A038194 -id:A038194 - cp's OEIS frontend

A078403 Primes whose digital root (A038194) is prime.

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 59, 61, 79, 83, 97, 101, 113, 131, 137, 149, 151, 167, 173, 191, 223, 227, 239, 241, 257, 263, 277, 281, 293, 311, 313, 317, 331, 347, 349, 353, 367, 383, 389, 401, 419, 421, 439, 443, 457, 461, 479, 491, 509, 547, 563, 569

Offset: 1

N. J. A. Sloane, Dec 26 2002

Union of A061238, A061240, A061241 and 3. - Ya-Ping Lu, Jan 03 2024

			59 is a term because 5+9=14, giving (final) iterated sum 1+4=5 and 5 is prime.

Cf. A038194, A061238, A061240, A061241, A070027, A078400.

Select[ Range[580], PrimeQ[ # ] && PrimeQ[Mod[ #, 9]] &]
Select[Prime[Range[200]],PrimeQ[Mod[#,9]]&] (* Harvey P. Dale, Aug 20 2017 *)

forprime(p=2,997,if(isprime(p%9),print1(p,",")))

from sympy import isprime, primerange; [print(p, end = ', ') for p in primerange(2, 570) if isprime(p%9)] # Ya-Ping Lu, Jan 03 2024

a(n) ~ 2n log n. - Charles R Greathouse IV, May 14 2025

Extended by Robert G. Wilson v, Klaus Brockhaus and Rick L. Shepherd, Dec 27 2002

A079130 Primes such that iterated sum-of-digits (A038194) is a square.

Original entry on oeis.org

13, 19, 31, 37, 67, 73, 103, 109, 127, 139, 157, 163, 181, 193, 199, 211, 229, 271, 283, 307, 337, 373, 379, 397, 409, 433, 463, 487, 499, 523, 541, 571, 577, 607, 613, 631, 643, 661, 733, 739, 751, 757, 769, 787, 811, 823, 829, 859, 877, 883, 919, 937, 967

Offset: 1

Klaus Brockhaus, Dec 28 2002

Primes which are 1 or 4 mod 9. - Charles R Greathouse IV, Sep 04 2014

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

Cf. A078403, A079131, A079132.

select(isprime,map(t -> (9*t+1,9*t+4),[$1..1000]));  # Robert Israel, Sep 04 2014

sQ[n_]:=MemberQ[{1,4,9},NestWhile[Total[IntegerDigits[#]]&,n,#>9&]]; Select[Prime[Range[300]],sQ] (* Harvey P. Dale, Dec 06 2012 *)

forprime(p=2,1000,if(issquare(p%9),print1(p,",")))

a(n) ~ 3n log n. - Charles R Greathouse IV, Sep 04 2014

A079131 Primes such that iterated sum-of-digits (A038194) is odd.

Original entry on oeis.org

3, 5, 7, 19, 23, 37, 41, 43, 59, 61, 73, 79, 97, 109, 113, 127, 131, 149, 151, 163, 167, 181, 199, 223, 239, 241, 257, 271, 277, 293, 307, 311, 313, 331, 347, 349, 367, 379, 383, 397, 401, 419, 421, 433, 439, 457, 487, 491, 509, 523, 541, 547, 563, 577, 599

Offset: 1

Klaus Brockhaus, Dec 28 2002

Subsequence of primes of A187318. - Michel Marcus, Jun 08 2015

Primes congruent to 1, 3, 5, 7 mod 18. - Robert Israel, Jun 08 2015

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

Cf. A038194, A078403, A079130, A079132, A187318.

[p: p in PrimesUpTo(600) | p mod 18 in [1,3,5,7]]; // Vincenzo Librandi, Jun 07 2015

[a: n in [0..1000] | IsPrime(a) where a is Floor(9*n/5)]; // Vincenzo Librandi, Jun 08 2015

select(isprime, [3, seq(seq(i*18+j, j=[1,5,7]),i=0..100)]); # Robert Israel, Jun 08 2015

Select[Prime[Range[120]], OddQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)

forprime(p=2,600,if((p%9)%2==1,print1(p,",")))

A079132 Primes such that iterated sum-of-digits (A038194) is even.

Original entry on oeis.org

2, 11, 13, 17, 29, 31, 47, 53, 67, 71, 83, 89, 101, 103, 107, 137, 139, 157, 173, 179, 191, 193, 197, 211, 227, 229, 233, 251, 263, 269, 281, 283, 317, 337, 353, 359, 373, 389, 409, 431, 443, 449, 461, 463, 467, 479, 499, 503, 521, 557, 569, 571, 587, 593, 607

Offset: 1

Klaus Brockhaus, Dec 28 2002

Cf. A078403, A079130, A079131.

