Displaying 1-10 of 12 results found.
Primes p where the digital sum is equal to 68.
+10
43
59999999, 69999899, 69999989, 78998999, 88989899, 88999979, 89699999, 89799989, 89989799, 89989979, 89997899, 89997989, 89999699, 89999969, 97889999, 98699999, 98879999, 98899799, 98979989, 98988899, 98989889, 98997989, 98998979, 98999969
EXAMPLE
69999899 is a prime with sum of the digits = 68, hence belongs to the sequence.
MATHEMATICA
Select[Prime[Range[10000000]], Total[IntegerDigits[#]]==68 &]
PROG
(Magma) [p: p in PrimesUpTo(100000000) | &+Intseq(p) eq 68];
(Python) # 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
CROSSREFS
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).
Primes whose sum of digits is 4.
+10
26
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
COMMENTS
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
EXAMPLE
3001 is a prime with sum of digits = 4, hence belongs to the sequence.
MAPLE
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:
MATHEMATICA
Union[FromDigits/@Select[Flatten[Table[Tuples[{0, 1, 2, 3}, k], {k, 9}], 1], PrimeQ[FromDigits[#]]&&Total[#]==4&]] (* Jayanta Basu, May 19 2013 *)
PROG
(PARI) 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 (PARI) 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 (PARI) 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 (Magma) [p: p in PrimesUpTo(800000000) | &+Intseq(p) eq 4]; // Vincenzo Librandi, Jul 08 2014
CROSSREFS
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. 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).
EXTENSIONS
Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
Primes whose sum of digits is 8.
+10
13
17, 53, 71, 107, 233, 251, 431, 503, 521, 701, 1061, 1151, 1223, 1511, 1601, 2141, 2213, 2411, 3023, 3041, 3203, 3221, 4013, 4211, 5003, 5021, 6011, 6101, 7001, 10007, 10061, 10133, 10151, 10223, 10313, 10331, 10601, 11213, 11321, 11411
EXAMPLE
1151 belongs to the sequence since it is a prime with sum of digits = 8.
MATHEMATICA
Select[Prime[Range[500000]], Total[IntegerDigits[#]]==8 &] (* Vincenzo Librandi, Jul 08 2014 *)
PROG
(Magma) [p: p in PrimesUpTo(20000) | &+Intseq(p) eq 8]; // Vincenzo Librandi, Jul 08 2014 (PARI) select( {is_ A062343(p, s=8)=sumdigits(p)==s&&isprime(p)}, primes([1, 12345])) \\ 2nd optional parameter for similar sequences with other digit sums. (PARI) { A062343_upto_length(L, s=8, 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)} \\ (End)
CROSSREFS
Subsequence of A062342 (primes with digit sum divisible by 8).
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001
Primes p with digital sum equal to 11.
+10
12
29, 47, 83, 137, 173, 191, 227, 263, 281, 317, 353, 443, 461, 641, 821, 911, 1019, 1091, 1109, 1163, 1181, 1217, 1307, 1361, 1433, 1451, 1523, 1613, 1721, 1811, 1901, 2027, 2063, 2081, 2153, 2207, 2243, 2333, 2351, 2423, 2441, 2531, 2621, 2711, 2801, 3251
MATHEMATICA
Select[Prime[Range[100000]], Total[IntegerDigits[#]]==11 &] (* Vincenzo Librandi, Jul 08 2014 *)
PROG
(Magma) [p: p in PrimesUpTo(10000) | &+Intseq(p) eq 11]; // Vincenzo Librandi, Jul 08 2014 (PARI) select( {is_ A106754(n)=sumdigits(n)==11&&isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022
CROSSREFS
Subsequence of A119891 (prime trios: chain of prime sums of digits; also has as subsequence A106762 (s = 23), A106774 (s = 41), etc).
Primes with digit sum = 26.
+10
4
1889, 1979, 1997, 2699, 2789, 2879, 2897, 2969, 3779, 3797, 4679, 4787, 4877, 4967, 5399, 5669, 5849, 5867, 5939, 6299, 6389, 6569, 6659, 6857, 6947, 6983, 7487, 7559, 7577, 7649, 7757, 7793, 7829, 7883, 7919, 7937, 8297, 8369, 8387, 8693, 8747, 8783
MATHEMATICA
Select[Prime[Range[1300]], Total[IntegerDigits[#]]==26&] (* Harvey P. Dale, Feb 14 2011 *)
PROG
(Magma) [p: p in PrimesUpTo(9000) | &+Intseq(p) eq 26]; // Vincenzo Librandi, Jul 08 2014 (PARI) select(x->sumdigits(x)==26, primes(1000)) \\ Michel Marcus, Jul 08 2014 (Python)
a= A107579(p=1889); [next(a) for _ in range(50)] # providing optional 1st arg = initial term, to "universal" code in A107579. - M. F. Hasler, Mar 16 2022
CROSSREFS
Cf. similar sequences listed in A244918.
Primes whose digit sum contains one or more digits of the same prime.
