A052034 -id:A052034 - cp's OEIS frontend

A259418 Primes p such that p plus the cube of sum of digits of p is a perfect square.

17, 131, 863, 1031, 1481, 3011, 3449, 3881, 3923, 5903, 16649, 17921, 22643, 26249, 26687, 30113, 30809, 33629, 48473, 56009, 58049, 60623, 70163, 71933, 75521, 94109, 109331, 129209, 134129, 155387, 179909, 193601, 194003, 195401, 219647, 239807, 258233, 263411

Offset: 1

K. D. Bajpai, Jun 26 2015

All the terms are congruent to 2 (mod 3).

			a(2) = 131 is prime: 131 + (1 + 3 + 1)^3 = 256 = 16^2.
a(3) = 863 is prime: 863 + (8 + 6 + 3)^3 = 5776 = 76^2.

K. D. Bajpai, Table of n, a(n) for n = 1..2610

Cf. A052034, A061309, A091365, A091366.

[p: p in PrimesUpTo(2*10^6) |  IsSquare(&+Intseq(p)^3 + p)] ;

Select[Prime[Range[100000]], IntegerQ[Sqrt[# + Plus @@ (IntegerDigits[#])^3]] &]

forprime(p=1, 10^6, if(issquare(sumdigits(p)^3 + p), print1(p, ", ")))

A091368 Primes p such that the sum of the digits of p is not prime, but the sum of each digit raised to the 4th power is prime.

Original entry on oeis.org

1699, 2689, 6199, 6829, 6991, 7477, 8089, 8269, 8629, 9619, 12589, 15289, 19069, 19609, 20599, 20959, 21589, 21859, 23857, 25189, 25819, 25873, 25981, 27259, 27529, 27583, 28069, 28537, 28573, 28591, 28753, 29059, 29527, 29581, 29851

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently, for primes such that each digit raised to the 4th power sum to a prime, it is more likely that the digits themselves also sum to a prime. In the first 10,000 primes there are 760 primes whose digits raised to the 4th power sum to a prime. Of these, only 106 are such that the sums of the digits are not prime. Interestingly, all of these primes have a digit sum of 25 or 35. Essentially this sequence is the terms of A091367 (primes whose digits raised to the 4th power sum to a prime) that do not also appear in A046704 (primes whose digits sum to a prime).

			a(1)=1699 because 1+6+9+9 = 25 which is not prime, but 1^4 + 6^4 + 9^4 + 9^4 = 14419 which is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046704 (primes whose digits sum to a prime) A091367 (primes whose digits raised to the 4th power sum to a prime) A052034 and A091362 (same observation for digits squared) A091366 and A091365 (same observation for digits cubed).

pnpQ[n_]:=Module[{idn=IntegerDigits[n]},!PrimeQ[Total[idn]]&&PrimeQ[ Total[ idn^4]]]; Select[Prime[Range[4000]],pnpQ] (* Harvey P. Dale, Apr 26 2018 *)

A176196 Primes such that the sum of k-th powers of digits, for each of k = 1, 2, 3, and 4, is also a prime.

Original entry on oeis.org

11, 101, 113, 131, 223, 311, 353, 461, 641, 661, 883, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1697, 1741, 2111, 2203, 3011, 3347, 3491, 3659, 4139, 4337, 4373, 4391, 4733, 4931, 5303, 5639, 5693, 6197, 6359, 6719, 6791, 6917, 6971, 7411, 7433

Offset: 1

Michel Lagneau, Apr 11 2010

For k = 1, 2, and 3 see A176179

			For the prime number n=14549 we obtain :
1 + 4 + 5 + 4 + 9 = 23 ;
1^2 +4^2 + 5^2 +4^2 + 9^2 = 139 ;
1^3 +4^3 + 5^3 +4^3 + 9^3 = 983 ;
1^4 +4^4 + 5^4 +4^4 + 9^4 = 7699 ;

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A176179, A109181, A046704, A052034, A091366.

with(numtheory):for n from 2 to 20000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:s4:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:s4:=s4+u^4:od:if type(n,prime)=true and type(s1,prime)=true and type(s2,prime)=true and type(s3,prime)=true and type(s4,prime)=true then print(n):else fi:od:

Select[Prime[Range[1000]],And@@PrimeQ[Total/@Table[IntegerDigits[#]^n,{n,4}]]&] (* Harvey P. Dale, Jun 16 2013 *)

from sympy import isprime, primerange
def ok(p):
    return all(isprime(sum(int(d)**k for d in str(p))) for k in range(1, 5))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(7443)) # Michael S. Branicky, Nov 23 2021

A180404 Primes p such that the sum of fifth power of their digits is a prime.

Original entry on oeis.org

11, 101, 191, 223, 227, 229, 281, 313, 331, 337, 359, 373, 379, 397, 463, 487, 557, 577, 593, 643, 683, 733, 739, 757, 773, 821, 863, 881, 911, 937, 953, 1019, 1033, 1091, 1109, 1123, 1129, 1181, 1213, 1231, 1259, 1277, 1291, 1303, 1321, 1381, 1433, 1439

Offset: 1

Carmine Suriano, Sep 02 2010

			a(5) = 227 since 2^5+2^5+7^5 = 32+32+16807 = 16871 is a prime.

Cf. A046704, A052034, A091366, A091367.

