Vincenzo Librandi - cp's OEIS frontend

A386618 Primes of the form 2^k + 13^k.

2, 173, 815730977

Offset: 1

Vincenzo Librandi, Aug 17 2025

If 13^k + 2^k is prime then k is either 0 or a power of 2. The corresponding values of k for a(1)-a(4) are 0, 2, 8 and 512. The fourth value is too long to enter.

Vincenzo Librandi, Table of n, a(n) for n = 1..4

Cf. A082101, A094475, A094476, A094477, A176946, A176947.

[a: n in [0..200] | IsPrime(a) where a is 13^n+2^n ];

Select[Table[2^n+13^n,{n,0,600}],PrimeQ]

A385951 Number of digits in 5^(n!).

Original entry on oeis.org

1, 1, 2, 5, 17, 84, 504, 3523, 28183, 253643, 2536423, 27900646, 334807751, 4352500756, 60935010579, 914025158672, 14624402538737, 248614843158529, 4475067176853513, 85026276360216746, 1700525527204334919, 35711036071291033288, 785642793568402732327

Offset: 0

Vincenzo Librandi, Jul 27 2025

Cf. A055642, A220078.

Subsequence of A210435.

Mathematica
```
Array[Floor[#! Log10@5+1]&,22]
```

a(n) = A055642(A220078(n)).

A385950 Number of digits in 3^(n!).

Original entry on oeis.org

1, 1, 1, 3, 12, 58, 344, 2405, 19238, 173138, 1731378, 19045154, 228541845, 2971043978, 41594615682, 623919235225, 9982707763599, 169706031981174, 3054708575661119, 58039462937561246, 1160789258751224920, 24376574433775723304, 536284637543065912676

Offset: 0

Vincenzo Librandi, Jul 27 2025

Cf. A055642, A100731.

Subsequence of A034888.

Mathematica
```
Array[Floor[#! Log10@3+1]&,22]
```

a(n) = A055642(A100731(n)).

A385949 Number of digits in 7^(n!).

Original entry on oeis.org

1, 1, 2, 6, 21, 102, 609, 4260, 34075, 306670, 3066692, 33733610, 404803314, 5262443074, 73674203025, 1105113045374, 17681808725979, 300590748341642, 5410633470149548, 102802035932841396, 2056040718656827910, 43176855091793386107, 949890812019454494354

Offset: 0

Vincenzo Librandi, Jul 24 2025

Cf. A055642, A220079, A317873.

Mathematica
```
Array[ Floor[#! Log10@7 + 1] &, 22]
```

a(n) = A055642(A220079(n)).

A384699 Triples of distinct primes whose sum is a perfect square ordered by increasing sum and then lexicographically.

Original entry on oeis.org

2, 3, 11, 3, 5, 17, 5, 7, 13, 2, 3, 31, 2, 5, 29, 2, 11, 23, 3, 5, 41, 3, 17, 29, 5, 7, 37, 5, 13, 31, 7, 11, 31, 7, 13, 29, 7, 19, 23, 13, 17, 19, 2, 3, 59, 2, 19, 43, 3, 5, 73, 3, 7, 71, 3, 11, 67, 3, 17, 61, 3, 19, 59, 3, 31, 47, 3, 37, 41, 5, 17, 59, 5, 23, 53, 5, 29, 47, 7, 13, 61, 7, 31, 43, 11, 17, 53

Offset: 1

Vincenzo Librandi, Jun 09 2025

The list is a flattened sequence of all distinct unordered triples of primes (p_1, p_2, p_3), with p_1 < p_2 < p_3, such that p_1 + p_2 + p_3 is a perfect square. The triples are sorted in ascending order of the square value. The flattened list lists the primes in each triple in ascending order.

			The first triples are:
  (2, 3, 11) with sum 16.
  (3, 5, 17) with sum 25.
  (5, 7, 13) with sum 25.
  (2, 3, 31) with sum 36.

Vincenzo Librandi, Table of n, a(n) for n = 1..1839

Cf. A000040, A000290, A183168.

limit := 400; triplette := []; for k in [4..Isqrt(limit)] do s := k^2; P := PrimesUpTo(s); for i in [1..#P-2] do for j in [i+1..#P-1] do for l in [j+1..#P] do  if P[i] + P[j] + P[l] eq s then Append(~triplette, [P[i], P[j], P[l]]); end if; end for; end for; end for; end for; flat := &cat triplette;

maxSquare=400; triplette=Reap[Do[s=k^2; primes=Select[Prime[Range[PrimePi[s]]], #

A384697 Primes of the form floor(2^k / 5).

Original entry on oeis.org

3, 409, 6553, 1677721, 6871947673, 472236648286964521369, 7922816251426433759354395033, 2451992865385422173373355243440494693789982595493763481

Offset: 1

Vincenzo Librandi, Jun 07 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..15

Cf. A383966 (corresponding k).

[ a: k in [0..500] | IsPrime(a) where a is 2^k div 5 ];

Select[Table[Quotient[(2^n), 5],{n,1,250}],PrimeQ]

a(n) = floor(2^A383966(n)/5).

A383966 Numbers k such that floor(2^k / 5) is a prime.

Original entry on oeis.org

4, 11, 15, 23, 35, 71, 95, 183, 475, 579, 631, 759, 1519, 1771, 3031, 6035, 6951, 11423, 37451, 51935, 68051

Offset: 1

Vincenzo Librandi, Jun 07 2025

From David A. Corneth, Jun 07 2025: (Start)
Observation from Hugo Pfoertner: For k > 4, terms are equivalent to 3 (mod 4).
Proof: We may write 2^k = 2^(4*m + r) = 16^m * 2^r = (15 + 1)^m * 2^r with 0 <= r <= 3.
Floor dividing by 5 gives 3*t + floor(2^r) for some positive integer t. floor(2^r) = 0 for r in {0, 1, 2} so then floor(2^k) is a multiple of 3. For k > 4 and r < 3 we definitely do get a multiple of 3 that is larger than 3 hence composite. (End)
a(22) > 150000. - Hugo Pfoertner, Jun 08 2025

			From _David A. Corneth_, Jun 07 2025: (Start)
4 is in the sequence as floor(2^4/5) = 3 is prime.
5 is not in the sequence as floor(2^5/5) = 6 which is not prime.
7 is not in the sequence as floor(2^7/5) = 25 is not prime.
8 is not in the sequence as 8 > 4 and 8 is not equivalent to 3 (mod 4).
11 is in the sequence as floor(2^11/5) = 409 which is prime. (End)

[n: n in [1..1000] | IsPrime(2^n div 5)];

Select[Range[1,20000],PrimeQ[Quotient[2^#,5]]&]

a(19)-a(21) from Hugo Pfoertner, Jun 07 2025

A383914 Primes p such that 12*2^p + 1 is also prime.

Original entry on oeis.org

3, 199, 3187, 44683, 59971, 213319, 303091, 916771

Offset: 1

Vincenzo Librandi, May 17 2025

If k is a term in A002253 and k-2 is prime, then k-2 is a term. - Amiram Eldar, May 17 2025

			3 is a term because 12*2^3+1 = 97 (prime).

Cf. A002253, A175172, A322301, A322302.

[p: p in PrimesUpTo (3500) | IsPrime(12*2^p+1)];

Select[Prime[Range[3500]],PrimeQ[12 2^#+1]&]

a(4)-a(8) from the b-file at A002253 added by Amiram Eldar, May 17 2025

A382118 Prime indices k such that prime(k) and prime(k) + 9 are anagrams.

Original entry on oeis.org

19, 73, 79, 163, 197, 241, 269, 281, 431, 439, 619, 647, 691, 739, 751, 761, 823, 877, 953, 1019, 1051, 1109, 1223, 1259, 1291, 1307, 1423, 1471, 1723, 1741, 1747, 1847, 1949, 1979, 2213, 2371, 2473, 2503, 2647, 2789, 2803, 2819, 2879, 2903, 2909, 3019, 3163, 3361

Offset: 1

Vincenzo Librandi, Apr 15 2025

Primes in A379208.

			The prime 19 is a term of the sequence because prime(19)= 67 and 67 + 9 = 76 are anagrams.

Vincenzo Librandi, Table of n, a(n) for n = 1..19861

Cf. A140353, A228157, A379208.

[n: n in [0..10000] | IsPrime(n) and Sort(Intseq(NthPrime(n))) eq Sort(Intseq(NthPrime(n) + 9))];

Select[Prime[Range[500]],Sort[IntegerDigits[Prime[#]]]==Sort[IntegerDigits[Prime[#]+9]]&]

A382082 F(k) such that F(k) + (F(k) reversed) is a palindrome, where F(k) is a Fibonacci number.

Original entry on oeis.org

0, 1, 2, 3, 13, 21, 34, 144, 233, 610, 4181, 832040, 102334155, 1134903170, 20365011074, 12200160415121876738

Offset: 1

Vincenzo Librandi, Mar 21 2025

Conjecture: The sequence appears to be finite.

The next term, F(k), has k > 3*10^5, if it exists. - Amiram Eldar, Mar 21 2025

			144 is in the sequence because 144 + 441 = 585 is a palindrome.

Intersection of A000045 and A015976.

Cf. A002113, A004086, A004091, A352124.

Rev := func;
[0] cat  [Fibonacci(n): n in [2..2*10^4] | q eq Rev(q) where q is Fibonacci(n)+Rev(Fibonacci(n))];

Mathematica

DeleteDuplicates@ Select[Fibonacci[Range[0, 100]], PalindromeQ[# + IntegerReverse[#]] &] (* Amiram Eldar, Mar 21 2025 *)

cp's OEIS Frontend

User: Vincenzo Librandi

A386618 Primes of the form 2^k + 13^k.

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A385951 Number of digits in 5^(n!).

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A385950 Number of digits in 3^(n!).

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A385949 Number of digits in 7^(n!).

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A384699 Triples of distinct primes whose sum is a perfect square ordered by increasing sum and then lexicographically.

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Magma

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A384697 Primes of the form floor(2^k / 5).

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A383966 Numbers k such that floor(2^k / 5) is a prime.

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A383914 Primes p such that 12*2^p + 1 is also prime.

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Magma

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Extensions

A382118 Prime indices k such that prime(k) and prime(k) + 9 are anagrams.

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