A119449 -id:A119449 - cp's OEIS frontend

A119450 Primes with odd digit sum.

3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 241, 263, 269, 281, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 401, 409, 421, 443, 449, 461, 463, 467, 487, 557, 571, 577, 593

Offset: 1

Zak Seidov, May 20 2006

On average, there are as many prime numbers for which the sum of decimal digits is even as prime numbers for which it is odd [A119450]. This hypothesis, first made in 1968, has recently been proved by researchers from the Institut de Mathematiques de Luminy.
Also primes such that absolute value of difference between largest digit and the sum of all the other digits is an odd integer. This is in accordance with hypothesis of Alexandre Gelfond, proved by C. Mauduit and J. Rivat as stated in Links section. - Osama Abuajamieh, Feb 10 2017
Considering the sequence digit sums, when prime, new maximum digit sums encounter the prime numbers themselves in order. This of course implies that, for any largest considered prime Pmax in this sequence, there will exist a larger entry P2 with digit sum = Pmax. Note the data available for such scrutiny grows very slowly - considering primes through 10^12 only attains digit sum to (prime) 97. Additionally, a parallel observation can be drawn about the behavior of companion sequence A119449. Also, this sequence appears to be a subset of A156756. - Bill McEachen, Mar 26 2017

T. D. Noe, Table of n, a(n) for n = 1..10000
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 13, 2010.

Primes with even digit sum A119449.

Cf. A000040, A200260.

select(t -> isprime(t) and convert(convert(t,base,10),`+`)::odd, [seq(i,i=3..1000,2)]); # Robert Israel, Feb 13 2017

Select[Prime@ Range@ 108, OddQ@ Total@ IntegerDigits@ # &] (* Michael De Vlieger, Feb 11 2017 *)

is(n)=isprime(n) && sumdigits(n)%2 \\ Charles R Greathouse IV, Feb 14 2017

a(n) = A000040(A200260(n)). - Jon Maiga, Jul 03 2021

{A000040(k) : A104638(k) odd}. - R. J. Mathar, Jul 13 2025

A243586 Primes p such that sum of digits + 3 is prime.

Original entry on oeis.org

2, 11, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 101, 103, 107, 109, 127, 149, 163, 167, 181, 211, 233, 239, 251, 257, 271, 277, 293, 307, 347, 349, 367, 383, 389, 419, 431, 433, 439, 457, 479, 491, 503, 509, 521, 523, 541, 547, 563, 569, 587, 613, 617

Offset: 1

Vincenzo Librandi, Jun 07 2014

Naturally, this sequence is not the same as A119449. First disagreement at index 44: a(44)=503, A119449(44)=499.

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

Cf. primes p such that sum of digits + k is prime: A166561 (k=1), this sequence (k=3), A176985 (k=5), A243587 (k=7), A243588 (k=9).

[p: p in PrimesUpTo(700) | IsPrime(q) where q is 3+&+Intseq(p)];

Select[Prime[Range[200]], PrimeQ[Plus@@IntegerDigits[#] + 3]&]

A062338 Primes whose sum of digits is a multiple of 4.

Original entry on oeis.org

13, 17, 31, 53, 71, 79, 97, 103, 107, 211, 233, 251, 277, 349, 367, 389, 431, 439, 457, 479, 503, 521, 547, 569, 587, 619, 659, 673, 677, 691, 701, 709, 727, 839, 853, 857, 907, 929, 947, 983, 1021, 1061, 1069, 1087, 1151, 1201, 1223, 1249, 1289, 1429

Offset: 1

Amarnath Murthy, Jun 21 2001

			349 is a prime with sum of digits =16=4*4, hence belongs to the sequence.

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

Subsequence of A119449.

Cf. A062336, A062337.

