A091366 -id:A091366 - cp's OEIS frontend

A180421 Members p of A091366 such that digit-reverse(p) is also in A091366.

11, 101, 113, 131, 151, 199, 311, 337, 353, 359, 373, 733, 757, 919, 953, 991, 1031, 1103, 1213, 1217, 1231, 1237, 1259, 1301, 1321, 1381, 1439, 1471, 1499, 1619, 1723, 1741, 1831, 1949, 3011, 3019, 3109, 3121, 3163, 3257, 3271, 3299, 3347, 3527, 3583, 3613, 3767

Offset: 1

Carmine Suriano, Sep 03 2010

			a(10) = A091366(19) = 359 and reverse(359) = 953 = A091366(42) is again in A091366.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A091366

filter:= proc(n) local i,L,d,y;
  if not isprime(n) then return false fi;
  L:= convert(n,base,10);
  d:= nops(L);
  isprime(add(L[i]^3,i=1..d)) and isprime(add(L[-i]*10^(i-1),i=1..d))
end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Apr 06 2021

{A091366(i): A004086(A091366(i)) in A091366}.

Keyword:base added and definition shortened by R. J. Mathar, Sep 23 2010

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.

Original entry on oeis.org

997, 2797, 3499, 4993, 7297, 7477, 7927, 8089, 8999, 9277, 9349, 9439, 9907, 11689, 12697, 12967, 14479, 14767, 14929, 14947, 16189, 16477, 16729, 16747, 16927, 16981, 17449, 17467, 18169, 18691, 19249, 19267, 19429, 19447, 19681, 19861

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently if the cubes of the digits of a prime 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 1969 primes p such that the cubes of the digits of p sum to a prime. Of these, only 358 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 A091366 (primes whose digits cubed sum to a prime) that do not also appear in A046704 (primes whose digits sum to a prime).

			a(1)=997 because 9+9+7 = 25 which is not prime, but 9^3+9^3+7^3 = 1801 which is prime.

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

Cf. A046704 (primes whose digits sum to a prime) A091366 (primes whose digits squared sum to a prime).

ssdQ[n_]:= Module[{idn = IntegerDigits[n]}, !PrimeQ[Total[idn]]&&PrimeQ[Total[idn^3]]]; Select[Prime[Range[4000]], ssdQ] (* Vincenzo Librandi, Apr 17 2013 *)

A091367 Primes p such that the sum of the digits raised to the 4th power is prime.

Original entry on oeis.org

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 179, 191, 197, 223, 269, 311, 313, 331, 353, 379, 397, 401, 443, 461, 601, 607, 641, 661, 719, 739, 809, 883, 911, 937, 971, 1013, 1019, 1031, 1033, 1091, 1097, 1103, 1109, 1181, 1301, 1303, 1367, 1433

Offset: 1

Chuck Seggelin, Jan 03 2004

			a(1) = 11 because 1^4 + 1^4 = 2 which is prime.
a(10) = 89 because 8^4 + 9^4 = 10657 which is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A046704 (primes whose digits sum to a prime), A052034 (primes whose digits squared sum to a prime), A091366 (primes whose digits cubed sum to a prime).

upto=500;Select[Prime[Range[upto]],PrimeQ[Total[IntegerDigits[#]^4]]&] (* Paolo Xausa, Nov 23 2021 *)

from sympy import isprime, primerange
def ok(p): return isprime(sum(int(d)**4 for d in str(p)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(1433)) # Michael S. Branicky, Nov 23 2021

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 *)

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, ", ")))

cp's OEIS Frontend

A180421 Members p of A091366 such that digit-reverse(p) is also in A091366.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A091367 Primes p such that the sum of the digits raised to the 4th power is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica