A108662 -id:A108662 - cp's OEIS frontend

A052034 Primes such that the sum of the squares of their digits is also a prime.

11, 23, 41, 61, 83, 101, 113, 131, 137, 173, 179, 191, 197, 199, 223, 229, 311, 313, 317, 331, 337, 353, 373, 379, 397, 401, 409, 443, 449, 461, 463, 467, 601, 641, 643, 647, 661, 683, 719, 733, 739, 773, 797, 829, 863, 883, 911, 919, 937, 971, 977, 991, 997, 1013

Offset: 1

Patrick De Geest, Dec 15 1999

Primes p such that the sum of the squared digits of p is a prime q. For the values of q see A109181.

			p = 23 is in the sequence because q = 2^2 + 3^2 = 13 is a prime.
9431 -> 9^2 + 4^2 + 3^2 + 1^2 = 107 (which is prime).

Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, p. 89.
Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Zak Seidov, 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. A003132, A052035, A091367, A108662, A109181.

a:=proc(n) local nn, L: nn:=convert(n,base,10): L:=nops(nn): if isprime(n)= true and isprime(add(nn[j]^2,j=1..L))=true then n else end if end proc: seq(a(n),n=1..1000); # Emeric Deutsch, Jan 08 2008

Select[Prime[Range[250]],PrimeQ[Total[IntegerDigits[#]^2]]&]  (* Harvey P. Dale, Dec 19 2010 *)

from sympy import isprime, primerange
def ok(p): return isprime(sum(int(d)**2 for d in str(p)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(1013)) # Michael S. Branicky, Nov 23 2021

Edited by N. J. A. Sloane, Dec 15 2007 and again on Dec 05 2008 at the suggestion of Zak Seidov

A182404 Numbers whose digit sum as well as sum of the squares of the digits is a prime.

Original entry on oeis.org

11, 12, 14, 16, 21, 23, 25, 32, 38, 41, 49, 52, 56, 58, 61, 65, 83, 85, 94, 101, 102, 104, 106, 110, 111, 113, 119, 120, 131, 133, 137, 140, 146, 160, 164, 166, 173, 179, 191, 197, 199, 201, 203, 205, 210, 223, 229, 230, 232, 250, 289, 292, 298, 302, 308

Offset: 1

Sumit Maheshwari, May 09 2010

			25 is here because 2 + 5 = 7 and 2*2 + 5*5 = 29 both are prime.

Cf. A028834, A108662.

fQ[n_] := Module[{d = IntegerDigits[n]}, PrimeQ[Total[d]] && PrimeQ[Total[d^2]]]; Select[Range[500], fQ] (* T. D. Noe, May 09 2012 *)

Intersection of A028834 and A108662. - Jason Yuen, Oct 15 2024

Incorrect comment removed by Jason Yuen, Oct 15 2024

A234021 Numbers with digits in nondecreasing order such that sum of squares of digits is a prime.

Original entry on oeis.org

11, 12, 14, 16, 23, 25, 27, 38, 45, 49, 56, 58, 78, 111, 113, 119, 126, 133, 137, 146, 159, 166, 168, 179, 199, 223, 229, 234, 249, 267, 289, 335, 337, 344, 346, 348, 355, 357, 368, 377, 379, 388, 449, 467, 559, 566, 678, 689, 779, 799, 1112, 1114, 1118, 1125

Offset: 1

Zak Seidov, Dec 31 2013

Primitive solutions of A108662. Intersection of A009994 and A108662.

Zak Seidov, Table of n, a(n) for n = 1..2091 (all terms up to 10^7)

Cf. A009994, A108662.

Select[Range[1125],LessEqual@@(id=IntegerDigits[#])&&PrimeQ[Total[id^2]]&] (* Ray Chandler, Dec 31 2013 *)

A307735 Integers k such that if m = k + A003132(k) then k = m - A003132(m).