Select[Prime[Range[500]],PrimeQ[Total[IntegerDigits[#]^5]]&] (* Harvey P. Dale, May 25 2011 *)

If a prime p = abcdef... (each letter being a single digit) then sum = a^5+b^5+... belongs to this sequence if sum is a prime.

A259489 Numbers n such that n plus the cube of sum of digits of n is a perfect square.

Original entry on oeis.org

17, 38, 131, 171, 360, 392, 500, 512, 605, 729, 863, 1031, 1035, 1481, 1737, 1994, 2156, 2268, 2483, 2513, 2520, 2732, 2817, 3011, 3240, 3384, 3449, 3710, 3881, 3923, 4032, 4100, 4112, 4145, 4572, 5193, 5456, 5598, 5720, 5832, 5903, 5924, 7164, 7388, 7625, 7631

Offset: 1

K. D. Bajpai, Jun 28 2015

All the terms are congruent to 2 or 0 (mod 3).

			a(3) = 131: 131 + (1 + 3 + 1)^3 = 256 = 16^2.
a(4) = 171: 171 + (1 + 7 + 1)^3 = 900 = 30^2.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A007494, A052034, A061309, A091365, A091366.

[ n: n in [1..10^4] |  IsSquare(&+Intseq(n)^3 + n) ] ;

Select[Range[50000], IntegerQ[Sqrt[# +Plus@@(IntegerDigits[#])^3]]&]

for(n = 1, 10^5, if(issquare(sumdigits(n)^3 + n), print1(n, ", ")))

A296187 Yarborough primes that remain Yarborough primes when each of their digits are replaced by their squares.

Original entry on oeis.org

73, 223, 233, 283, 337, 383, 523, 733, 773, 823, 2333, 2683, 2833, 2857, 3323, 3583, 3673, 3733, 3853, 5333, 6673, 6737, 6883, 7333, 7673, 7727, 7877, 8233, 8563, 8623, 22277, 22283, 22727, 23333, 23833, 25237, 25253, 25633, 26227, 26833, 27583, 27827, 27883, 32257

Offset: 1

K. D. Bajpai, Feb 14 2018

A Yarborough prime is a prime that does not contain digits 0 or 1.

Terms t of A106116 such that A048385(t) is also a term of A106116. - Felix Fröhlich, Feb 14 2018

			a(1) = 73 is a prime, and replacing each of its digits by its square yields 499, which is also prime. Neither 73 nor 499 contains digits 0 or 1, so both are Yarborough primes.
a(10) = 823 is a prime, and replacing each of its digits by its square gives 6449, another prime. Neither 823 nor 6449 contains digits 0 or 1, so both are Yarborough primes.

Chris C. Caldwell, Yarborough prime

Cf. A106116 (Yarborough primes), A048385, A052034, A296563 (digits to cubes).

k = 2; Select[Prime[Range[1000000]], Min[IntegerDigits[#]] > 1 &&  Min[IntegerDigits[Flatten[IntegerDigits[(IntegerDigits[#]^k)]]]] > 1 && PrimeQ[FromDigits[Flatten[IntegerDigits[(IntegerDigits[#]^k)]]]] &]

eva(n) = subst(Pol(n), x, 10)
is_a106116(n) = ispseudoprime(n) && vecmin(digits(n)) > 1
a048385(n) = my(d=digits(n), e=[]); for(k=1, #d, d[k]=d[k]^2); for(k=1, #d, my(dd=digits(d[k])); for(t=1, #dd, e=concat(e, dd[t]))); eva(e)
is(n) = is_a106116(n) && is_a106116(a048385(n)) \\ Felix Fröhlich, Mar 26 2018

{A106116(k): A048385(A106116(k)) in A106116}. - Felix Fröhlich, Feb 14 2018

A381878 Prime numbers p such that the sum of the d_i-th prime numbers, where (d_i) are the nonzero digits of p, is also a prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 71, 103, 107, 181, 211, 223, 227, 229, 233, 239, 257, 277, 293, 347, 383, 389, 433, 443, 449, 467, 479, 487, 499, 523, 563, 569, 587, 647, 653, 659, 677, 683, 701, 727, 743, 769, 787, 811, 839, 857, 859, 863, 877, 883, 947, 967, 983

Offset: 1

Jean-Marc Rebert, Mar 09 2025

			13 is a term, because prime(1) + prime(3) = 2 + 5 = 7 is prime.

Cf. A046704, A052034.

Select[Prime[Range[170]], PrimeQ[Total[Prime[DeleteCases[IntegerDigits[#],0]]]] &] (* Stefano Spezia, Mar 09 2025 *)

from sympy import isprime, prime
def ok(p): return isprime(p) and isprime(sum(prime(di) for di in map(int, str(p)) if di))
print([k for k in range(1, 999) if ok(k)]) # Michael S. Branicky, Mar 09 2025

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A259418 Primes p such that p plus the cube of sum of digits of p is a perfect square.

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A091368 Primes p such that the sum of the digits of p is not prime, but the sum of each digit raised to the 4th power is prime.

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A176196 Primes such that the sum of k-th powers of digits, for each of k = 1, 2, 3, and 4, is also a prime.

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A180404 Primes p such that the sum of fifth power of their digits is a prime.

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A259489 Numbers n such that n plus the cube of sum of digits of n is a perfect square.

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A296187 Yarborough primes that remain Yarborough primes when each of their digits are replaced by their squares.

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A381878 Prime numbers p such that the sum of the d_i-th prime numbers, where (d_i) are the nonzero digits of p, is also a prime.

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