A129613 -id:A129613 - cp's OEIS frontend

A050474 Solutions to 2*phi(x) = x+1.

1, 3, 15, 255, 65535, 83623935, 4294967295, 6992962672132095

Offset: 1

Jud McCranie, Dec 24 1999

If n is in the sequence and n+2 is prime then m=n*(n+2) is in the sequence because 2*phi(m) = 2*phi(n*(n+2)) = 2*phi(n)*(n+1) = (n+1)^2 = m+1. We can obtain the terms 3, 15, 255, 65535 & 4294967295 from 1 (the first term) in this way. Also since 83623935 is a term and 83623935+2 is prime 83623935*(83623935+2)=6992962672132095 is in the sequence. So 1 and 83623935 are the only known independent terms and next term of this sequence if it exists is the third such term. - Farideh Firoozbakht, May 01 2007
The next term, if it exists, has at least 7 distinct prime factors (see Beiler, p. 92). - Jud McCranie, Dec 13 2012
From Chris Boyd, Mar 22 2015: (Start)
Solutions to k*phi(x) = x + 1, including a(1) - a(8), were published in 1932 by D. H. Lehmer. In the paper's summing up, "3*5*353*929" (= 4919055) was printed in error; it should have read "3*5*17*353*929" (= 83623935), i.e., a(6). This error has been propagated in several subsequent texts, including Wong's thesis.
Lehmer identified solutions where x has fewer than 7 distinct prime factors. Wong showed that no additional solutions exist unless x has at least 8 distinct prime factors. It appears not to be excluded by either author that an unidentified solution < a(8) with 8 or more distinct prime factors may exist. (End)
There are no other terms below 10^25. - Max Alekseyev, May 04 2024

			2*phi(15) = 2*8 = 15 + 1, so 15 is a member of the sequence.

A. H. Beiler, Recreations in the Theory of Numbers, page 92.

D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
E. Wong, Computations on normal families of primes, Simon Fraser University, 1997, MSc thesis.

Subsequence of A203966.

Cf. A000010, A129613, A129614, A129615, A202855.

[n: n in [1..2*10^6] | 2*EulerPhi(n) eq (n+1)]; // Vincenzo Librandi, Mar 22 2015

Select[Range[700000], (# + 1)== 2 EulerPhi[#] &] (* Vincenzo Librandi, Mar 22 2015 *)

is_A050474(n)=if(2*eulerphi(n)==n+1,1,0) \\ Chris Boyd, Mar 22 2015

A number n is in the sequence iff phi(n^2)=1+2+3+...+n because n is in the sequence <=> 2*phi(n)=n+1 <=> n*phi(n)=n*(n+1)/2 <=> phi(n^2)=1+2+3++...+n. For n=1,2,...,5, a(n)=2^2^(n-1)-1. - Farideh Firoozbakht, Jan 26 2006

A226281 Least number k such that 3^(2^i) + k is prime for i = 0,1,..,n-1.

Original entry on oeis.org

2, 2, 2, 2, 58, 440, 18248, 2024098, 4263330280, 22836544460, 40728071843930

Offset: 1

Robin Garcia, Jun 02 2013

Generalized Fermat primes of the form b^(2^i) + k.

			a(5) = 58 because k = 58 is the minimal k such that  N = 3^(2^i) + k is prime for i = 0, 1, 2 ,3 ,4; N = 61, 67, 139, 6619, 43046779. But 3^(2^5) + 58 is divisible by 37 and three other primes.

Cf. A129613, A225560.

a(11) from Giovanni Resta, Jun 17 2013

A129614 a(n) is the smallest natural number m such that 3^(3^k) + m is prime for k=0,1,...,n.

Original entry on oeis.org

2, 2, 4, 16, 346, 390410, 390410, 34712918410, 40501164004

Offset: 0

Farideh Firoozbakht, May 13 2007, Jun 01 2007

Next term is greater than 2*10^8.

			For k=0,1,...,6 3^3^k + 390410 are prime and 390410 is the smallest number with this property so a(6)=390410.

Cf. A129613, A129615.

A129615 a(n) is the smallest natural number m such that 4^4^k + m is prime for k=0,1,...,n.

Original entry on oeis.org

1, 1, 15, 385, 137593, 106908087

Offset: 0

Farideh Firoozbakht, May 13 2007

Next term is greater than 3*10^8.

			For k=0,1,...,5 4^4^k + 106908087 are prime and 106908087 is the smallest number with this property so a(5)=106908087.

Cf. A129613, A129614.

A226707 Least number k such that 5^(2^i) + k is prime for i = 0,1,..,n-1.

Original entry on oeis.org

2, 6, 6, 48, 384, 1518, 2037018, 2037018, 44279730804, 424427587626

Offset: 1

Robin Garcia, Jun 15 2013

			a(5) = 384 because k = 384 is the minimal k such that  N = 5^(2^i) + k is prime for i = 0, 1, 2, 3, 4; N = 389, 409, 1009, 391009, 152587891009. But 5^(2^5) + 384 is divisible by 797 and another prime.

Cf. A129613, A225560 A226281.

a(10) from Giovanni Resta, Jun 16 2013

A370523 Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.

Original entry on oeis.org

4, 7, 9, 15, 21, 33, 45, 63, 75, 105, 153, 183, 195, 243, 273, 285, 435, 525, 573, 603, 813, 825, 1065, 1233, 1305, 1623, 2145, 2595, 2715, 2805, 3375, 3465, 3933, 4023, 4245, 4275, 4653, 4803, 4935, 5655, 6303, 6705, 7563, 8865, 10095, 10503, 10863, 12165, 12243, 12825, 13713, 13725, 14013

Offset: 1

Thomas Ordowski, Feb 22 2024

nonn

If k > 4 is a term of this sequence, then (k-2, k-4) is a twin prime pair.
So all terms k > 7 are divisible by 3, and k = 7 is the only prime here.
It seems that there are infinitely many such numbers.
Note that A039669 is finite and probably complete.

			The number 15 is a term, since 15-2^(2^0) and 15-2^(2^1) are primes 13 and 11.

Cf. A039669, A129613.

q[k_] := Module[{m = 0}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)], m++]; 2^(2^m) >= k]; Select[Range[4, 15000], q] (* Amiram Eldar, Feb 22 2024 *)

More terms from Amiram Eldar, Feb 22 2024

A161719 Least number k such that 7^(2^i) + k is prime for i = 0,1,..,n-1.

Original entry on oeis.org

4, 4, 10, 40, 990, 8712, 1691820, 69466840, 6173190532, 4139897208540

Offset: 1

Robin Garcia, Jun 16 2013

			a(5) = 990 because k = 990 is the minimal k such that N = 7^(2^i) + k is prime for i = 0, 1, 2, 3, 4; N = 997, 1039, 3391, 5765791, 33232930570591. But 7^(2^5) + 990 is divisible by 37 and four other primes.

Cf. A129613, A225560 A226281, A226707.

a(10) from Giovanni Resta, Jun 16 2013

A226708 Least number k such that 6^(2^i) + k is prime for i = 0,1,..,n-1.

Original entry on oeis.org

1, 1, 1, 11, 77, 1081, 142531, 41758027, 1206670783, 10654858617041

Offset: 1

Robin Garcia, Jun 15 2013

a(5) = 77 because k = 77 is the minimal k such that N = 6^(2^i) + k is prime for i = 0, 1, 2, 3, 4; N = 83, 113, 1373, 1679693, 2821109907533. But 6^(2^5) + 77 is divisible by 79 and another prime.

Cf. A129613, A225560, A226281, A226707.

a(10) from Giovanni Resta, Jun 16 2013

A226745 Least number k such that 9^(2^i) + k is prime for i = 0,1,..,n-1.

Original entry on oeis.org

2, 2, 2, 58, 440, 18248, 1659002, 71997050, 6776406070, 2038979576678

Offset: 1

Robin Garcia, Jun 16 2013

Cf. A129613, A225560, A226281, A226707.

a(10) from Giovanni Resta, Jun 16 2013

A226766 Least number k such that 4^(2^i) + k is prime for i = 0,1,..,n-1.

Original entry on oeis.org

1, 1, 1, 1, 15, 93, 53097, 3014907, 2295032545, 46868723367

Offset: 1

Robin Garcia, Jun 17 2013

a(7) = 53097 < A129613(7) (base-2 version).

Cf. A129613, A225560, A226281, A226707

a(10) from Giovanni Resta, Jun 17 2013

cp's OEIS Frontend

A050474 Solutions to 2*phi(x) = x+1.

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A226281 Least number k such that 3^(2^i) + k is prime for i = 0,1,..,n-1.

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A129614 a(n) is the smallest natural number m such that 3^(3^k) + m is prime for k=0,1,...,n.

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A129615 a(n) is the smallest natural number m such that 4^4^k + m is prime for k=0,1,...,n.

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A226707 Least number k such that 5^(2^i) + k is prime for i = 0,1,..,n-1.

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A370523 Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.

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A161719 Least number k such that 7^(2^i) + k is prime for i = 0,1,..,n-1.

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A226708 Least number k such that 6^(2^i) + k is prime for i = 0,1,..,n-1.

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A226745 Least number k such that 9^(2^i) + k is prime for i = 0,1,..,n-1.

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A226766 Least number k such that 4^(2^i) + k is prime for i = 0,1,..,n-1.

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