Original entry on oeis.org

0, 9, 205, 212, 217, 366, 457, 663, 1314, 1315, 1348, 1672, 1742, 1792, 1797, 2005, 2012, 2017, 2129, 2201, 2208, 2213, 2216, 2305, 2404, 2405, 2465, 2564, 2565, 2671, 2741, 2748, 2789, 2829, 3114, 3115, 3205, 3303, 3306, 3394, 3436, 3475, 3696, 3819, 4204, 4205, 4245, 4347, 4475, 4542, 4629, 4647, 4688

Offset: 1

Antonio Roldán, Apr 25 2019

A003132(n) is the sum of the squares of the digits of n.

			205 is in the sequence because 205 + 2^2 + 0^2 + 5^2 = 234 and 234 - 2^2 - 3^2 - 4^2 = 205.

Cf. A003132, A108662, A175396, A209303, A223035, A225049, A258881.

sod2[n_] := Total @ (IntegerDigits[n]^2); aQ[n_] := sod2[n + (s=sod2[n])] == s; Select[Range[0, 4700], aQ] (* Amiram Eldar, Jul 03 2019 *)

for(i = 0 , 5000 , a = i + norml2(digits(i)) ; b = a - norml2(digits(a)) ; if(i == b , print1(i , ", ")))

A177149 Indices n such that the sums of the squares of the digits of prime(n) are prime.

Original entry on oeis.org

5, 9, 13, 18, 23, 26, 30, 32, 33, 40, 41, 43, 45, 46, 48, 50, 64, 65, 66, 67, 68, 71, 74, 75, 78, 79, 80, 86, 87, 89, 90, 91, 110, 116, 117, 118, 121, 124, 128, 130, 131, 137, 139, 145, 150, 153, 156, 157, 159, 164, 165, 167, 168, 170, 171, 173, 174, 182, 184, 185

Offset: 1

Michel Lagneau, May 03 2010

n such that prime(n) is in A108662. - Robert Israel, Aug 05 2019

			5 is in the sequence because the 5th prime is 11, and 1^2 + 1^2 = 2 prime;
9 is in the sequence because the 9th prime is 23, and 2^2 + 3^2 = 13 prime;
139 is in the sequence because the 139th prime is 797, and 7^2 + 9^2 + 7^2 =179 prime.

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

Cf. A108662.

with(numtheory): nn:= 150: T:=array(1..nn):k:=1:for n from 1 to 731 do:p:=ithprime(n):l:=evalf(floor(ilog10(p))+1):n0:=p:s:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s:=s+u^2:od:if type(s,prime)=true then T[k]:=n:k:=k+1: else fi:od:print(T):
# Simpler:
filter:= proc(n) isprime(add(t^2,t=convert(ithprime(n),base,10))) end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 05 2019

Select[Range[200],PrimeQ[Total[IntegerDigits[Prime[#]]^2]]&] (* Harvey P. Dale, Jan 10 2021 *)

A181364 Fibonacci numbers whose digits, when squared, sum to a prime.

Original entry on oeis.org

21, 377, 610, 2584, 17711, 75025, 196418, 514229, 63245986, 701408733, 1134903170, 1836311903, 2971215073, 17167680177565, 72723460248141, 117669030460994, 2880067194370816120, 19740274219868223167, 354224848179261915075, 1500520536206896083277

Offset: 1

Carmine Suriano, Oct 15 2010

			a(5) = 17711 = Fibonacci(22) since 1^2+7^2+7^2+1^2+1^2 = 1+49+49+1+1 = 101 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Intersection of A000045 and A108662.

Cf. A178838, A065398.

Select[Fibonacci[Range[150]],PrimeQ[Total[IntegerDigits[#]^2]]&] (* Harvey P. Dale, Feb 27 2012 *)

a(n) = A000045(A178838(n)). - Michel Marcus, Sep 01 2025

Corrected and extended by Harvey P. Dale, Feb 27 2012

A245475 Numbers n such that the sum of digits, sum of squares of digits, and sum of cubes of digits are all prime.

