A091365 -id:A091365 - cp's OEIS frontend

A091366 Primes p such that the sum of the cubes of the digits of p is prime.

11, 101, 113, 131, 139, 151, 193, 199, 223, 227, 241, 263, 269, 281, 283, 311, 337, 353, 359, 373, 421, 449, 461, 463, 487, 557, 577, 593, 599, 641, 643, 661, 733, 757, 821, 823, 827, 829, 883, 887, 919, 953, 991, 997, 1013, 1031, 1039, 1051, 1093, 1103, 1123

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently, in most cases the sum of the digits of such primes is also prime, see A091365 for the exceptions.

I conjecture the contrary: the relative density of numbers in this sequence with prime digit sum is 0. - Charles R Greathouse IV, Sep 08 2010

			a(1)=11 because 1^3 + 1^3 = 2 which is prime. a(10)=227 because 2^3 + 2^3 + 7^3 = 359 which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A046704 (primes whose digits sum to a prime) A052034 (primes whose digits squared sum to a prime) A091365 (primes whose digits cubed sum to a prime but whose digits do not sum to a prime).

Select[Prime[Range[2, 200]], PrimeQ[Total[IntegerDigits[#]^3]]&] (* Vincenzo Librandi, Apr 13 2013 *)

is(n)=my(v);if(!isprime(n),return(0));v=eval(Vec(Str(n)));isprime(sum(i=1,#v,v[i]^3)) \\ Charles R Greathouse IV, Sep 08 2010

a(44) = 997 inserted by Charles R Greathouse IV, Sep 08 2010

A176179 Primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

Original entry on oeis.org

11, 101, 113, 131, 199, 223, 311, 337, 353, 373, 449, 461, 463, 641, 643, 661, 733, 829, 883, 919, 991, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1499, 1697, 1741, 1949, 2089, 2111, 2203, 2333, 2441, 2557, 3011, 3037, 3307, 3323, 3347, 3491, 3583, 3637, 3659, 3673, 3853, 4049, 4111, 4139, 4241, 4337, 4373, 4391, 4409

Offset: 1

Michel Lagneau, Apr 10 2010

See A091365 for the exceptions for the case where the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.

			For the prime number n =5693 we obtain :
5 + 6 + 9 + 3 = 23 ;
5^2 + 6^2 + 9^2 + 3^2 = 151 ;
5^3 + 6^3 + 9^3 + 3^3 = 1097.

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

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A109181, A046704, A052034, A091366.

with(numtheory):for n from 2 to 10000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=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:od:if type(n,prime)=true and type(s1,prime)=true and type(s2,prime)=true and type(s3,prime)=true then print(n):else fi:od:

okQ[n_]:=Module[{idn=IntegerDigits[n]}, And@@PrimeQ[Total/@{idn,idn^2,idn^3}]]; Select[Prime[Range[600]],okQ]  (* Harvey P. Dale, Jan 18 2011 *)

from sympy import isprime, primerange
def ok(p):
    return all(isprime(sum(int(d)**k for d in str(p))) for k in [1, 2, 3])
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(4409)) # Michael S. Branicky, Nov 23 2021

Corrected and extended by Harvey P. Dale, Jan 18 2011

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

Original entry on oeis.org

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 *)

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, ", ")))

A121642 Numbers with composite sum of digits and prime sum of cubes of digits.

Original entry on oeis.org

799, 889, 898, 979, 988, 997, 2779, 2797, 2977, 3499, 3949, 3994, 4399, 4588, 4777, 4858, 4885, 4939, 4993, 5488, 5668, 5686, 5848, 5866, 5884, 6568, 6586, 6658, 6667, 6676, 6685, 6766, 6856, 6865, 7099, 7279, 7297, 7477, 7666, 7729, 7747, 7774, 7792

Offset: 1

Tanya Khovanova, Sep 08 2006

			For example the sum of digits of 799 is 25 which is composite; the sum of the cubes of its digits is 343 + 729 + 729 = 1801, which is prime.

Cf. A091365 (Primes p such that the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.) is a subsequence of this sequence.

sod[k_, m_] := Plus @@ (IntegerDigits[k]^m); Select[ Table[n, {n, 10000}], (! PrimeQ[sod[ #, 1]] && PrimeQ[sod[ #, 3]]) &]

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A091366 Primes p such that the sum of the cubes of the digits of p is prime.

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A176179 Primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

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

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A121642 Numbers with composite sum of digits and prime sum of cubes of digits.

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