A106766 -id:A106766 - 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

A119892 Prime quartet leaders: largest number of a prime quartet.

Original entry on oeis.org

2999, 3989, 4799, 4889, 5879, 5897, 5987, 6599, 6689, 6779, 6869, 6959, 6977, 7499, 7589, 7877, 7949, 8597, 8669, 8849, 8867, 9479, 9497, 9587, 9677, 9749, 9767, 9839, 9857, 9929, 12899, 13799, 13997, 14699, 14879, 14897, 14969, 15797, 15887, 15959

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 27 2006

A prime quartet is a set of four different prime numbers such that the fourth number is a 1-digit number which is the sum of the digits of the third number, the third number is the sum of the digits of the second number and the second number is the sum of the digits of the first number.
Different from A106766.
Comment from Joshua Zucker, Apr 24 2007, on the difference between this sequence and A106766: The digit sum must be the largest member of a prime trio, so the first number where the sequences differ must be with digit sum 47 and thus have at least 6 digits - so until then you get all the primes with 4 or 5 digits that have digit sum 29.
a(2322)=389999 is the first value different from A106766, where A106766(2322)=390359. See also A106778 = primes with digit sum = 47: A106778(1)=389999. - Martin Fuller and Ray Chandler, Apr 24 2007
The sequence of prime quintet leaders is probably too large for the OEIS; its first term is the 334-digit prime 5*10^333-10^330-10^328-1 with sum of digits a(1) = 2999. - Charles R Greathouse IV, Mar 11 2022

			2999 is in the sequence because it is the largest number of the prime quartet (2999,29,11,2).

Harvey P. Dale, Table of n, a(n) for n = 1..2000
L. Stevens, Prime ensembles

Cf. A119889, A119890, A119891.

pqQ[n_]:=Module[{p1=NestList[Total[IntegerDigits[#]]&,n,3]},AllTrue[ Take[ p1,3],#>9&]&&AllTrue[p1,PrimeQ]]; Select[Range[16000],pqQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 02 2020 *)

DigitSum(n,b=10)=local(x);x=0;while(n,x+=n%b;n\=b);x
PrimeEnsemble(n,b=10)=local(x);x=1;while(ispseudoprime(n),if(n=4, print1(p", "))); \\ Martin Fuller

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

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A062339 Primes whose sum of digits is 4.

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A119892 Prime quartet leaders: largest number of a prime quartet.

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