+10
4
2, 3, 5, 7, 19, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 271, 281, 283, 307, 317, 337, 347, 367, 373, 379, 397, 419, 461, 463, 467, 491, 541, 557, 571, 613, 617, 619, 631, 641, 643, 647, 661, 673, 691, 719, 733, 739, 743, 751
EXAMPLE
139 is a prime whose digit sum of 13 contains the digits 1 and 3 which are also in the prime.
149 is a prime whose digit sum of 14 contains the digits 1 and 4 which are also in the prime.
419 is a prime whose digit sum of 14 contains the digits 1 and 4 which are also in the prime.
MAPLE
filter:= proc(n) local L, s;
L:= convert(n, base, 10);
s:= convert(L, `+`);
convert(convert(s, base, 10), set) intersect convert(L, set) <> {}
end proc:
select(filter, [seq(ithprime(i), i=1..100)]); # Robert Israel, Feb 27 2023
PROG
(PARI) isok(p) = isprime(p) && (#setintersect(Set(digits(p)), Set(digits(sumdigits(p)))) >= 1); \\ Michel Marcus, Nov 12 2017
Primes with digital sum 70.
+10
4
189997999, 199799989, 199898899, 199997899, 199997989, 199998889, 268999999, 269998999, 278989999, 278999989, 279889999, 279988999, 287998999, 287999989, 288998989, 288999889, 288999979, 289699999, 289789999, 289889989
COMMENTS
The sequence begins with 8438 9-digit numbers.
Then there are 739572 10-digit numbers.
All terms == 7 (mod 18).
MATHEMATICA
Select[Prime[Range[3*10^8]], Total[IntegerDigits[#]]==70 &] (* Vincenzo Librandi, Jul 09 2014 *)
PROG
(Magma) [p: p in PrimesUpTo(3*10^8) | &+Intseq(p) eq 70]; // Vincenzo Librandi, Jul 09 2014 (Python) # see code in A107579 which can be used to produce this sequence by giving the initial term p = 189997999 (or 8*10**7-1, for digit sum 70). - M. F. Hasler, Mar 16 2022
CROSSREFS
Cf. similar sequences listed in A244918.
Primes whose sum of digits is a multiple of 5.
+10
3
5, 19, 23, 37, 41, 73, 109, 113, 127, 131, 163, 181, 271, 307, 311, 389, 401, 433, 479, 523, 541, 569, 587, 613, 631, 659, 677, 811, 839, 857, 929, 947, 983, 997, 1009, 1013, 1031, 1063, 1103, 1117, 1153, 1171, 1289, 1301, 1423, 1487, 1531, 1559, 1621, 1667
EXAMPLE
569 is a prime with sum of digits = 20, hence belongs to the sequence.
MATHEMATICA
Select[Prime[Range[300]], Divisible[Total[IntegerDigits[#]], 5]&] (* Harvey P. Dale, Jul 06 2020 *)
PROG
(Magma) [ p: p in PrimesUpTo(10000) | &+Intseq(p) mod 5 eq 0 ]; // Vincenzo Librandi, Apr 02 2011 (Python)
from sympy import primerange as primes
def ok(p): return sum(map(int, str(p)))%5 == 0
(PARI) select( {is_ A062340(n)=sumdigits(n)%5==0&&isprime(n)}, primes([1, 2000])) \\ M. F. Hasler, Mar 10 2022
EXTENSIONS
Corrected and extended by Harvey P. Dale and Larry Reeves (larryr(AT)acm.org), Jul 04 2001
Palindromic primes with digit sum 20.
+10
3
929, 16661, 17471, 36263, 70607, 72227, 73037, 91019, 1074701, 1082801, 1180811, 1262621, 1328231, 1360631, 1508051, 1532351, 1630361, 1712171, 1802081, 3160613, 3218123, 7014107, 7300037, 9002009, 102383201, 102707201, 103282301
COMMENTS
Cf. A070250 Palindromic primes with digit sum 10, A107579 Primes with digit sum = 10, A106760 Primes with digit sum = 20, A109185 Palindromic primes with digit sum 40.
MATHEMATICA
Select[Prime[Range[5940000]], PalindromeQ[#]&&Total[IntegerDigits[#]]==20&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 19 2021 *)
Palindromic primes with digit sum = 40.
+10
3
97879, 98689, 1878781, 1968691, 1976791, 1984891, 3768673, 3784873, 3792973, 3858583, 3948493, 3964693, 7278727, 7392937, 7466647, 7564657, 7654567, 7662667, 7850587, 7916197, 9078709, 9446449, 9470749, 9626269, 9634369
COMMENTS
Cf. A070250 Palindromic primes with digit sum = 10, A107579 Primes with digit sum = 10, A106760 Primes with digit sum = 20, A109184 Palindromic primes with digit sum = 20, A109207 Palindromic primes with digit sum = 50.
MATHEMATICA
Select[Prime@ Range[9000, 10^6], And[# == Reverse@ #, Total@ # == 40] &@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 18 2015 *)
PROG
(PARI) isok(n) = isprime(n) && (d=digits(n)) && (Vecrev(d)==d) && (sumdigits(n)==40); \\ Michel Marcus, Dec 18 2015
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