A028834 -id:A028834 - cp's OEIS frontend

A277995 Primes with prime subscripts whose digits are primes, whose digital root is prime, whose sum of digits is prime and whose reversal is also prime.

3, 5, 353, 32732237, 35225327, 75527537, 75535277, 75557723, 75737723, 75755257, 77322233, 77752733, 322375577, 322775737, 325725577, 325773727, 337735553, 352272233, 355322777, 357333377, 357735773, 372577727, 372753727, 375577733, 375722377, 375727237, 377725723, 377752723

Offset: 1

Ilya Gutkovskiy, Nov 08 2016

Intersection of A006450, A007500, A019546, A028834 and A078403.

			32732237 is in the sequence because 32732237 is the 2016197-th prime number, 2016197 is prime, digits 2, 3 and 7 are primes, 32732237 -> 3 + 2 + 7 + 3 + 2 + 2 + 3 + 7 = 29 (is prime) -> 2 + 9 = 11 -> 1 + 1 = 2, 2 is prime and 73223723 is also prime.

Cf. A000040, A006450, A007500, A019546, A028834, A078403, A087368.

Select[Table[Prime[Prime[n]], {n, 1500000}], Complement[IntegerDigits[#1], {2, 3, 5, 7}] == {} && PrimeQ[#1 - 9 Floor[(#1 - 1)/9]] && PrimeQ[Total[IntegerDigits[#1]]] && PrimeQ[FromDigits[Reverse[IntegerDigits[#1]]]] & ]

A346489 a(n) is the index of the smallest prime whose digits sum to n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 10, 19, 24, 46, 109, 430, 669, 3245, 6057, 7837, 33067, 77777, 476643, 601855, 3556550, 5216954, 15739663, 41146179, 189961757, 882206624, 3325059246, 15581005618, 23007498153, 37607875618

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

			a(10) = 430 because prime(430) = 2999, 2 + 9 + 9 + 9 = 29 = prime(10) and this is the smallest such number.

Cf. A000040, A000720, A007605, A028834, A046704, A046864, A054750, A075177.

Module[{nn=522*10^4,tbl},tbl=Table[{n,Total[IntegerDigits[Prime[n]]]},{n,nn}];Table[SelectFirst[tbl,#[[2]]==Prime[n]&],{n,20}]][[;;,1]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Feb 19 2023 *)

a(n) = A000720(A054750(n)).

a(n) = min {k : A007605(k) = prime(n)}.

A359449 Positive integers in which the sum of the k-th powers of their digits is a prime number for k = 1, 2, 3, 4, 5, and 6 but not for k=7.

Original entry on oeis.org

223, 232, 322, 1349, 1394, 1439, 1493, 1934, 1943, 2023, 2032, 2203, 2230, 2302, 2320, 3022, 3149, 3194, 3202, 3220, 3419, 3491, 3914, 3941, 4139, 4193, 4319, 4391, 4913, 4931, 9134, 9143, 9314, 9341, 9413, 9431, 10349, 10394, 10439, 10493, 10934, 10943, 13049, 13094, 13409, 13490, 13904, 13940

Offset: 1

José Hernández, Jan 02 2023

			223 belongs to this sequence because 2+2+3=7, 2^2+2^2+3^2=17, 2^3+2^3+3^3=43, 2^4+2^4+3^4=113, 2^5+2^5+3^5=307, and 2^6+2^6+3^6=857 are prime numbers whereas 2^7+2^7+3^7 is a composite number.

Cf. A028834, A108662, A225534, A210767, A245358.

filter:= proc(n) local L,t,k;
  L:= convert(n,base,10);
  andmap(isprime, [seq(add(t^k,t=L),k=1..6)]) and not isprime(add(t^7,t=L))
end proc:
select(filter, [$1..20000]); # Robert Israel, Jan 03 2023

For[a = 0, a <= 9, a++,
 For[b = 0, b <= 9, b++,
 For[c = 0, c <= 9, c++,
 For[d = 0, d <= 9, d++,
   If[PrimeQ[a + b + c + d] == True &&
      PrimeQ[a^2 + b^2 + c^2 + d^2] == True &&
      PrimeQ[a^3 + b^3 + c^3 + d^3] == True &&
      PrimeQ[a^4 + b^4 + c^4 + d^4] == True &&
      PrimeQ[a^5 + b^5 + c^5 + d^5] == True &&
      PrimeQ[a^6 + b^6 + c^6 + d^6] == True &&
      PrimeQ[a^7 + b^7 + c^7 + d^7] == False, Print[a, b, c, d]]]]]]
(* This code outputs all the terms of the sequence in the interval [1,10^4]. *)

isok(n) = my(d=digits(n)); for (i=1, 6, if (!isprime(sum(k=1,#d, d[k]^i)), return(0))); !isprime(sum(k=1,#d, d[k]^7)); \\ Michel Marcus, Jan 02 2023

A359610 Numbers k such that the sum of the 5th powers of the digits of k is prime.

Original entry on oeis.org

11, 101, 110, 111, 119, 128, 133, 182, 188, 191, 218, 223, 227, 229, 232, 247, 272, 274, 281, 292, 313, 322, 331, 337, 346, 359, 364, 368, 373, 377, 379, 386, 395, 397, 427, 436, 463, 472, 478, 487, 539, 557, 568, 575, 577, 586, 593, 634, 638, 643, 658, 667

Offset: 1

José Hernández, Jan 06 2023

It is easy to establish that the sequence is infinite: if x is in the sequence, so is 10*x.

Alternatively: the sequence is infinite as the sequence contains all numbers consisting of a prime number of 1s and an arbitrary number of 0s. - Charles R Greathouse IV, Jan 06 2023

			11 is a term since 1^5 + 1^5 = 2 is prime.

A031974 is a subsequence.
Cf. A055014 (sum of the 5th powers of digits).
Cf. A028834, A108662, A225534, A210767, A245358.

top = 999; (* Find all terms <= top *)
For[t = 11, t <= top, t++, k = IntegerLength[t]; sum = 0;
   For[e = 0, e <= k - 1, e++, sum = sum + NumberDigit[t, e]^5];
      If[PrimeQ[sum] == True, Print[t]]]
Select[Range[670],PrimeQ[Total[IntegerDigits[#]^5]] &] (* Stefano Spezia, Jan 08 2023 *)

isok(k) = isprime(vecsum(apply(x->x^5, digits(k)))); \\ Michel Marcus, Jan 07 2023

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A277995 Primes with prime subscripts whose digits are primes, whose digital root is prime, whose sum of digits is prime and whose reversal is also prime.

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A346489 a(n) is the index of the smallest prime whose digits sum to n-th prime.

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A359449 Positive integers in which the sum of the k-th powers of their digits is a prime number for k = 1, 2, 3, 4, 5, and 6 but not for k=7.

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A359610 Numbers k such that the sum of the 5th powers of the digits of k is prime.

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Mathematica

PARI