Original entry on oeis.org

11, 101, 110, 111, 113, 131, 146, 164, 166, 199, 223, 232, 289, 298, 311, 322, 335, 337, 346, 353, 355, 364, 373, 388, 416, 436, 449, 461, 463, 494, 533, 535, 553, 566, 614, 616, 634, 641, 643, 656, 661, 665, 733, 829, 838, 883, 892, 919, 928, 944, 982, 991, 1001, 1010, 1011, 1013, 1031, 1046, 1064, 1066, 1099

Offset: 1

Derek Orr, Jul 23 2014

There are infinitely many numbers in this sequence; 0's can be added to any number any number of times in any logical order (i.e., the number doesn't start with a zero).

			1^1 + 4^1 + 6^1 = 11 is prime.
1^2 + 4^2 + 6^2 = 53 is prime.
1^3 + 4^3 + 6^3 = 281 is prime.
Thus 146, 164, 416, 461, 641, and 614 are members of this sequence.

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

Cf. A182404, A225534, A108662.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  isprime(convert(L,`+`)) and
  isprime(convert(map(`^`,L,2),`+`)) and
  isprime(convert(map(`^`,L,3),`+`))
end proc:
select(filter, [$1..2000]); # Robert Israel, Dec 04 2024

sdpQ[n_]:=Module[{idn=IntegerDigits[n]},AllTrue[{Total[idn], Total[ idn^2], Total[ idn^3]}, PrimeQ]]; Select[Range[1100],sdpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 06 2018 *)

for(n=1,10^3,d=digits(n);s1=sum(i=1,#d,d[i]);s2=sum(j=1,#d,d[j]^2);s3=sum(k=1,#d,d[k]^3);if(isprime(s1)&&isprime(s2)&&isprime(s3),print1(n,", ")))

A344366 Integers k such that the sum of squares of digits of both k and k-1 are prime.

Original entry on oeis.org

12, 102, 111, 120, 160, 230, 250, 380, 410, 450, 520, 560, 720, 780, 830, 870, 1002, 1011, 1020, 1060, 1100, 1101, 1110, 1370, 1640, 1680, 1910, 1950, 1970, 1990, 2030, 2050, 2340, 2670, 2920, 3080, 3170, 3240, 3420, 3460, 3550, 3570, 3710, 3840, 3860, 4010

Offset: 1

Charles U. Lonappan, May 19 2021

Integers k such that k and k-1 are both in A108662.
Terms are never prime. They cannot end in the digits 3,4,5,6,7,8,9.
If k is a term, phi(k) is divisible by 4.
The set of such numbers is infinite.

			12 is in the sequence because the sum of squares of digits of 12 is 5 and that of 11 is 2, and both 5 and 2 are prime numbers.

Charles U. Lonappan, Consecutive natural numbers whose sum of squares of digits is prime, International Journal of Research in Engineering and Science (IJRES), Volume 9 Issue 1 (2021) pp. 14-16.

Cf. A108662.

q[n_] := PrimeQ[Plus @@ (IntegerDigits[n]^2)]; Select[Range[2, 5000], q[#-1] && q[#] &] (* Amiram Eldar, May 19 2021 *)

isok(k) = isprime(norml2(digits(k-1))) && isprime(norml2(digits(k))); \\ Michel Marcus, May 24 2021

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

cp's OEIS Frontend

A052034 Primes such that the sum of the squares of their digits is also a prime.

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A182404 Numbers whose digit sum as well as sum of the squares of the digits is a prime.

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A234021 Numbers with digits in nondecreasing order such that sum of squares of digits is a prime.

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A307735 Integers k such that if m = k + A003132(k) then k = m - A003132(m).

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A177149 Indices n such that the sums of the squares of the digits of prime(n) are prime.

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A181364 Fibonacci numbers whose digits, when squared, sum to a prime.

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A245475 Numbers n such that the sum of digits, sum of squares of digits, and sum of cubes of digits are all prime.

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A344366 Integers k such that the sum of squares of digits of both k and k-1 are prime.

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