A104072 -id:A104072 - cp's OEIS frontend

A157006 Numbers k such that 2^k + 25 is prime.

2, 4, 6, 8, 10, 20, 22, 34, 70, 92, 112, 118, 236, 250, 378, 438, 570, 654, 800, 1636, 2848, 4948, 5670, 6772, 7494, 8006, 9056, 11038, 16268, 21416, 21738, 33370, 78706, 112130, 126446, 164046, 219250, 236432, 368048, 524154, 530810, 640854, 699740, 746302, 754038, 754376, 931976, 989562

Offset: 1

Edwin Dyke (ed.dyke(AT)btinternet.com), Feb 20 2009

nonn

a(40) > 5*10^5. - Robert Price, Oct 15 2015

Since each term is even (n = 2*k), prime numbers of the form 2^k + 25 (see A104072) also have the form 4^k + 25. Those values of k are given in A204388. - Timothy L. Tiffin, Aug 06 2016

			For k = 2, 2^2 + 25 = 29.
For k = 4, 2^4 + 25 = 41.
For k = 6, 2^6 + 25 = 89.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+25, PRP Top Records.

Cf. A094076, A104072, A123250, A204388.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23), this sequence (2^k+25), A157007 (2^k+27), A156982 (2^k+29), A247952 (2^k+31), A247953 (2^k+33), A220077 (2^k+35).

[n: n in [1..1000] | IsPrime(2^n+25)]; // Vincenzo Librandi, Aug 07 2016

Delete[Union[Table[If[PrimeQ[2^n + 25], n, 0], {n, 1, 1000}]], 1]
Select[Range[0, 10000], PrimeQ[2^# + 25] &] (* Vincenzo Librandi, Aug 07 2016 *)

is(n)=ispseudoprime(2^n+5^2) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = 2*A204388(n). - Timothy L. Tiffin, Aug 09 2016

Extended by Vladimir Joseph Stephan Orlovsky, Feb 27 2011
a(29)-a(39) from Robert Price, Oct 15 2015
a(40)-a(48) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 25 2023

A243429 Primes of the form 2^n + 39.

Original entry on oeis.org

41, 43, 47, 71, 103, 167, 1063, 2087, 8231, 131111, 536870951, 8589934631, 549755813927, 8796093022247, 154742504910672534362390567, 40564819207303340847894502572071, 162259276829213363391578010288167, 2722258935367507707706996859454145691687

Offset: 1

Vincenzo Librandi, Jun 05 2014

nonn

Associated n: 1, 2, 3, 5, 6, 7, 10, 11, 13, 17, 29, 33, 39, 43, 87, 105, 107, 131, 253, 329, ....

Vincenzo Librandi, Table of n, a(n) for n = 1..34

Cf. primes of the form 2^n+k: A092506 (k=1), A057733 (k=3), A123250 (k=5), A104066 (k=7), A104070 (k=9), A156940 (k=11), A104067 (k=13), A144487 (k=15), A156973 (k=17), A104068 (k=19), A156983 (k=21), A176922 (k=23), A104072 (k=25), A104071 (k=27), A156974 (k=29), A104069 (k=31), A176926 (k=33), A176927 (k=35), A176924 (k=37), this sequence (k=39), A176925 (k=41), A243430 (k=43), A243431 (k=45), A243432 (k=47), A104073 (k=49).

[a: n in [0..500] | IsPrime(a) where a is 2^n+39];

Select[Table[2^n + 39, {n, 0, 500}], PrimeQ]

A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

Original entry on oeis.org

58, 75, 328, 850, 1210, 2848, 35968, 537088, 549768921088, 8796145451008

Offset: 1

Timothy L. Tiffin, Aug 05 2016

Any term x = a(m) can be combined with any term y = A275701(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2.  Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one.  So far, these two sequences have produced only one amicable pair: (1210,1184) = (a(5),A275701(2)) = (A063990(4),A063990(3)).  If more are ever found, then they will also exhibit x-y = 26.
Notice that:
a(1) =     58 =   29*  2 = (4^1+25)*(4^1)/2
a(3) =    328 =   41*  8 = (4^2+25)*(4^2)/2
a(6) =   2848 =   89* 32 = (4^3+25)*(4^3)/2
a(7) =  35968 =  281*128 = (4^4+25)*(4^4)/2
a(8) = 537088 = 1049*512 = (4^5+25)*(4^5)/2.
If p = 4^k+25 is prime and n = p*(p-25)/2, then it is not hard to show that 2*n - sigma(n) = 26. The values of k in A204388 will guarantee that p is prime (A104072). Similarly, if q = 2*4^k+25 is prime and n = q*(q-25)/2, then 2*n - sigma(n) = 26. However, q will never be prime since it will always be divisible by 3: 2*4^k+25 == (2*1^k+25) mod 3 == 27 mod 3 == 0 mod 3. So, the following values will be in this sequence and provide upper bounds for the next seven terms:
(4^10+25)*(4^10)/2 = 549768921088 >= a(9)
(4^11+25)*(4^11)/2 = 8796145451008 >= a(10)
(4^17+25)*(4^17)/2 = 147573952804424777728 >= a(11)
(4^35+25)*(4^35)/2 = 696898287454081973187748591279228938354688 >= a(12)
(4^46+25)*(4^46)/2 = 12259964326927110866866776279099475433218926722425028608 >= a(13)
(4^56+25)*(4^56)/2 = 13479973333575319897333507543509880240529303896615642871755920375808 >= a(14)
(4^59+25)*(4^59)/2 = 55213970774324510299478046898216207773446358605225195265697257166471168 >= a(15).
The rightmost digit of n = p*(p-25)/2 will always be 8. [Proof: If k is odd, then 4^k+25 == 9 mod 10 and (4^k)/2 == 2 mod 10, which implies that p*(p-25)/2 == 8 mod 10. If k is even, then 4^k+25 == 1 mod 10 and (4^k)/2 == 8 mod 10, which implies that p*(p-25)/2 == 8 mod 10.]
a(10) > 2.3*10^12. - Giovanni Resta, Aug 07 2016
a(11) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018

			a(1) = 58, since 2*58-sigma(58) = 116-90 = 26.
a(2) = 75, since 2*75-sigma(75) = 150-124 = 26.
a(3) = 328, since 2*328-sigma(328) = 656-630 = 26.

D. Alpern, Factorization using the Elliptic Curve Method.

Cf. A033879, A063990, A104072, A204388, A275701 (abundance 26).

[n: n in [1..2*10^6] | (2*n-SumOfDivisors(n)) eq 26]; // Vincenzo Librandi, Aug 06 2016

Select[Range[10^6], 2 # - (DivisorSigma[1, #]) == 26 &] (* Vincenzo Librandi, Aug 06 2016 *)

is(n) = 2*n-sigma(n)==26 \\ Felix Fröhlich, Aug 06 2016

a(9) from Giovanni Resta, Aug 07 2016

a(10) from Hiroaki Yamanouchi, Aug 21 2018

A204388 Numbers k such that 4^k + 25 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 11, 17, 35, 46, 56, 59, 118, 125, 189, 219, 285, 327, 400, 818, 1424, 2474, 2835, 3386, 3747, 4003, 4528, 5519, 8134, 10708, 10869, 16685, 39353, 56065, 63223, 82023, 109625, 118216, 184024, 262077, 265405, 320427, 349870, 373151, 377019, 377188, 465988, 494781

Offset: 1

Juri-Stepan Gerasimov, Jan 15 2012

nonn

a(40) > 2.5*10^5. - Robert Price, Oct 17 2015

Since 4^(6k) + 25 = 4096^k + 25 == (1^k + 25) mod 13 = 26 mod 13 == 0 mod 13, no multiple of 6 will be in this sequence. - Timothy L. Tiffin, Aug 07 2016

			From _Timothy L. Tiffin_, Aug 09 2016: (Start)
 a(1) = 1, since 4^1 + 25 = 4 + 25 = 29, which is prime.
 a(2) = 2, since 4^2 + 25 = 16 + 25 = 41, which is prime.
 a(3) = 3, since 4^3 + 25 = 64 + 25 = 89, which is prime.
 a(4) = 4, since 4^4 + 25 = 256 + 25 = 281, which is prime.
 a(5) = 5, since 4^5 + 25 = 1024 + 25 = 1049, which is prime.
 a(6) = 10, since 4^10 + 25 = 1048576 + 25 = 1048601, which is prime. (End)

Cf. A057196, A104072, A157006.

[n: n in [1..1000] | IsPrime(4^n+25)]; // Vincenzo Librandi, Aug 07 2016

Select[Range[2000], PrimeQ[4^# + 25] &] (* T. D. Noe, Feb 03 2012 *)

for(n=1, 1e5, if(isprime(4^n + 25), print1(n", "))) \\ Altug Alkan, Oct 17 2015

a(n) = A157006(n)/2. - Robert Price, Oct 17 2015

a(22)-a(39) derived from A157006 by Robert Price, Oct 17 2015

a(40)-a(48) derived from A157006 by Elmo R. Oliveira, Nov 28 2023

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A157006 Numbers k such that 2^k + 25 is prime.

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A243429 Primes of the form 2^n + 39.

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A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

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A204388 Numbers k such that 4^k + 25 is prime.

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