Select[Prime[Range[120]], EvenQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)

forprime(p=2,600,if((p%9)%2==0,print1(p,",")))

A079155 The number of primes less than 10^n whose digital root (A038194) is also prime.

Original entry on oeis.org

4, 15, 85, 619, 4800, 39266, 332276, 2880818, 25423985, 227527467

Offset: 1

Robert G. Wilson v, Dec 27 2002

			a(2) = 15 because the only primes less than 100 whose have digital roots are also prime are {2,3,5,7,11,23,29,41,43,47,59,61,79,83,97}.

The primes are in A078403, their digital roots are in A078400.

c = 0; k = 1; Do[ While[ k < 10^n, If[ PrimeQ[k] && PrimeQ[ Mod[k, 9]], c++ ]; k++ ]; Print[c], {n, 1, 8}]

# use primerange (slower) vs. sieve.primerange (>> memory) for larger terms
from sympy import isprime, sieve
def afind(terms):
  s = 0
  for n in range(1, terms+1):
    s += sum(isprime(p%9) for p in sieve.primerange(10**(n-1), 10**n))
    print(s, end=", ")
afind(7) # Michael S. Branicky, Apr 15 2021

a(9)-a(10) from Michael S. Branicky, Apr 15 2021

A007605 Sum of digits of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 2, 4, 8, 10, 5, 11, 4, 10, 5, 7, 11, 8, 14, 7, 13, 8, 10, 16, 11, 17, 16, 2, 4, 8, 10, 5, 10, 5, 11, 13, 14, 7, 13, 10, 14, 11, 17, 10, 11, 13, 17, 19, 4, 7, 11, 13, 8, 14, 7, 8, 14, 11, 17, 10, 16, 11, 13, 14, 10, 5, 7, 11, 7, 13, 14, 16, 11, 17, 16, 13, 19, 14, 20, 19, 5

Offset: 1

N. J. A. Sloane, Mira Bernstein, and Robert G. Wilson v

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
CNRS press release, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers (French) [On a problem posed by Gelfond: the sum of digits of primes] Ann. of Math. (2) 171(2010), no. 3, 1591--1646. MR2680394 (2011j:11137)
Enrique Navarrete, Distributions of Sums of Digits of Primes
Robert G. Wilson v, Letter to C. A. Pickover, Mar. 1993

Cf. A000040, A007953, A038194, A065073, A068395, A133223, A376714, A380192.

For inverse, see A067180 and A067523.

a007605_list = map a007953 a000040_list -- Reinhard Zumkeller, Aug 04 2011

[ &+Intseq(NthPrime(n), 10): n in [1..80] ]; // Klaus Brockhaus, Jun 13 2009

map(t -> convert(convert(t,base,10),`+`), select(isprime, [2,(2*i+1 $ i=1..1000)])); # Robert Israel, Aug 16 2015

Table[Apply[Plus, RealDigits[Prime[n]][[1]]], {n, 1, 100}]
Plus@@ IntegerDigits[Prime[Range[100]]] (* Zak Seidov *)

dsum(n)=my(s);while(n,s+=n%10;n\=10);s
forprime(p=2,1e3,print1(dsum(p)", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = sumdigits(prime(n)); \\ Michel Marcus, Dec 20 2017

from sympy import prime
def a(n): return sum(map(int, str(prime(n))))
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Feb 03 2021

a(n) = A007953(A000040(n)) = A007953(prime(n)).

A007652 Final digit of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 3, 7, 1, 7, 9, 9, 1, 7, 3, 7, 3, 9, 1, 1, 3, 7, 9, 1, 3, 7, 9, 3, 9, 1, 1, 7, 3, 9, 1, 7, 1, 3, 3, 7, 1, 3, 7, 1, 7, 7, 9, 3, 9, 7, 3, 9, 3, 9, 7, 1, 9, 9, 1, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 7, 1, 9, 3, 9, 1, 3, 1, 7, 7, 3, 9, 1

Offset: 1

N. J. A. Sloane, Simon Plouffe

Primes modulo 10.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Alberto Fraile et al., Prime numbers and random walks in a square grid, Phys. Rev. E, 104(5) (2021), 054114.
Evelyn Lamb, "Peculiar Pattern Found in 'Random' Prime Numbers", Nature, March 14, 2016, republished by Scientific American.
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv preprint arXiv:1603.03720 [math.NT], 2016.
Index entries for sequences related to final digits of numbers.

Cf. A000040, A010879, A110923, A039701-A039706, A038194, A039709-A039715, A290450.

