n*p=2*(2^(n-1)*(p-1))= 2*phi(m), so sigma(m)=m mod(phi(m)). Hence for n>0, 2^A002235(n)* (3*2^A002235(n)-1) is in the sequence and 2^164987*(3*2^164987-1) is the largest known term of the sequence. - Farideh Firoozbakht, Apr 15 2006

			sigma(10) = 18 is congruent to 10 mod phi(10) = 4, so 10 is a term of the sequence.

Jud McCranie, Table of n, a(n) for n = 1..100 (First 71 terms from Donovan Johnson, a(72)-a(93) from Giovanni Resta).
Douglas E. Iannucci, On the Equation sigma(n) = n + phi(n), Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.2.

Cf. A000010, A002235.

Select[ Range[ 1, 10^5 ], Mod[ DivisorSigma[ 1, # ], EulerPhi[ # ] ] == Mod[ #, EulerPhi[ # ] ] & ]

is(n)=sigma(n)==Mod(n,eulerphi(n)) \\ Charles R Greathouse IV, Feb 19 2013

More terms from Jason Earls, Jan 14 2002

More terms from Farideh Firoozbakht, Apr 15 2006

A091997 Numbers n such that 3*2^(n-1) - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 12, 19, 35, 39, 44, 56, 65, 77, 95, 104, 144, 207, 217, 307, 325, 392, 459, 471, 828, 1275, 3277, 4205, 5135, 7560, 12677, 14899, 18124, 18820, 25691, 26460, 41629, 51388, 71784, 80331, 85688, 88172, 97064, 123631, 155931, 164988, 234761, 414841

Offset: 1

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 17 2004

nonn

			3*2^(3-1) - 1 = 11 so a(1) = 3.

Eric Weisstein's World of Mathematics, Thabit ibn Kurrah's rules

Cf. A002235.

Do[If[PrimeQ[3*2^(n - 1) - 1], Print[n]], {n, 1, 8000}] (* Mohammed Bouayoun (Mohammed.Bouayoun(AT)yahoo.fr), Apr 13 2006 *)

for (i=1,1000,if(isprime(3*2^(i-1)-1),print1(i,",")))

a(n) = A002235(n) + 1. - Jinyuan Wang, Jan 30 2020

More terms from Mohammed Bouayoun (Mohammed.Bouayoun(AT)yahoo.fr), Apr 13 2006

More terms from Jinyuan Wang, Jan 30 2020

A262994 Smallest number k>2 such that k*2^n-1 is a prime number.

Original entry on oeis.org

3, 3, 3, 3, 4, 3, 3, 5, 7, 5, 3, 5, 9, 5, 4, 8, 4, 3, 28, 14, 7, 26, 13, 39, 22, 11, 16, 8, 4, 20, 10, 5, 6, 3, 24, 12, 6, 3, 25, 24, 12, 6, 3, 14, 7, 20, 10, 5, 19, 11, 21, 20, 10, 5, 3, 32, 16, 8, 4, 17, 24, 12, 6, 3, 67, 63, 43, 63, 40, 20, 10, 5, 15, 12, 6, 3

Offset: 1

Pierre CAMI, Oct 07 2015

nonn

If k=2^j then n+j is a Mersenne exponent.

a(n)=3 if and only if 3*2^n-1 is a prime; that is, n belongs to A002235. - Altug Alkan, Oct 08 2015

			3*2^1-1=5 prime so a(1)=3;
3*2^2-1=11 prime so a(2)=3;
3*2^3-1=23 prime so a(3)=3.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A247202, A002235.

a[n_] := For[k = 3, True, k++, If[PrimeQ[k*2^n - 1], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 07 2015 *)

a(n) = {k=3; while (! isprime(k*2^n-1), k++); k;} \\ Michel Marcus, Oct 08 2015

A272057 Numbers n such that 3*4^n - 1 is prime.

Original entry on oeis.org

1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859

Offset: 1

Tim Johannes Ohrtmann, Apr 19 2016

These are Williams primes to base 3.

Half of the even terms of A002235.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A002235, A000043, A003307, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907.

Select[Range[0, 100000], PrimeQ[3*4^# - 1] &]

for(n=1,10000, if(isprime(3*4^n-1), print1(n,", ")))

a(25) corrected and a(33)-a(36) added by Giovanni Resta, Apr 19 2016, using data from A002235.

A115747 Numbers n such that phi(n) + sigma(n) = 5/2*n.

Original entry on oeis.org

18, 20, 88, 368, 1504, 24448, 98048, 5238976, 25161728, 2730992944, 33995232256, 412316336128, 1391737114624, 7732492570624

Offset: 1

Farideh Firoozbakht, Feb 12 2006

If p = 3*2^(m-1)-1 is an odd prime then 2^m*p is in the sequence because phi(2^m*p) = 2^(m-1)*(3*2^(m-1)-2), sigma(2^m*p) = (2^(m+1)-1)*(3*2^(m-1)) so phi(2^m*p)+sigma(2^m*p) = 2^(m-1)*(3* 2^(m-1)-2)+(2^(m+1)-1)*(3*2^(m-1)) = 3*2^(2m-2)-2^m+3*2^(2m)-3*2^ (m-1) = 2^(m-1)*(3*2^(m-1)-2+3*2^(m+1)-3) = 2^(m-1)*(3*5*2^(m-1)-5) = 5/2*2^m*(3*2^(m-1)-1) = 5/2*(2^m*p). Except 18 & 5238976 all known terms of the sequence are of the form 2^m*(3*2^(m-1)-1), where (3*2^(m-1)-1) is prime.

a(15) > 10^13. - Giovanni Resta, Jul 13 2015

			25161728 is in the sequence because phi(25161728) + sigma(25161728) = 12578816 + 50325504 = 5/2*25161728.

Cf. A002235.

Do[If[DivisorSigma[1,n]+EulerPhi[n]==5/2*n,Print[n]],{n,200000000}]

isok(n) = eulerphi(n) + sigma(n) == 5*n/2; \\ Michel Marcus, Jul 14 2015

a(10)-a(12) from Donovan Johnson, Feb 29 2012
a(13) from Donovan Johnson, Apr 04 2012
a(14) from Giovanni Resta, Jul 13 2015

A164523 Nonnegative numbers n such that 6*2^n-1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 10, 17, 33, 37, 42, 54, 63, 75, 93, 102, 142, 205, 215, 305, 323, 390, 457, 469, 826, 1273, 3275, 4203, 5133, 7558, 12675, 14897, 18122, 18818, 25689, 26458, 41627, 51386, 71782, 80329, 85686, 88170, 97062

Offset: 1

Vincenzo Librandi, Aug 15 2009

nonn

The associated primes are in A007505.

			n=0 is in the sequence because 6*2^0-1=5 is prime. n=1 is in the sequence because 6*2^1-1=11 is prime.

Cf. A157341.

is(n)=ispseudoprime(6*2^n-1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A002235(n+1)-1. - R. J. Mathar, Aug 17 2009

Extended by R. J. Mathar, Aug 17 2009

a(28)-a(43) from Donovan Johnson, Jul 09 2010

cp's OEIS Frontend

A230537 Numbers n such that 3^7*2^n - 1 is prime.

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A238797 Smallest k such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime, k <= 2*n+1, or 0 if no such k exists.

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A245241 Integers n such that 6 * 7^n + 1 is prime.

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A247202 Smallest odd k > 1 such that k*2^n - 1 is a prime number.

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A066679 Numbers n such that sigma(n) is congruent to n mod phi(n).

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A091997 Numbers n such that 3*2^(n-1) - 1 is prime.

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A262994 Smallest number k>2 such that k*2^n-1 is a prime number.

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A238797 Smallest k such that 2^k - (2n+1) and (2n+1)2^k - 1 are both prime, k <= 2n+1, or 0 if no such k exists.