[ p: p in PrimesUpTo(10000) | &+Intseq(p) mod 4 eq 0 ]; /* Vincenzo Librandi, Apr 02 2011 */

filter:= x -> (convert(convert(x,base,10),`+`) mod 4 = 0) and isprime(x);
A062338:= select(filter, [seq(2*i+1,i=0..1000)]); # Robert Israel, Apr 20 2014

is(n)=isprime(n) && sumdigits(n)%4==0 \\ Charles R Greathouse IV, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001

A200259 Numbers k such that k-th prime has an even digit sum.

Original entry on oeis.org

1, 5, 6, 7, 8, 11, 12, 16, 17, 20, 21, 22, 25, 26, 27, 28, 29, 31, 35, 38, 39, 42, 47, 51, 52, 54, 55, 58, 59, 62, 63, 69, 70, 73, 76, 77, 81, 83, 84, 85, 88, 92, 94, 95, 96, 97, 98, 99, 100, 101, 103, 104, 107, 112, 113, 114, 115, 119, 120, 122, 123, 125

Offset: 1

N. J. A. Sloane, Nov 15 2011

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A119449, A200260.

[ n: n in [1..169] | IsEven(&+Intseq(NthPrime(n)))]; // Bruno Berselli, Nov 15 2011

Select[Range[1000], EvenQ[Total[IntegerDigits[Prime[#]]]] &] (* T. D. Noe, Nov 21 2011 *)

A186647 Even numbers whose decimal digits sum to a prime.

Original entry on oeis.org

2, 12, 14, 16, 20, 30, 32, 34, 38, 50, 52, 56, 58, 70, 74, 76, 92, 94, 98, 102, 104, 106, 110, 120, 122, 124, 128, 140, 142, 146, 148, 160, 164, 166, 182, 184, 188, 200, 210, 212, 214, 218, 230, 232

Offset: 1

Giovanni Teofilatto, Feb 25 2011

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

Cf. A119449.

Select[2*Range[0,200],PrimeQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Oct 05 2019 *)

A247797 Lexicographically earliest permutation of prime numbers, such that adjacent terms have coprime sums of digits in decimal representation.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 13, 29, 17, 41, 31, 43, 19, 47, 37, 61, 53, 67, 59, 83, 71, 89, 73, 137, 79, 113, 97, 131, 101, 139, 103, 151, 107, 157, 109, 173, 127, 179, 149, 191, 163, 193, 167, 197, 181, 199, 211, 223, 227, 229, 233, 241, 251, 263, 239, 269, 257

Offset: 1

Reinhard Zumkeller, Nov 25 2014

A049084(a(n)) defines a permutation of the positive integers, cf. A250552.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals of Mathematics, Vol. 171, No. 3, 1591-1646, 2010.

Cf. A000040, A007953, A007605, A049084, A250552, A119449, A119450.

import Data.List (delete)
a247797 n = a247797_list !! (n-1)
a247797_list = f 1 $ zip a000040_list a007605_list where
   f q' vws = g vws where
     g  ((p,q):pqs) = if gcd q q' == 1
                         then p : f q (delete (p,q) vws) else g pqs

A338976 Primes p such that p*A007953(p)+1 is prime.

Original entry on oeis.org

2, 11, 13, 17, 19, 59, 71, 97, 107, 109, 149, 167, 181, 239, 271, 419, 431, 499, 509, 523, 547, 563, 613, 631, 691, 727, 811, 853, 859, 983, 1009, 1063, 1087, 1117, 1151, 1193, 1229, 1409, 1427, 1487, 1559, 1579, 1601, 1759, 1823, 1913, 1973, 2039, 2099, 2161, 2237, 2251, 2309, 2411, 2437, 2473

Offset: 1

J. M. Bergot and Robert Israel, Dec 18 2020

			a(3) = 13 is a term because 13 and 13*(1+3)+1 = 53 are prime.

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

Cf. A007953, A057147.

Subsequence of A119449.

select(t -> isprime(t) and isprime(t*convert(convert(t,base,10),`+`)+1), [$2..10^4]);

isok(p) = isprime(p) && isprime(p*sumdigits(p)+1); \\ Michel Marcus, Dec 18 2020

A199339 a(n) = number of primes with an even digit sum among the first n primes minus the number with an odd digit sum.

