A106758 -id:A106758 - cp's OEIS frontend

A244918 Primes p where the digital sum is equal to 68.

59999999, 69999899, 69999989, 78998999, 88989899, 88999979, 89699999, 89799989, 89989799, 89989979, 89997899, 89997989, 89999699, 89999969, 97889999, 98699999, 98879999, 98899799, 98979989, 98988899, 98989889, 98997989, 98998979, 98999969

Offset: 1

Vincenzo Librandi, Jul 08 2014

			69999899 is a prime with sum of the digits = 68, hence belongs to the sequence.

Vincenzo Librandi and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..45 from Vincenzo Librandi.

Cf. Primes p where the digital sum is equal to k: 2, 11 and 101 for k=2; A062339 (k=4), A062341 (k=5), A062337 (k=7), A062343 (k=8), A107579 (k=10), A106754 (k=11), A106755 (k=13), A106756 (k=14), A106757 (k=16), A106758 (k=17), A106759 (k=19), A106760 (k=20), A106761 (k=22), A106762 (k=23), A106763 (k=25), A106764 (k=26), A048517 (k=28), A106766 (k=29), A106767 (k=31), A106768 (k=32), A106769 (k=34), A106770 (k=35), A106771 (k=37), A106772 (k=38), A106773 (k=40), A106774 (k=41), A106775 (k=43), A106776 (k=44), A106777 (k=46), A106778 (k=47), A106779 (k=49), A106780 (k=50), A106781 (k=52), A106782 (k=53), A106783 (k=55), A106784 (k=56), A106785 (k=58), A106786 (k=59), A106787 (k=61), A107617 (k=62), A107618 (k=64), A107619 (k=65), A106807 (k=67), this sequence (k=68), A181321 (k=70).

[p: p in PrimesUpTo(100000000) | &+Intseq(p) eq 68];

Select[Prime[Range[10000000]], Total[IntegerDigits[#]]==68 &]

# see code in A107579: the same code can be used to produce this sequence, by giving the initial term p = 6*10**7-1, for digit sum 68. - M. F. Hasler, Mar 16 2022

A062339 Primes whose sum of digits is 4.

Original entry on oeis.org

13, 31, 103, 211, 1021, 1201, 2011, 3001, 10111, 20011, 20101, 21001, 100003, 102001, 1000003, 1011001, 1020001, 1100101, 2100001, 10010101, 10100011, 20001001, 30000001, 101001001, 200001001, 1000000021, 1000001011, 1000010101, 1000020001, 1000200001, 1002000001, 1010000011

Offset: 1

Amarnath Murthy, Jun 21 2001

Is this sequence (and its brothers A062337, A062341 and A062343) infinite?

10^A049054(m)+3 and 3*10^A056807(m)+1 are subsequences. A107715 (primes containing only digits from set {0,1,2,3}) is a supersequence. Terms not containing the digit 3 are either terms of A020449 (primes that contain digits 0 and 1 only) or of A106100 (primes with maximal digit 2) - and thus terms of these sequences' union A036953 (primes containing only digits from set {0,1,2}). - Rick L. Shepherd, May 23 2005

			3001 is a prime with sum of digits = 4, hence belongs to the sequence.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe)
Amin Witno, Numbers which factor as their digital sum times a prime, International Journal of Open Problems in Computer Science and Mathematics 3:2 (2010), pp. 132-136.

