A136539 -id:A136539 - cp's OEIS frontend

A001770 Numbers k such that 5*2^k - 1 is prime.

2, 4, 8, 10, 12, 14, 18, 32, 48, 54, 72, 148, 184, 248, 270, 274, 420, 1340, 1438, 1522, 1638, 1754, 1884, 2014, 2170, 2548, 2622, 2652, 2704, 13510, 21738, 25624, 41934, 51478, 52540, 53230, 172300, 245728, 350028, 1194164

Offset: 1

N. J. A. Sloane

A084213(a(n)+1) is in A136539, for all n. - Farideh Firoozbakht and M. F. Hasler, Nov 03 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms M = (6a+1)*2^(2m-1)-1 and (6a-1)*2^(2m)-1, Math. Comp., 22 (1968), 420-422.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254 (5*2^n+1 is prime), A050522 (primes of the form 5*2^n - 1).

A001770:=n->`if`(isprime(5*2^n-1),n,NULL): seq(A001770(n), n=1..1000); # Wesley Ivan Hurt, Oct 15 2014

Select[Range[1000], PrimeQ[5*2^# - 1] &] (* Vaclav Kotesovec, Apr 28 2014 *)

is(n)=isprime(5*2^n-1) \\ Charles R Greathouse IV, Feb 07 2017

More terms from Hugo Pfoertner, Jun 23 2004

a(40) from the Wilfrid Keller link by Robert Price, Dec 22 2018

A084213 Binomial transform of A081250.

Original entry on oeis.org

1, 4, 18, 76, 312, 1264, 5088, 20416, 81792, 327424, 1310208, 5241856, 20969472, 83881984, 335536128, 1342160896, 5368676352, 21474770944, 85899214848, 343597121536, 1374389010432, 5497557090304, 21990230458368, 87960926027776

Offset: 0

Paul Barry, May 19 2003

When 5*2^n - 1 is prime, that is, n is in A001770, then a(n+1) is in A136539. - Farideh Firoozbakht and M. F. Hasler, Nov 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (6,-8)

[5*4^n/4-2^n/2+0^n/4: n in [0..30]]; // Vincenzo Librandi, Jun 15 2011

seq(coeff(series((1-2*x+2*x^2)/((1-2*x)*(1-4*x)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 09 2018

Table[If[n==0, 1, 2^(n-2)*(5*2^n - 2)], {n,0,30}] (* G. C. Greubel, Oct 08 2018 *)
CoefficientList[Series[(1 - 2*x + 2*x^2)/((1-2*x)*(1-4*x)), {x, 0, 50}], x] (* or *)
CoefficientList[Series[(5*Exp[4*x] - 2*Exp[2*x] + 1)/4, {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* Stefano Spezia, Oct 11 2018 *)

vector(30, n, n--; (5*4^n - 2^(n+1) + 0^n)/4) \\ G. C. Greubel, Oct 08 2018

a(n) = (5*4^n - 2^(n+1) + 0^n)/4.
G.f.: (1 - 2*x + 2*x^2)/((1-2*x)*(1-4*x)).
E.g.f.: (5*exp(4*x) - 2*exp(2*x) + 1)/4.
a(n+1) = 2^n*(5*2^n - 1) for all n >= 0. - M. F. Hasler, Nov 03 2012

A114930 Numbers n such that phi(n)=2*reversal(n).

Original entry on oeis.org

6180, 27630, 2914830, 4471740, 27000630, 637062480, 27000000630, 679410757980, 4412687534631, 4421625783741

Offset: 1

Farideh Firoozbakht, Jan 29 2006

If m>1 and p=3*10^m+7 is prime then 90*p is in the sequence because phi(90*p)=phi(90)*phi(p)=24*(3*10^m+6)=2*(36*10^m+72) =2*reversal(27*10^m+63)=2*reversal(9*p)=2*reversal(90*p). Note that 30 divides all known terms of this sequence. Next term is greater than 11*10^7.

a(11) > 10^13. - Giovanni Resta, Aug 12 2019

			637062480 is a term because phi(637062480) = 2*84260736 = 2*reversal(637062480).

Cf. A000010, A004086, A069215, A114930, A136539, A114931, A136538.

Do[If[EulerPhi[n]==2*FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 110000000}]

a(6)-a(8) from Giovanni Resta, Oct 28 2012

a(9)-a(10) from Giovanni Resta, Aug 12 2019

A136538 Numbers n such that reversal(n)=2*phi(n).

Original entry on oeis.org

2, 4, 8, 42, 84, 2763, 4032, 8064, 67314, 86558, 291483, 2700063, 2700000063, 4039603962, 46420566582, 6739054689866

Offset: 1

Farideh Firoozbakht, Jan 04 2008

If m>1 and p=3*10^m+7 is prime then n=9*p is in the sequence (the proof is easy). If n is an even term of the sequence and the largest digit of n is less than 5(3) then 2n is (both numbers 2n & 4n are) in the sequence (the proof is easy).

a(17) > 10^13. - Giovanni Resta, Aug 12 2019

			Reversal(42)=24=2*12=2*phi(42), so 42 is in the sequence. [Example corrected Jan 25 2008]

Cf. A000010, A004086, A069215, A114930, A114931, A136539.

Do[If[FromDigits@Reverse@IntegerDigits@n==2*EulerPhi[n], Print[n]],{n,100000000}]

a(13)-a(15) from Giovanni Resta, Oct 28 2012

a(16) from Giovanni Resta, Aug 12 2019

A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

Original entry on oeis.org

34, 568, 147328, 603971584, 9663643648, 39582416502784, 696341272098017608537735168, 765635325572111542783369494684623699968, 3615610599582728119969414707766982030374842621310535527825408, 3791242500068058721125048996612134914443116117566314438843154038784

Offset: 1

Jinyuan Wang, Feb 12 2020

nonn

This is the case h = 8 of the h-perfect numbers as defined in the Harborth link.

			34 is a term since sigma(34) + sigma(8*34) = 612, that is 18*34.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A002236, A050524, A136539, A281993, A330413.

isok(m) = sigma(m) + sigma(8*m) == 18*m;

a(n) = 2^A002236(n) * A050524(n).

A330413 Integers m such that sigma(m) + sigma(16m) = 34m.

Original entry on oeis.org

268, 4336, 69568, 73014378496, 18691696623616, 80280230208715249156096, 5516815412193254337299253840314368, 22596875928343569838211798520159010816, 106710729501573572985208420194451100911225778218295042768896, 7689318425915528602346510723233181380859942271270135051778769275060995751936

Offset: 1

Jinyuan Wang, Feb 12 2020

nonn

This is the case h = 16 of the h-perfect numbers as defined in the Harborth link.

			268 is a term since sigma(268) + sigma(16*268) = 9112, that is 34*268.

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.

Cf. A000203, A000396, A001774, A136539, A281993, A330412.

isok(m) = sigma(m) + sigma(16*m) == 34*m;

a(n) = 2^A001774(n) * (17*2^A001774(n) - 1).

cp's OEIS Frontend

A001770 Numbers k such that 5*2^k - 1 is prime.

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A084213 Binomial transform of A081250.

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A114930 Numbers n such that phi(n)=2*reversal(n).

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A136538 Numbers n such that reversal(n)=2*phi(n).

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A330412 Integers m such that sigma(m) + sigma(8*m) = 18*m.

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A330413 Integers m such that sigma(m) + sigma(16*m) = 34*m.

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A330412 Integers m such that sigma(m) + sigma(8m) = 18m.

A330413 Integers m such that sigma(m) + sigma(16m) = 34m.