Original entry on oeis.org

1, 0, -1, -2, -1, 0, 1, 2, 1, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 5, 4, 5, 4, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, -1, -2, -1, -2, -3, -4, -3, -2, -3, -2, -1, -2, -3, -2, -1, -2, -3, -2, -1, -2, -3, -4, -5, -6, -5, -4, -5, -6, -5, -6

Offset: 1

M. F. Hasler, Nov 14 2011

			a(1)=1 because the first prime has an even sum of digits.
a(2)=0, a(3)=-1, a(4)=-2 because the following primes (3,5,7) have odd sum of digits.
a(5)=-1, a(6)=0, a(7)=1, a(8)=2 because the 5th, 6th, 7th and 8th prime (11, 13, 17, 19) have an even sum of digits.

T. D. Noe, Table of n, a(n) for n = 1..10000
CNRS Press release, The sum of digits of prime numbers is evenly distributed, May 12, 2010.
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 12, 2010.

Cf. A007605, A007953, A130911, A200259, A200260, A119449, A119450, A200261, A200262, A200263, A200264.

a[1] := 1; a[n_] := a[n] = a[n - 1] + (-1)^(Plus@@IntegerDigits[Prime[n]]); Table[a[n], {n, 74}] (* Alonso del Arte, Nov 14 2011 *)

s=0;vector(90,n,s+=(-1)^A007953(prime(n)))

a(n)=sum_{k=1...n} (-1)^A007605(n).

Equals A200262 - A200264.

A227785 Primes p such that p - ssd(p) is the square of a prime, where ssd(k) is the sum of the squared decimal digits of k.

Original entry on oeis.org

11, 2903, 3533, 3803, 5197, 9533, 18973, 24763, 37321, 73561, 96953, 113621, 124777, 129097, 134837, 139241, 398341, 830003, 1100509, 1585201, 1661789, 2211257, 4541309, 4871077, 4897709, 5340949, 5958751, 7393123, 8185501, 8744003, 11485559, 15343039, 15343079

Offset: 1

Underwood Dudley, Will Gosnell, Charles R Greathouse IV, and Robert Israel, Aug 09 2013

			11 is a member since 11 - 2 = 3^2 where 3 is prime. 2903 is a member since 2903 - 94 = 53^2 where 53 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2900

Subsequence of A119449. ssd(n) = A003132(n).

ssd:= n->add(d^2,d=convert(n,base,10));
S:= select(t -> type(sqrt(t - ssd(t)),prime), [seq(ithprime(j),j=1..10^5)]);

fQ[n_] := PrimeQ[ Sqrt[ n - Total[ IntegerDigits[ n]^2]]]; p = 2; lst = {}; While[p < 15500000, If[ fQ@ p, AppendTo[ lst, p]]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jun 01 2014 *)

ssd(n)=my(d=digits(n));sum(i=1,#d,d[i]^2)
v=List();forprime(p=2,1e5,if(issquare(p-ssd(p),&t) && isprime(t), listput(v,p))); Vec(v)

cp's OEIS Frontend

A119450 Primes with odd digit sum.

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A243586 Primes p such that sum of digits + 3 is prime.

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Magma

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A062338 Primes whose sum of digits is a multiple of 4.

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Magma

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Extensions

A200259 Numbers k such that k-th prime has an even digit sum.

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Magma

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A186647 Even numbers whose decimal digits sum to a prime.

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Mathematica

A247797 Lexicographically earliest permutation of prime numbers, such that adjacent terms have coprime sums of digits in decimal representation.

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Haskell

A338976 Primes p such that p*A007953(p)+1 is prime.

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Maple

PARI

A199339 a(n) = number of primes with an even digit sum among the first n primes minus the number with an odd digit sum.

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Mathematica