Subsequence of A062338, A107288, and A107715 (primes with digits <= 3).
A159352 is a subsequence.
Cf. A000040 (primes), A052218 (digit sum = 4), A061239 (primes == 4 (mod 9)).
Cf. Primes p with digital sum equal to k: {2, 11 and 101} for k=2; this sequence (k=4), A062341 (k=5), A062337 (k=7), A062343 (k=8), A107579 (k=10), A106754 (k=11), A106755 (k=13), A106756 (k=14), A106757 (k=16), A106758 (k=17), A106759 (k=19), A106760 (k=20), A106761 (k=22), A106762 (k=23), A106763 (k=25), A106764 (k=26), A048517 (k=28), A106766 (k=29), A106767 (k=31), A106768 (k=32), A106769 (k=34), A106770 (k=35), A106771 (k=37), A106772 (k=38), A106773 (k=40), A106774 (k=41), A106775 (k=43), A106776 (k=44), A106777 (k=46), A106778 (k=47), A106779 (k=49), A106780 (k=50), A106781 (k=52), A106782 (k=53), A106783 (k=55), A106784 (k=56), A106785 (k=58), A106786 (k=59), A106787 (k=61), A107617 (k=62), A107618 (k=64), A107619 (k=65), A106807 (k=67), A244918 (k=68), A181321 (k=70).
Cf. A049054 (10^k+3 is prime), A159352 (these primes).
Cf. A056807 (3*10^k+1 is prime), A259866 (these primes).
Cf. A020449 (primes with digits 0 and 1), A036953 (primes with digits <= 2), A106100 (primes with largest digit = 2), A069663, A069664 (smallest resp. largest n-digit prime with minimum digit sum).

[p: p in PrimesUpTo(800000000) | &+Intseq(p) eq 4]; // Vincenzo Librandi, Jul 08 2014

N:= 20: # to get all terms < 10^N
B[1]:= {1}:
B[2]:= {2}:
B[3]:= {3}:
A:= {}:
for d from 2 to N do
   B[4]:= map(t -> 10*t+1,B[3]) union  map(t -> 10*t+3,B[1]);
   B[3]:= map(t -> 10*t, B[3]) union map(t -> 10*t+1,B[2]) union map(t -> 10*t+2,B[1]);
   B[2]:= map(t -> 10*t, B[2]) union map(t -> 10*t+1,B[1]);
   B[1]:= map(t -> 10*t, B[1]);
   A:= A union select(isprime,B[4]);
od:
sort(convert(A,list)); # Robert Israel, Dec 28 2015

Union[FromDigits/@Select[Flatten[Table[Tuples[{0,1,2,3},k],{k,9}],1],PrimeQ[FromDigits[#]]&&Total[#]==4&]] (* Jayanta Basu, May 19 2013 *)
FromDigits/@Select[Tuples[{0,1,2,3},10],Total[#]==4&&PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Jul 23 2025 *)

for(a=1,20,for(b=0,a,for(c=0,b,if(isprime(k=10^a+10^b+10^c+1),print1(k", "))))) \\ Charles R Greathouse IV, Jul 26 2011

select( {is_A062339(p,s=4)=sumdigits(p)==s&&isprime(p)}, primes([1,10^7])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

A062339_upto_length(L,s=4,a=List(),u=[10^(L-k)|k<-[1..L]])=forvec(d=[[1,L]|i<-[1..s]], isprime(p=vecsum(vecextract(u,d))) && listput(a,p),1); Vecrev(a) \\ M. F. Hasler, Mar 09 2022

Intersection of A052218 (digit sum 4) and A000040 (primes). - M. F. Hasler, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Rick L. Shepherd, May 23 2005
More terms from Lekraj Beedassy, Dec 19 2007

A061242 Primes of the form 9*k - 1.

Original entry on oeis.org

17, 53, 71, 89, 107, 179, 197, 233, 251, 269, 359, 431, 449, 467, 503, 521, 557, 593, 647, 683, 701, 719, 773, 809, 827, 863, 881, 953, 971, 1061, 1097, 1151, 1187, 1223, 1259, 1277, 1367, 1439, 1493, 1511, 1583, 1601, 1619, 1637, 1709, 1871, 1889, 1907

