A001770 -id:A001770 - cp's OEIS frontend

A002253 Numbers k such that 3*2^k + 1 is prime.

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353

Offset: 1

N. J. A. Sloane

From Zak Seidov, Mar 08 2009: (Start)
List is complete up to 3941000 according to the list of RB & WK.
So far there are only 4 primes: 2, 5, 41, 353. (End)

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.
H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
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).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
PrimeGrid, 321 Prime Search statistics
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254, A004119, A181565.

See A039687 for the actual primes.

is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<M. F. Hasler, Mar 03 2023

a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023

Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 2009
Edited by N. J. A. Sloane, Mar 13 2009
a(47) and a(48) from the Ballinger & Keller web page, Joerg Arndt, Apr 07 2013
a(49) from https://t5k.org/primes/page.php?id=116922, Fabrice Le Foulher, Mar 09 2014
Terms moved from Data to b-file (Links), and additional term appended to b-file, by Jeppe Stig Nielsen, Oct 30 2020

A002254 Numbers k such that 5*2^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 4687, 5947, 13165, 23473, 26607, 125413, 209787, 240937, 819739, 1282755, 1320487, 1777515

Offset: 1

N. J. A. Sloane

H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
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).

Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681.
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050526.

lst={};Do[If[PrimeQ[5*2^n+1], AppendTo[lst, n]], {n, 0, 2*10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

is(n)=isprime(5*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

Corrected (removed incorrect term 40937) and added more terms (from http://web.archive.org/web/20161028080239/http://www.prothsearch.net/riesel.html), Joerg Arndt, Apr 07 2013

A136539 Numbers n such that n=6*phi(n)-sigma(n).

Original entry on oeis.org

76, 1264, 327424, 5241856, 83881984, 1342160896, 343597121536

Offset: 1

Farideh Firoozbakht, Jan 05 2008, Feb 01 2008

If 5*2^n-1 is prime (that is, n is in A001770) then m = 2^n*(5*2^n-1) is in the sequence. Proof: 6*phi(m)-sigma(m) = 6*2^(n-1)*(5*2^n-2) -(2^(n+1)-1)*5*2^n = 30*2^(2n-1)-6*2^n-5*2^(2n+1)+5*2^n = 5*2^(2n)-2^n = 2^n(5*2^n-1) = m.
The first seven terms of the sequence are of such form, with n=2, 4, 8, 10, 12, 14, 18. Are all terms of the sequence of this form?
a(8) > 10^12. - Giovanni Resta, Nov 03 2012

			6*phi(76)-sigma(76)=6*36-140=76 so 76 is in the sequence.

Cf. A001770, A136540.

Do[If[n==6*EulerPhi[n]-DivisorSigma[1,n],Print[n]],{n,85000000}]

a(n) = 2^k*(5*2^k-1) = A084213(k+1) with k = A001770(n), for n = 1,...,7. - M. F. Hasler, Nov 03 2012

a(7) from Giovanni Resta, Nov 03 2012

A050522 Primes of form 5*2^n-1.

Original entry on oeis.org

19, 79, 1279, 5119, 20479, 81919, 1310719, 21474836479, 1407374883553279, 90071992547409919, 23611832414348226068479, 1784059615882449851322857461811868920478433279

Offset: 1

N. J. A. Sloane, Dec 29 1999

Heiko Harborth, On h-perfect numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 57-62.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

See A001770 for more terms.

Select[Table[5*2^n-1,{n,0,150}],PrimeQ] (* Jean-François Alcover, Mar 21 2011 *)

a(n) = 5*2^A001770(n) - 1. - Max Alekseyev, Feb 20 2024

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

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

A099650 Solutions to x+phi(x) = sigma(x)/2.

Original entry on oeis.org

456, 828, 7584, 33462, 1596048, 1964544, 19800384, 26211264, 31451136, 106805184, 156868224, 316113024, 502349274, 503291904, 1557940992, 8024671392, 8052965376, 11697091872, 22149447168, 87877745664, 443520605184, 626058783744

Offset: 1

Labos Elemer, Nov 05 2004

nonn

If 5*2^n-1 is prime then m=3*2^(n+1)*(5*2^n-1) is in the sequence because m+phi(m)=2^(n+1)*3*(5*2^n-1)+2^(n+1)*(5*2^n-2)=2^(n+1) *(20*2^n-5)=2^(n+1)*5*(2^(n+2)-1)=1/2*4*(2^(n+2)-1)*(5*2^n)= 1/2*sigma(3)*sigma(2^(n+1))*sigma(5*2^n-1)=1/2*sigma(3*2^(n+1) *(5*2^n-1))=1/2*sigma(m). So 3*2^(A001770+1)*(5*2^A001770-1) is a subsequence of this sequence. A110084 is this subsequence. Next term is greater than 10^8. - Farideh Firoozbakht, Aug 04 2005

a(23) > 10^12. - Donovan Johnson, Feb 29 2012

			n=456: phi(456) = 144, sigma(456) = 1200.

Cf. A000010, A000203, A001770, A110084.

Do[If[DivisorSigma[1, m] == 2m + 2 EulerPhi[m], Print[m]], {m, 100000000}] (Firoozbakht)

Two more terms from Farideh Firoozbakht, Aug 04 2005

a(10)-a(22) from Donovan Johnson, Feb 29 2012

A235989 sigma(n) is an additive inverse of n modulo phi(n).

Original entry on oeis.org

1, 2, 6, 10, 12, 28, 76, 120, 312, 588, 672, 888, 1060, 1264, 1656, 14496, 17900, 22896, 44676, 71712, 77688, 95040, 183600, 233088, 327424, 411264, 425376, 446016, 453258, 655776, 1041120, 1253304, 2708640, 5241856, 5468352, 8676576, 9738912, 12536640, 59489184

Offset: 1

Joseph L. Pe, Jan 27 2014

nonn

sigma(10) = 18 is congruent to 2 = -10 mod 4 and phi(10) = 4; so 10 is a term of the sequence.

If p = 5*2^k-1 is a prime, as it happens for k = 2, 4, 8, 10, 12, 14,... (A001770), then n = 2^k*p is in the sequence, since n+sigma(n) = 6*phi(n). - Giovanni Resta, Jan 27 2014

Giovanni Resta, Table of n, a(n) for n = 1..59 (terms < 10^11)

Cf. A001770.

t = {1}; For[i = 1, i <= 10^6, i++; If[Mod[DivisorSigma[1, i] + i, EulerPhi[i]] == 0, AppendTo[t, i]]]; t

isok(n) = !((sigma(n) + n) % eulerphi(n)); \\ Michel Marcus, Jan 27 2014

More terms from Michel Marcus, Jan 27 2014

A238749 Exponents of third Mersenne prime pair: numbers n such that 2^n - 5 and 5*2^n - 1 are both prime.

Original entry on oeis.org

4, 8, 10, 12, 18, 32

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

a(7) > 350028.
Intersection of A059608 and A001770.
Exponents of Mersenne prime pairs {2^n - (2k + 1), (2k + 1)*2^n - 1}:
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, ...
for k = 14:

			a(1) = 4 because 2^4 - 5 = 11 and 5*2^4 - 1 = 79 are both primes.

Cf. A000043, A001770, A059608, A237422, A238694, A238251, A238751, A238797.

[n: n in [0..100] | IsPrime(2^n-5) and IsPrime(5*2^n-1)]; // Vincenzo Librandi, May 17 2015

fQ[n_] := PrimeQ[2^n - 5] && PrimeQ[5*2^n - 1]; k = 1; While[ k < 15001, If[fQ@ k, Print@ k]; k++] (* Robert G. Wilson v, Mar 05 2014 *)
Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5 2^# - 1] &] (* Vincenzo Librandi, May 17 2015 *)

isok(n) = isprime(2^n - 5) && isprime(5*2^n - 1); \\ Michel Marcus, Mar 04 2014

A238751 Lesser prime of third Mersenne prime pair {2^m - 5, 5*2^m - 1}.

Original entry on oeis.org

11, 251, 1019, 4091, 65531, 4294967291

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 04 2014

By comparing A059608 and A001770, the next term, if it exists, is larger than 2^350028. - Giovanni Resta, Mar 06 2014
Lesser prime of Mersenne prime pair of order k and of the form {2^m - (2k - 1), (2k - 1)*2^m - 1}:
for order k = 1: 3, 7, 31, 127, 8191, 131071, ... (Mersenne primes A000668)
for order k = 2: 5, 13, 61, ...
for order k = 3: 11, 251, 1019, 4091, 655531, 4294967291, ... (this sequence)
for order k = 4:
for order k = 5: 2097143, ...
for order k = 6: 3, ...
for order k = 7:
for order k = 8: 17, 1009, 16369, ...
for order k = 9: 47, 65519, 1048559, 68719476719, ...
for order k = 10: 13, 2097133, ...
for order k = 11: 107, 8171, ...
for order k = 12: 41, 233, 4073, ...
for order k = 13: 487, ...
for order k = 14: 5, 229, 997, ...
for order k = 15: 97, ...

			11 is in this sequence because Mersenne prime pair {2^4-(2*3-1) = 11, (2*3-1)*2^4-1 = 79} where 3 is order and 11 is lesser prime (for m = 4).

Cf. A237422, A238694, A238749, A059608, A001770, A156560, A050522.

2^Select[Range[1000], PrimeQ[2^# - 5] && PrimeQ[5*2^# - 1] &] - 5 (* Giovanni Resta, Mar 06 2014 *)

Numbers 2^m - 5 for m belonging to the intersection of A001770 and A059608. - Max Alekseyev, Feb 20 2024

cp's OEIS Frontend

A002253 Numbers k such that 3*2^k + 1 is prime.

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A002254 Numbers k such that 5*2^k + 1 is prime.

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A136539 Numbers n such that n=6*phi(n)-sigma(n).

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A050522 Primes of form 5*2^n-1.

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A084213 Binomial transform of A081250.

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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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A099650 Solutions to x+phi(x) = sigma(x)/2.

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A235989 sigma(n) is an additive inverse of n modulo phi(n).

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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.