A119890 -id:A119890 - cp's OEIS frontend

A119891 Prime trio leaders: largest number of a prime trio.

29, 47, 83, 137, 173, 191, 227, 263, 281, 317, 353, 443, 461, 599, 641, 797, 821, 887, 911, 977, 1019, 1091, 1109, 1163, 1181, 1217, 1307, 1361, 1433, 1451, 1499, 1523, 1613, 1697, 1721, 1787, 1811, 1877, 1901, 1949, 2027, 2063, 2081, 2153, 2207, 2243

Offset: 1

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

A prime trio is a set of three distinct prime numbers such that the third number is a 1-digit number which is the sum of the digits of the second number and the second number is the sum of the digits of the first number.

			443 is in the sequence because it is the largest number of the prime trio (443, 11, 2).
599 is the first term with sum of digits different from 11 (cf. A106754), namely 23 (cf. A106762). This sequence contains also all primes with sum of digits equal to 41, 43, 61 etc., but not 29, 47, ... since the second digit sum must be a single-digit prime, i.e., 2, 3, 5 or 7. - _M. F. Hasler_, Mar 09 2022

Robert Israel, Table of n, a(n) for n = 1..10000
Luc Stevens, Prime ensembles

Subsequence of A304367.
Cf. A000040 (primes), A007953 (sum of digits), A106754 (primes with s.o.d. = 11), A106762 (s.o.d.(p) = 23), A106774 (s.o.d.(p) = 41), A106775 (s.o.d.(p) = 43), A106787 (s.o.d.(p) = 61): subsequences.
Cf. A119889, A119890, A119892.

filter:= proc(n) local x,y;
  if not isprime(n) then return false fi;
  x:= convert(convert(n,base,10),`+`);
  if x < 10 or not isprime(x) then return false fi;
  y:= convert(convert(x,base,10),`+`);
  member(y,{2,3,5,7})
end proc:
select(filter, [seq(i,i=11..10000,2)]); # Robert Israel, May 21 2021

ptQ[n_]:=Module[{c=NestList[Total[IntegerDigits[#]]&,n,2]},Length[ Union[c]] == 3&&And@@PrimeQ[c]]; Select[Prime[Range[500]],ptQ] (* Harvey P. Dale, Aug 15 2012 *)

select( {is_A119891(n, s=sumdigits(n))=bittest(172, sumdigits(s)) && isprime(s) && s>9 && isprime(n)}, primes([1,2345])) \\ M. F. Hasler, Mar 09 2022

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

A119889 Prime soloist : prime number which is no member of any prime ensemble.

Original entry on oeis.org

3, 13, 17, 19, 31, 37, 53, 59, 67, 71, 73, 79, 89, 97, 103, 107, 109, 127, 139, 149, 157, 163, 167, 179, 181, 193, 197, 199, 211, 229, 233, 239, 251, 257, 269, 271, 277, 283, 293, 307, 337, 347, 349, 359, 367, 373, 379, 383, 389, 397, 409, 419, 431, 433, 439

Offset: 1

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

			17 is in the sequence because the sum of the digits (8) is a composite number. 449 is in the sequence because the sum of the digits (17) is another prime soloist.

L. Stevens, Prime ensembles

Cf. A119890, A119891, A119892.

Corrected by T. D. Noe, Oct 25 2006

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A119891 Prime trio leaders: largest number of a prime trio.

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

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A119889 Prime soloist : prime number which is no member of any prime ensemble.

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