A028834 -id:A028834 - cp's OEIS frontend

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

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

A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

Original entry on oeis.org

11, 14, 41, 49, 94, 111, 119, 122, 128, 133, 155, 166, 182, 188, 191, 199, 212, 218, 221, 229, 236, 263, 281, 289, 292, 298, 313, 326, 331, 362, 368, 386, 449, 494, 515, 551, 559, 595, 616, 623, 632, 638, 661, 683, 779, 797, 812, 818, 821, 829, 836, 863, 881

Offset: 1

Ryan Bresler, May 29 2021

If k is in the sequence then all anagrams of k are in the sequence. - David A. Corneth, May 29 2021

Trivially, this sequence has infinite elements. A031974 is an infinite sequence that is found in this sequence - Ryan Bresler, May 30 2021

			11 is a term because its digit sum is 2 (prime) and its digit product is 1 (perfect square > 0).

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A028834 and A050626.
Subsequence of A052382.
A031974 is a subsequence of this sequence.

q:= n-> (l-> not 0 in l and isprime(add(i, i=l)) and
         issqr(mul(i, i=l)))(convert(n, base, 10)):
select(q, [$0..999])[];  # Alois P. Heinz, May 29 2021

from math import prod
from sympy import isprime, integer_nthroot
def ok(n):
  d = list(map(int, str(n)))
  return 0 not in d and isprime(sum(d)) and integer_nthroot(prod(d), 2)[1]
print(list(filter(ok, range(1000)))) # Michael S. Branicky, May 29 2021

A084995 Numbers which can be written as the product of two different primes and the sum of digits is also prime.

Original entry on oeis.org

14, 21, 34, 38, 58, 65, 74, 85, 94, 106, 111, 115, 119, 122, 133, 142, 146, 155, 166, 201, 203, 205, 209, 214, 218, 221, 247, 254, 265, 274, 278, 287, 298, 302, 319, 326, 335, 346, 355, 362, 371, 377, 382, 386, 391, 395, 403, 407, 427, 445, 454, 458, 469, 478, 481, 485

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 30 2003

			E.g., 14 = 7*2 and 1+4 = 5 is also prime.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A006881, A028834, A108606.

Module[{nn=60},Select[Union[Times@@@Subsets[Prime[Range[nn]],{2}]],PrimeQ[ Total[ IntegerDigits[#]]]&&#<=2Prime[nn]&]] (* Harvey P. Dale, Feb 28 2022 *)

is(n)={bigomega(n)==2 && !issquare(n) && isprime(sumdigits(n))}
select(is, [1..500]) \\ Andrew Howroyd, Jan 05 2020

Intersection of A028834 and A006881. - Andrew Howroyd, Jan 05 2020

Terms a(14) and beyond from Andrew Howroyd, Jan 05 2020

A129630 Numbers k such that sum of digits of (k+1) is a prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 11, 13, 15, 19, 20, 22, 24, 28, 29, 31, 33, 37, 40, 42, 46, 48, 49, 51, 55, 57, 60, 64, 66, 69, 73, 75, 82, 84, 88, 91, 93, 97, 100, 101, 103, 105, 109, 110, 112, 114, 118, 119, 121, 123, 127, 130, 132, 136, 138, 139, 141, 145, 147, 150, 154, 156, 159

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 31 2007

Cf. A007953, A028834.

P:=proc(n) local i,k,w; for i from 1 by 1 to n do w:=0; k:=binomial(i+1,i); while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; if isprime(w) then print(i); fi; od; end: P(1000);

Select[Range[200], PrimeQ[Total[IntegerDigits[# + 1]]] &]

a(n) = A028834(n) - 1.

A167631 Emirps A006567 with a sum of digits and a number of digits which are both primes.

Original entry on oeis.org

113, 157, 179, 199, 311, 337, 359, 733, 739, 751, 937, 953, 971, 991, 10039, 10079, 10091, 10253, 10273, 10321, 10343, 10453, 10457, 10459, 10499, 10613, 10639, 10651, 10781, 10853, 10859, 10891, 10909, 11003, 11083, 11159, 11197, 11243

Offset: 1

Claudio Meller, Nov 07 2009

Subsequence of A006567 and A028834. - R. J. Mathar, Nov 12 2009

			113 and 311 are distinct primes, and 3 (number of digits) and 5 (1+1+3) are also primes.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

sod(n)=my(s=0);while(n>9,s+=n%10;n\=10);s+n
R(n)=my(v=eval(Vec(Str(n))),s=0);forstep(i=#v,1,-1,s=10*s+v[i]);s
isA167631(n)=my(r); isprime(#Str(n)) && isprime(sod(n)) && isprime(n) && isprime(r=R(n)) && n!=r \\ Charles R Greathouse IV, Nov 10 2009

from sympy import primerange, isprime
A167631 =[]
for power_of_ten in [2,3,5]: # (7 can be added (12 sec. and 26790 terms), 11 not recommended)
    primes = list(primerange(10**(power_of_ten-1),10**power_of_ten))
    for p in primes:
        if str(p) != (p_rev:=str(p)[::-1]):
            if isprime(int(p_rev)):
                if isprime(sum(list(map(int, p_rev.strip())))): A167631.append(p)
print(A167631) # Karl-Heinz Hofmann, Feb 19 2025

Edited by Charles R Greathouse IV and R. J. Mathar, Nov 10 2009

A295389 Numbers whose sum of digits is squarefree.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 14, 15, 16, 19, 20, 21, 23, 24, 25, 28, 29, 30, 32, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 50, 51, 52, 55, 56, 58, 59, 60, 61, 64, 65, 67, 68, 69, 70, 73, 74, 76, 77, 78, 82, 83, 85, 86, 87, 89, 91, 92, 94, 95, 96, 98, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 114

Offset: 1

Nile Nepenthe Wynar, Nov 21 2017

Numbers k such that A007953(k) is a term of A005117. - Felix Fröhlich, Nov 21 2017

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

Cf. A005117, A007953. Includes A028834.

select(n -> numtheory:-issqrfree(convert(convert(n,base,10),`+`)), [$1..200]); # Robert Israel, Nov 21 2017

is(n) = issquarefree(sumdigits(n));
for(n = 1, 200, if(is(n), print1(n,", ")))

A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 3, 5, 6, 7, 7, 8, 7, 8, 5, 11, 9, 10, 8, 13, 8, 12, 9, 13, 11, 12, 10, 15, 11, 16, 10, 17, 10, 20, 12, 20, 14, 18, 13, 21, 13, 22, 13, 20, 14, 25, 14, 22, 18, 22, 15, 26, 12, 29, 17, 25, 15, 27, 15, 30, 19, 26, 14, 32, 17, 33, 19, 27, 19, 31, 18, 34, 19, 29, 19, 37, 16, 33, 21, 30, 24, 39, 20, 38

Offset: 2

Samuel Harkness, Jun 15 2022

The graph of (n,a(n)) shows an interesting structure, somewhat resembling a comet with four tails. Starting at the bottom tail and going upwards:
Observations:
The bottom "tail" contains all n with both 2 and 3 as prime factors, i.e., numbers n in A008588 (1/6 of all n).
The second "tail" contains all n with 2 as a prime factor but not 3, i.e., numbers n in A047235 (1/3 of all n).
The third "tail" contains all n with 3 as a prime factor but 2, i.e., numbers n in A016945 (1/6 of all n).
The top "tail" contains all n with neither 2 nor 3 as a prime factor, i.e., numbers n in A007310 (1/3 of all n).
The bottom of each "tail" contains n with 5 as a prime factor. Moving up within each "tail," the prime factors of each n tend to increase.

			For n=7, express 7 in all bases from 2 to 7, then add the numbers, counting those which are prime:
  base 2: 1 1 1 --> 1+1+1=3 prime
  base 3: 2 1   --> 2+1=3   prime
  base 4: 1 3   --> 1+3=4   nonprime
  base 5: 1 2   --> 1+2=3   prime
  base 6: 1 1   --> 1+1=2   prime
  base 7: 1     --> 1=1     nonprime
The sum of the digits of the base-b expansion of 7 in 4 different bases b (2, 3, 5, and 6) from base 2 to 7 is prime, so a(7)=4.

Samuel Harkness, Table of n, a(n) for n = 2..10000
Samuel Harkness, Colored Scatterplot of the first 50000 terms, with n as multiples of 2 or 3
Samuel Harkness, Colored Scatterplot of the first 50000 terms, with the lowest factor of n among 5, 7, 11, or 13
Rémy Sigrist, Scatterplot of (n, b) such that the sum of digits of n in base b is prime (with n = 1..1000, b = 2..n).
Rémy Sigrist, Colored scatterplot of the first 50000 terms (where the color is function of gcd(n, 2*3*5)).

Cf. A008588, A047235, A016945, A007310, A028834, A052294.

a[n_] := Count[Range[2, n], ?(PrimeQ[Plus @@ IntegerDigits[n, #]] &)]; Array[a, 84, 2] (* _Amiram Eldar, Jun 17 2022 *)

a(n) = sum(b=2, n, isprime(sumdigits(n, b))); \\ Michel Marcus, Jun 16 2022

from sympy.ntheory import isprime, digits
def A355017(n): return sum(1 for b in range(2,n) if isprime(sum(digits(n,b)[1:]))) # Chai Wah Wu, Jun 17 2022

A372897 Count of n-digit numbers whose sum of digits is a prime.

Original entry on oeis.org

4, 33, 303, 2670, 23741, 222638, 2211826, 22325173, 220321667, 2128051302, 20606839279, 203631013986, 2048538361591, 20655036405780, 205672896661755, 2012878671315492, 19505453673514959, 190027534666354756, 1884928265282803982, 19032829919297816897, 193085599933330233795

Offset: 1

Antoine Mathys, May 15 2024

a(n) is the number of terms in A028834 with n digits.
Sum of digits s in n digits is a composition of s into n parts the first of which ranges 1 to 9 and the rest 0 to 9.  The number of such compositions is the coefficient of x^s in polynomial (x^1 + ... + x^9)*(x^0 + ... + x^9)^(n-1) and a(n) is the sum of those coefficients where s is prime. - Kevin Ryde, May 19 2024
a(554) is the first term for which number_of_digits(a(n)) != n. - Antoine Mathys, May 22 2024

			For n=1 the a(1)=4 numbers are 2,3,5,7.

Antoine Mathys, Table of n, a(n) for n = 1..1000

Cf. A000720, A028834.

a[n_]:=Sum[Coefficient[Sum[x^i,{i,9}]Sum[x^i,{i,0,9}]^(n-1),x^i],{i,Prime[Range[PrimePi[9n]]]}]; Array[a,21] (* Stefano Spezia, May 16 2024 *)

a(n)=my(p=sum(i=1,9,x^i)*sum(i=0,9,x^i)^(n-1),s=0);forprime(q=2,9*n,s+=polcoef(p,q));s;

a(12)-a(21) from Stefano Spezia, May 16 2024

A162250 Values of the form prime(prime(i)) with a prime digital sum.

Original entry on oeis.org

3, 5, 11, 41, 67, 83, 157, 179, 191, 241, 283, 331, 353, 401, 461, 599, 739, 773, 797, 919, 991, 1031, 1217, 1297, 1433, 1471, 1499, 1523, 1723, 1741, 1787, 2027, 2063, 2081, 2221, 2269, 2351, 2609, 2647, 2683, 2719, 2803, 3019, 3109, 3169, 3259, 3299

Offset: 1

Cino Hilliard, Jun 28 2009

			Prime(prime(6)) = 41. 4+1=5, prime. So 41 is in the sequence.

Intersection of A006450 and A028834.

sodip(n) = {
local(s=0,a,x,y,j,p);
for(x=1,n, p=prime(prime(x)); a=eval(Vec(Str(p))); y=sum(j=1,length(a),a[j]); if(isprime(y),print1(p","));)
}

{A006450(i) : A007953(A006450(i)) in A000040}. [From R. J. Mathar, Aug 03 2009]

Definition rephrased by R. J. Mathar, Sep 11 2009

A162253 Smallest value of the n-fold nesting prime(prime(...(k)...)) with a prime digital sum.

Original entry on oeis.org

2, 3, 5, 11, 1787, 5381, 5381, 5381, 648391, 648391, 414507281407, 414507281407

Offset: 1

Cino Hilliard, Jun 29 2009

n-deep nestings prime(prime(...(prime(k))...)) = prime^n(k) can be arranged in a table T(n,k),
    2   3    5     7    11    13 : A000040, n=0
    3   5   11    17    31    41 : A006450, n=1
    5  11   31    59   127   179 : A038580, n=2
   11  31  127   277   709  1063 : A049090
   31 127  709  1787  5381  8527 : A049203
  127 709 5381 15299 52711 87803 : A049202
a(n) is the leftmost value in the n-th row (the one with the smallest k) with a digit sum which is prime.
In order to generate the entries a(11) and a(12), prime2() was used which reads a large 880 gigabyte file of all primes < 10^12.

			1st nesting is prime(1) = 2 which has a prime digit sum: a(0). The second nesting is prime(prime(1)) = 3, which has a prime digits sum: a(1)=3. The 3rd and 4th nesting also succeed for k=1 while the fifth nesting prime(prime(prime(prime(prime(4))))) = 1787 is the first occurrence of sum of digits is prime. Here nesting for k = 1,2,3 does not sum to a prime number.

for(j=1,12,print(j","sodip2(100,j)","));
sodip2(n,m) = \\multiple nesting of prime(prime(prime..(n)
{
local(s=0,a,x,y,j,p);
for(x=1,n,
for(i=1,m,p=prime2(p));
a=eval(Vec(Str(p)));
y=sum(j=1,length(a),a[j]);
if(isprime(y),return(p));
)
}

{min A000040^n(k): A000040^n(k) in A028834}. - R. J. Mathar, Jul 16 2009

Definition rephrased by R. J. Mathar, Jul 16 2009

cp's OEIS Frontend

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

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A344825 Integers whose digit sum is prime and whose digit product is a perfect square > 0.

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A084995 Numbers which can be written as the product of two different primes and the sum of digits is also prime.

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A129630 Numbers k such that sum of digits of (k+1) is a prime.

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A167631 Emirps A006567 with a sum of digits and a number of digits which are both primes.

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A295389 Numbers whose sum of digits is squarefree.

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A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.

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A372897 Count of n-digit numbers whose sum of digits is a prime.

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