Offset: 1

Amarnath Murthy, Apr 23 2001

Or, primes of the form 18k - 1. Corresponding values of k are in A138918. - Zak Seidov, Apr 03 2008
From Doug Bell, Mar 23 2009: (Start)
Conjecture: if a(n) = 9x - 1, the integer formed by the repeating digits in the decimal fraction x/a(n) is the smallest integer such that rotating the digits to the left produces a number which is (x+1)/x times larger.
Example: x = 2, a(n) = 17: 2/17 = 0.1176470588235294... repeating with a cycle of 16.
1176470588235294 * 3/2 = 1764705882352941, which is 1176470588235294 rotated to the left.
An additional conjecture is that the values of x from this sequence are the only values where rotating an integer one to the left produces a value (x+1)/x times as large. (End)
The last conjecture is false. For example, for x = 3 we have 230769*(4/3) = 307692, but 9*3-1 = 26 is not in the sequence. - Giovanni Resta, Jul 28 2015
Conjecture: Primes p such that ((x+1)^9-1)/x has 4 irreducible factors of degree 2 over GF(p). - Federico Provvedi, Jun 27 2018

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A061237, A061238, A061239, A061240, A061241 (p mod 9 = 1, 2, 4, 5 and 7), A138918 (18n - 1 is prime), A258663 (9n - 1 is prime).

Can be partitioned in disjoint subsequences A062343 (primes with sum of digits s = 8), A106758 (s = 17), A106764 (s = 26), A106770 (s = 35), A106776 (s = 44), A106782 (s = 53), A107617 (s = 62), etc.

[a: n in [0..250] | IsPrime(a) where a is 9*n - 1 ]; // Vincenzo Librandi, Jun 07 2015

select(isprime, [seq(18*i-1,i=1..1000)]); # Robert Israel, Sep 03 2014

Select[ Range[ 2500 ], PrimeQ[ # ] && Mod[ #, 9 ] == 8 & ]
Select[9*Range[300] - 1, PrimeQ]

select( {is(n)=n%9==8&&isprime(n)}, primes([1,2000])) \\ M. F. Hasler, Mar 10 2022

from sympy import prime
A061242 = [p for p in (prime(n) for n in range(1,10**3)) if not (p+1) % 18]
# Chai Wah Wu, Sep 02 2014

A010888(a(n)) = 8. - Reinhard Zumkeller, Feb 25 2005

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

More terms from Robert G. Wilson v, May 10 2001
Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 30 2008
Edited by M. F. Hasler, Mar 10 2022

A106757 Primes with digit sum = 16.

Original entry on oeis.org

79, 97, 277, 349, 367, 439, 457, 547, 619, 673, 691, 709, 727, 853, 907, 1069, 1087, 1249, 1429, 1447, 1483, 1609, 1627, 1663, 1753, 1861, 1933, 1951, 2239, 2293, 2347, 2383, 2437, 2473, 2617, 2671, 2707, 2833, 2851, 3049, 3067, 3229, 3319, 3373, 3391

Offset: 1

Zak Seidov, May 16 2005

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

Cf. A000040 (primes), A007953 (sum of digits), A235227 (digit sum = 16).
Cf. A062339 (same for digit sum s = 4), A106756 (s = 14), A106758 (s = 17), and others listed in A244918 (s = 68).
Subsequence of A062342 (primes whose sum of digits is a multiple of 8) and of A107288 (primes with sum of digits a square).

[p: p in PrimesUpTo(4000) | &+Intseq(p) eq 16]; // Vincenzo Librandi, Jul 08 2014

Reap[Do[If[16==Apply[Plus,IntegerDigits[p=Prime[n]]],Sow[p]],{n,1000}]][[2,1]] (* Zak Seidov, Oct 30 2009 *)
Select[Prime[Range[500]],Total[IntegerDigits[#]]==16&] (* Harvey P. Dale, Nov 14 2011 *)

select( {is_A106757(n)= sumdigits(n)==16 && isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A235227 (digit sum = 16); also equals { p in A000040 | A007953(p) = 16 }. - M. F. Hasler, Mar 09 2022

More terms from Zak Seidov, Oct 30 2009

cp's OEIS Frontend

A244918 Primes p where the digital sum is equal to 68.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

A062339 Primes whose sum of digits is 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A061242 Primes of the form 9*k - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A106757 Primes with digit sum = 16.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions