A079363 -id:A079363 - cp's OEIS frontend

A003307 Numbers k such that 2*3^k - 1 is prime.

1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104

Offset: 1

N. J. A. Sloane

R. K. Guy, Unsolved Problems in Theory of Numbers, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
Steven Harvey, Williams Primes
W. Keller and J. Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.
H. C. Williams and C. R. Zarnke, Some prime numbers of the forms 2*3^n+1 and 2*3^n-1, Math. Comp., 26 (1972), 995-998.

Cf. A002235, A046865, A079906, A046866, A001771, A005541, A056725, A046867, A079907.

Cf. A079363 (primes of the form 2*3^k - 1), A003306 (k such that 2*3^k + 1 is prime).

for(n=1,1e4,if(ispseudoprime(2*3^n-1),print1(n", "))) \\ Charles R Greathouse IV, Jul 16 2011

More terms from Douglas Burke (dburke(AT)nevada.edu)
More terms from T. D. Noe, Aug 24 2005
Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(35) from Borys Jaworski, Sep 02 2011
a(36) from Borys Jaworski, Feb 13 2012
a(37) from Jeppe Stig Nielsen, Sep 28 2018

A163667 Numbers n such that sigma(n) = 9*phi(n).

Original entry on oeis.org

30, 264, 714, 3080, 3828, 6678, 10098, 12648, 21318, 22152, 24882, 44660, 49938, 61344, 86304, 94944, 118296, 129504, 130356, 147560, 183396, 199386, 201756, 207264, 216936, 248710, 258440, 265914, 275196, 290290, 321204, 505164, 628776, 706266, 706836

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Aug 09 2009

This sequence is a subsequence of A011257 because sqrt(phi(n)*sigma(n)) = 3*phi(n).

If 2^p-1 and 2*3^k-1 are two primes greater than 5 then n = 2^(p-2)*(2^p-1)*3^(k-1)*(2*3^k-1) (the product of two relatively prime terms 2^(p-2)*(2^p-1) and 3^(k-1)*(2*3^k-1) of A011257) is in the sequence. The proof is easy.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Cf. A000010, A000043, A000203, A000668, A003307, A011257, A079363.

Select[Range[700000],DivisorSigma[1,# ]==9EulerPhi[ # ]&]

is(n)=sigma(n)==9*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A120378 Integers n such that 2*11^n-1 is prime.

Original entry on oeis.org

2, 8, 248, 2474, 2900, 6600, 24746, 105704

Offset: 1

Walter Kehowski, Jun 28 2006

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 241 is 181 in base 12.

a(9) > 2*10^5. - Robert Price, Nov 06 2015

			a(1)=2 since 2*11^2-1=241 is the first prime of this form.

Cf. A000043, A000668, A002957, A002959, A003307, A079363, A055558.

for w to 1 do for k from 1 to 2000 do n:=2*11^k-1; if isprime(n) then printf("%d, %d",k,n) fi od od;

Select[Range[0, 200000], PrimeQ[2*11^# - 1] &] (* Robert Price, Nov 06 2015 *)

a(n) = n-th integer k such that 2*11^k-1 is prime.

More terms from Ryan Propper, Jan 14 2008

a(7)-a(8) from Robert Price, Nov 06 2015

A120375 Integers k such that 2*5^k - 1 is prime.

Original entry on oeis.org

4, 6, 16, 24, 30, 54, 96, 178, 274, 1332, 2766, 3060, 4204, 17736, 190062, 223536, 260400, 683080

Offset: 1

Walter Kehowski, Jun 28 2006

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 1249 is 881 in base 12.

a(16) > 2*10^5. - Robert Price, Mar 14 2015

			a(1) = 4 since 2*5^4 - 1 = 1249 is the first prime.

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), this sequence (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).
Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), A120376 (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958.

[n: n in [0..2800] |IsPrime(2*5^n - 1)]; // Vincenzo Librandi, Sep 23 2018

for w to 1 do for k from 1 to 2000 do n:=2*5^k-1; if isprime(n) then printf("%d, %d ",k,n) fi od od;

Select[Range[0, 100], PrimeQ[2*5^# - 1] &] (* Robert Price, Mar 14 2015 *)

isok(k) = ispseudoprime(2*5^k-1); \\ Altug Alkan, Sep 22 2018

a(n) = 2*A002958(n).

More terms from Ryan Propper, Mar 28 2007
a(14) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 02 2007
a(15) from Robert Price, Mar 14 2015
a(16)-a(18) from Jorge Coveiro and Tyler NeSmith, Jun 14 2020

A234503 Number of ways to write n = k + m with k > 0 and m > 0 such that 3^(phi(k)/2 + phi(m)/12) + 2 is prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 4, 4, 2, 3, 2, 1, 3, 4, 8, 3, 4, 4, 4, 6, 3, 4, 6, 3, 5, 5, 3, 2, 2, 6, 5, 3, 2, 3, 7, 4, 3, 4, 4, 3, 4, 4, 4, 5, 2, 5, 2, 6, 5, 7, 3, 5, 7, 6, 13, 5, 7, 7, 10, 6, 8, 8, 9, 6, 7, 8, 6, 6, 5, 7, 9, 6, 7, 8, 10

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

It might seem that a(n) > 0 for all n > 14, but a(43905) = 0. If a(n) > 0 infinitely often, then there are infinitely many primes of the form 3^m + 2.
Similarly, it might seem that for n > 26 there is a positive integer k < n such that m = phi(k)/2 + phi(n-k)/12 is an integer with 3^m - 2 prime, but n = 41213 is a counterexample.
See also A234451 and A236358 for similar sequences.

			a(15) = 1 since 15 = 1 + 14 with 3^(phi(1)/2 + phi(14)/12) + 2 = 3 + 2 = 5 prime.
a(23) = 1 since 23 = 10 + 13 with 3^(phi(10)/2 + phi(13)/12) + 2 = 3^3 + 2 = 29 prime.
a(24) = 1 since 24 = 3 + 21 with 3^(phi(3)/2 + phi(21)/12) + 2 = 3^2 + 2 = 11 prime.
a(37) = 1 since 37 = 9 + 28 with 3^(phi(9)/2 + phi(28)/12) + 2 = 3^4 + 2 = 83 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Cf. A000010, A000040, A000244, A014232, A057735, A079363, A111974, A234344, A234346, A234347, A234361, A234451, A234470, A234475, A234504, A236358.

f[n_,k_]:=3^(EulerPhi[k]/2+EulerPhi[n-k]/12)+2
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A120377 Primes of the form 2*11^k-1.

Original entry on oeis.org

241, 428717761

Offset: 1

Walter Kehowski, Jun 28 2006

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 241 is 181 in base 12.

The values of k < 1000 that yield primes are 2, 8, 248. - T. D. Noe, Nov 16 2006

			a(1) = 241 since 2*11^2-1 = 241 is the first prime.

Amiram Eldar, Table of n, a(n) for n = 1..3

Cf. A000043, A000668, A002957, A002959, A003307, A055558, A079363, A120378.

for w to 1 do for k from 1 to 2000 do n:=2*11^k-1; if isprime(n) then printf("%d, %d",k,n) fi od od;

Select[2*11^Range[1000]-1, PrimeQ] (* T. D. Noe, Nov 16 2006 *)

a(n) = n-th number such that 2*11^k-1 that is prime for some k.

a(n) = 2*11^A120378(n)-1. - R. J. Mathar, Mar 06 2010

Corrected by T. D. Noe, Nov 16 2006

A236358 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2*3^m + 1 prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 0, 2, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 3, 2, 3, 2, 1, 4, 1, 4, 1, 4, 5, 3, 5, 7, 7, 8, 5, 5, 4, 4, 7, 7, 4, 7, 3, 6, 4, 5, 5, 6, 7, 6, 4, 5, 7, 6, 9, 5, 8, 7, 7, 4, 6, 5, 4, 6, 9, 8, 3, 6, 8, 9, 8, 8, 7, 8, 8, 9, 8, 4, 7, 4, 7, 7, 5, 4, 8, 6, 6, 7, 11

Offset: 1

Zhi-Wei Sun, Jan 23 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 26.
(ii) For any integer n > 37, there is a positive integer k < n such that m = phi(k)/2 + phi(n-k)/12 is an integer with 2*3^m - 1 prime.
We have verified both parts for n up to 50000. Clearly, part (i) implies that there are infinitely many positive integers m with 2*3^m + 1 prime, while part (ii) implies that there are infinitely many positive integers m with 2*3^m - 1 prime.

			 a(36) = 1 since phi(15)/2 + phi(21)/12 = 4 + 1 = 5 with 2*3^5 + 1 = 487 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000244, A079363, A111974, A234451, A234503, A234504.

p[n_]:=IntegerQ[n]&&PrimeQ[2*3^n+1]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A079362 Sequence of sums of alternating powers of 3.

Original entry on oeis.org

1, 4, 5, 14, 17, 44, 53, 134, 161, 404, 485, 1214, 1457, 3644, 4373, 10934, 13121, 32804, 39365, 98414, 118097, 295244, 354293, 885734, 1062881, 2657204, 3188645, 7971614, 9565937, 23914844, 28697813, 71744534, 86093441, 215233604

Offset: 1

Cino Hilliard, Feb 15 2003

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A079360, A079363, A028242, A048473 (bisection).

a:=[1,4,5];; for n in [4..30] do a[n]:=a[n-1]+3*a[n-2]-3*a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

I:=[1,4,5]; [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..40]]; // G. C. Greubel, Aug 07 2019

a[1]:=1:a[2]:=4:for n from 3 to 100 do a[n]:=3*a[n-2]+2 od: seq(a[n], n=1..33); # Zerinvary Lajos, Mar 17 2008

LinearRecurrence[{1,3,-3},{1,4,5},40] (* Harvey P. Dale, Oct 18 2016 *)

a(n)=if(n<1,0,1+sum(k=2,n,3^((k\2)-(k%2))))

a(n)=if(n<0,0,(5/3-3*n%2)*2^ceil(n/2)-1)

@CachedFunction
def a(n):
    if (n==0): return 1
    elif (1<=n<=2): return n+3
    else: return a(n-1) + 3*a(n-2) - 3*a(n-3)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 07 2019

G.f.: x*(1+3*x-2*x^2)/((1-x)*(1-3*x^2)). - Michael Somos, Feb 18 2003

For n >= 1, a(2n-1) = (2/3)*3^n - 1, a(2n) = (5/3)*3^n - 1. - Benoit Cloitre, Feb 16 2003

A120376 Primes of the form 2*5^k - 1.

Original entry on oeis.org

1249, 31249, 305175781249, 119209289550781249, 1862645149230957031249, 111022302462515654042363166809082031249, 25243548967072377773175314089049159349542605923488736152648925781249

Offset: 1

Walter Kehowski, Jun 28 2006

nonn

See comments for A057472. Examined in base 12, all n must be even and all primes must be 1-primes. For example, 1249 is 881 in base 12.

The next term has 125 digits. - Harvey P. Dale, Jan 26 2019

			a(1) = 4 since 2*5^4 - 1 = 1249 is the first prime.

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), A120375 (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).
Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), this sequence (b=5), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958.

for w to 1 do for k from 1 to 2000 do n:=2*5^k-1; if isprime(n) then printf("%d, %d",k,n) fi od od;

Select[2*5^Range[100]-1,PrimeQ] (* Harvey P. Dale, Jan 26 2019 *)

for(k=1, 1e3, if(ispseudoprime(p=2*5^k-1), print1(p, ", "))); \\ Altug Alkan, Sep 22 2018

a(n) = 2*5^A120375(n) - 1 = 2*5^(2*A002958(n)) - 1. - Jianing Song, Sep 22 2018

A293356 Even integers k such that lambda(sum of even divisors of k) = sum of odd divisors of k.

Original entry on oeis.org

2, 20, 40, 48, 68, 176, 212, 304, 328, 944, 1360, 1712, 1888, 2320, 2344, 2864, 4240, 7120, 7888, 7984, 8448, 8960, 11920, 12032, 14416, 14592, 15536, 17492, 20224, 21520, 23984, 24208, 24592, 25904, 26112, 28160, 29440, 30464, 34560, 35920, 36352, 40528, 41296

Offset: 1

Michel Lagneau, Oct 07 2017

nonn

Or even integers k such that A002322(A146076(k)) = A000593(k).
Observations:
The primes a(n)/4: {5, 17, 53, 4373, 13121, ...} are of the form 2*3^m - 1, m > 0 (A079363).
The primes a(n)/8: {5, 41, 293, 4941257, ...} are of the form 6*7^m - 1, m = 0, 1, ... (primes in A198688).
The set of the primes {a(n)/16} = {3, 11, 19, 59, 107, 179, 499, 971, 1499, 1619, ...} contains the primes of the form 4*3^(2m+1) - 1 = {11, 107, 971, ...}, m = 0, 1, ...

			68 is in the sequence because A002322(A146076(68)) = A002322(108) = 18 and A000593(68) = 18.

Robert Israel, Table of n, a(n) for n = 1..738

Cf. A000593, A000668, A002322, A146076, A281707.

with(numtheory):
for n from 2 by 2 to 10^6 do:
x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
   for k from 1 to n1 do:
    if type(x[k],even)
     then
     s0:=s0+ x[k]:
     else
     s1:=s1+ x[k]:
    fi:
  od:
    if s1=lambda(s0)
     then
     printf(`%d, `,n):
     else
    fi:
od:

fQ[n_] :=
Block[{d = Divisors@n},
  CarmichaelLambda[Plus @@ Select[d, EvenQ]] ==
Plus @@ Select[d, OddQ]]; Select[2 Range@2000, fQ] (* Robert G. Wilson v, Oct 07 2017 *)

is(n)=if(n%2, return(0)); my(s=valuation(n,2),d=sigma(n>>s)); lcm(znstar(d*(2^(s+1)-2))[2])==d \\ Charles R Greathouse IV, Dec 26 2017

Edited by Robert Israel, Dec 28 2017

cp's OEIS Frontend

A003307 Numbers k such that 2*3^k - 1 is prime.

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A163667 Numbers n such that sigma(n) = 9*phi(n).

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A120378 Integers n such that 2*11^n-1 is prime.

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Maple

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A120375 Integers k such that 2*5^k - 1 is prime.

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Magma

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A234503 Number of ways to write n = k + m with k > 0 and m > 0 such that 3^(phi(k)/2 + phi(m)/12) + 2 is prime, where phi(.) is Euler's totient function.

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A120377 Primes of the form 2*11^k-1.

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A236358 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2*3^m + 1 prime}|, where phi(.) is Euler's totient function.

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