[Intseq(p)[1]: p in PrimesUpTo(600)]; // Bruno Berselli, Feb 14 2013

[p mod 10: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 10, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 10], {n, 120}] (* Ray Chandler, Oct 01 2005 *)
Mod[Prime[Range[100]], 10] (* Vincenzo Librandi, May 06 2014 *)

a(n)=prime(n)%10 \\ Charles R Greathouse IV, Oct 13 2016

primes(100)%10 \\ Charles R Greathouse IV, Oct 13 2016

a(n) = A010879(A000040(n)). - Michel Marcus, May 06 2014

Sum_k={1..n} a(k) ~ 5*n. - Amiram Eldar, Dec 11 2024

Extended by Ray Chandler, Oct 01 2005

A039701 a(n) = n-th prime modulo 3.

Original entry on oeis.org

2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1

Offset: 1

Clark Kimberling

If n > 2 and prime(n) is a Mersenne prime then a(n) = 1. Proof: prime(n) = 2^p - 1 for some odd prime p, so prime(n) = 2*4^((p-1)/2) - 1 == 2 - 1 = 1 (mod 3). - Santi Spadaro, May 03 2002; corrected and simplified by Dean Hickerson, Apr 20 2003
Except for n = 2, a(n) is the smallest number k > 0 such that 3 divides prime(n)^k - 1. - T. D. Noe, Apr 17 2003
a(n) <> 0 for n <> 2; a(A049084(A003627(n))) = 2; a(A049084(A002476(n))) = 1; A134323(n) = (1 - 0^a(n)) * (-1)^(a(n)+1). - Reinhard Zumkeller, Oct 21 2007
Probability of finding 1 (or 2) in this sequence is 1/2. This follows from the Prime Number Theorem in arithmetic progressions. Examples: There are 4995 1's in terms (10^9 +1) to (10^9+10^4); there are 10^9/2-1926 1's in the first 10^9 terms. - Jerzy R Borysowicz, Mar 06 2022

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A091178 (indices of 1's), A091177 (indices of 2's).
Cf. A120326 (partial sums).
Cf. A185934, A217659.
Cf. A010872.
Other moduli: A039702-A039706, A038194, A007652, A039709-A039715.

a039701 = (`mod` 3) . a000040
a039701_list = map (`mod` 3) a000040_list
-- Reinhard Zumkeller, Nov 16 2012

[p mod(3): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 3, n=1..105); # Nathaniel Johnston, Jun 29 2011

Mathematica
```
Table[Mod[Prime[n], 3], {n, 100}]
```

primes(100)%3 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ (3/2)*n. - Amiram Eldar, Dec 11 2024

A039702 a(n) = n-th prime modulo 4.

Original entry on oeis.org

2, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 3, 1, 3, 3, 1, 3, 1, 3, 1, 1, 3, 3, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3

Offset: 1

Clark Kimberling

Except for the first term, A100672(n) = (a(n)-1)/2 = parity of A005097. - Jeremy Gardiner, May 17 2008

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A010873.
Cf. A005097, A100672.
Cf. A039701, A039703-A039706, A100672, A005097, A038194, A007652, A039709-A039715.

a039702 = (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 20 2012

[p mod 4: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 4, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 4], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 4] (* Vincenzo Librandi, May 06 2014 *)

a(n)=prime(n)%4 \\ Charles R Greathouse IV, Jun 13 2013

Sum_k={1..n} a(k) ~ 2*n. - Amiram Eldar, Dec 11 2024

A039706 a(n) = n-th prime modulo 8.

Original entry on oeis.org

2, 3, 5, 7, 3, 5, 1, 3, 7, 5, 7, 5, 1, 3, 7, 5, 3, 5, 3, 7, 1, 7, 3, 1, 1, 5, 7, 3, 5, 1, 7, 3, 1, 3, 5, 7, 5, 3, 7, 5, 3, 5, 7, 1, 5, 7, 3, 7, 3, 5, 1, 7, 1, 3, 1, 7, 5, 7, 5, 1, 3, 5, 3, 7, 1, 5, 3, 1, 3, 5, 1, 7, 7, 5, 3, 7, 5, 5, 1, 1, 3, 5, 7, 1, 7, 3, 1, 1, 5, 7, 3, 7, 7, 3, 3, 7, 5, 1, 3, 5, 3, 5, 3, 1, 3

Offset: 1

Clark Kimberling

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A039701-A039705, A038194, A007652, A039709-A039715.

[p mod(8): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 8, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 8], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 8] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%8 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ 4*n. - Amiram Eldar, Dec 11 2024

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A078403 Primes whose digital root (A038194) is prime.

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A079130 Primes such that iterated sum-of-digits (A038194) is a square.

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A079131 Primes such that iterated sum-of-digits (A038194) is odd.

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A079132 Primes such that iterated sum-of-digits (A038194) is even.

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A079155 The number of primes less than 10^n whose digital root (A038194) is also prime.

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A007605 Sum of digits of n-th prime.

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A007652 Final digit of prime(n).

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