A062088 -id:A062088 - cp's OEIS frontend

A253140 Smallest of three consecutive primes in arithmetic progression with common difference 24 and digit sum prime.

89, 373, 773, 863, 1279, 2063, 2089, 2399, 2663, 2753, 3299, 4153, 4373, 5879, 6173, 6263, 6779, 7079, 7499, 7853, 9473, 10453, 11399, 12253, 12479, 14699, 16763, 19379, 21163, 21563, 25073, 29363, 32189, 33599, 40063, 41879, 42773, 50053, 50363, 52673, 56453

Offset: 1

K. D. Bajpai, Dec 27 2014

			a(1) = 89: 89 + 24 = 113; 113 + 24 = 137; all three are prime. Their digit sums 8+9 = 17, 1+1+3 = 5 and 1+3+7 = 11 are also prime.
a(2) = 373: 373 + 24 = 397; 397 + 24 = 421; all three are prime. Their digit sums 3+7+3 = 13, 3+9+7 = 19 and 4+2+1 = 7 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A033447, A062088.

A253140 = {}; Do[d = 24; k = Prime[n]; k1 = k+d; k2 = k+2d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[Plus@@IntegerDigits[k]] && PrimeQ[Plus@@IntegerDigits[k1]] && PrimeQ[Plus@@IntegerDigits[k2]], AppendTo[A253140,k]], {n,20000}]; A253140
tcpQ[n_]:=Module[{a=n+24,b=n+48},AllTrue[{a,b},PrimeQ]&&AllTrue[Total/@ (IntegerDigits/@{n,a,b}),PrimeQ]]; Select[Prime[Range[6000]],tcpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 16 2016 *)

A110028 Primes with a prime number of digits, all of them prime, that add up to a prime.

Original entry on oeis.org

23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 22573, 23327, 25237, 25253, 25523, 27253, 27527, 32233, 32237, 32257, 32323, 32327, 33223, 33353, 33377, 33533, 33773, 35227, 35353, 35533, 35537, 35573, 35753, 37223, 37337, 52237, 52253, 52727, 53353

Offset: 1

Sergio Pimentel, Mar 31 2006

First 7-digit number in this sequence is 2222333.
From Michael De Vlieger, Feb 02 2019: (Start)
First p-digit number in this sequence:
                                     23
                                    223
                                  22573
                                2222333
                            22222222223
                          2222222225323
                      22222222222225237
                    2222222222222223527
                22222222222222222232723
          22222222222222222222222222577
        2222222222222222222222222232257
  2222222222222222222222222222222235773
  ...
(End)

			22573 is a term because 22573 is prime, it has five digits (5 is a prime), all digits (2,3,5,7) are prime, and the sum of the digits is 2+2+5+7+3 = 19, which is also a prime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A062088.

a:=proc(n) local nn: nn:=convert(n,base,10): if isprime(n) and isprime(nops(nn)) and map(isprime,nn)=[seq(true,i=1..nops(nn))] and isprime(add(nn[j],j=1..nops(nn))) then n fi end: seq(a(k),k=1..60000); # Emeric Deutsch, Apr 02 2006

Select[Prime@ Range@ 6000, And[PrimeQ@ Length@ #, AllTrue[#, PrimeQ], PrimeQ@ Total@ #] &@ IntegerDigits@ # &] (* or *)
With[{p = {2, 3, 5, 7}}, Table[Select[FromDigits /@ Select[Tuples[p, {q}], PrimeQ@ Total@ # &], PrimeQ], {q, Prime@ Range@ 3}]] // Flatten (* Michael De Vlieger, Feb 02 2019 *)

from sympy import isprime, nextprime
from itertools import islice, product
def agen(): # generator of terms
    p = 2
    while True:
        for d in product("2357", repeat=p-1):
            for last in "37":
                if isprime(sum(map(int, s:="".join(d) + last))):
                    if isprime(t:=int(s)):
                        yield t
        p = nextprime(p)
print(list(islice(agen(), 40))) # Michael S. Branicky, May 26 2023

A070029 Primes with only prime digits and whose initial, all intermediate and final iterated sums of digits are primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 223, 227, 353, 2333, 2777, 3323, 7727, 27527, 33377, 33773, 35537, 35573, 35753, 37337, 52727, 53777, 55337, 55373, 55733, 57557, 57737, 57773, 73553, 73757, 75227, 75353, 75377, 75533, 75557, 75773, 77573, 222557, 222773

Offset: 1

Rick L. Shepherd, Apr 21 2002

This sequence is the intersection of A062088 and A070027.

			53777 is a term because 53777 is a prime with only prime digits and 5+3+7+7+7=29, 2+9=11 and 1+1=2 are all prime.

Cf. A070027, A062088 (only first sum of digits is necessarily prime).

iifpQ[n_]:=AllTrue[NestWhileList[Total[IntegerDigits[#]]&,n,#>9&],PrimeQ]; Select[Prime[Range[20000]],AllTrue[IntegerDigits[#],PrimeQ]&&iifpQ[#]&] (* Harvey P. Dale, Jul 18 2021 *)

A117608 Let p be an element of A110028. Let L(p) be the sorted list of digits of p and let LL be the set of all L(p) with duplicates removed and ordered lexicographically. Then a(n) is the first element of A110028 such that L(a(n))=LL(n).

Original entry on oeis.org

23, 223, 227, 337, 353, 557, 577, 773, 22573, 23327, 25253, 27527, 32233, 33353, 33377, 35353, 35537, 53777, 57557, 75577, 77377, 2222333, 2222533, 2222537, 2227727, 2233337, 2233757, 2235353, 2235557, 2237773, 2277553, 2332333, 2525557, 2572777, 3333773, 3335537, 3335737

Offset: 1

Walter Kehowski, Apr 06 2006

Terms have at least one 3 or at least one 7. - David A. Corneth, May 26 2023

			a(4)=337 since 337, 373 and 733 all have the same sorted list of digits [3,3,7].

David A. Corneth, Table of n, a(n) for n = 1..10871 (terms <= 10^43)
David A. Corneth, PARI program

Cf. A000040, A062088, A110028.

a:=proc(b,n) local nn: nn:=convert(n, base, b): if isprime(n) and isprime(nops(nn)) and andmap(isprime,nn) and isprime(convert(nn,`+`)) then n else fi end: L:=[seq(a(10,k), k=1..10^5)]; U:=[]: for z to 1 do A:=L; for x in L do l:=sort(convert(x,base,10)); m:=[selectremove(proc(z) sort(convert(z,base,10))=l end, A)]; if not m[1]=[] then U:=[op(U),min(op(m[1]))]; fi; if m[2]=[] then break else A:=m[2]; fi od od; U;

A158205 Primes with every digit a nonprime and the sum of the digits a prime.

Original entry on oeis.org

11, 41, 61, 89, 101, 191, 199, 401, 409, 449, 461, 601, 641, 661, 809, 881, 911, 919, 991, 1019, 1091, 1109, 1181, 1499, 1619, 1811, 1901, 1949, 4001, 4049, 4111, 4409, 4441, 4481, 4649, 4801, 4861, 4889, 4919, 4999, 6089, 6449, 6481, 6661, 6689, 6841

Offset: 1

Juri-Stepan Gerasimov, Mar 13 2009

Nonprime digits are 0, 1, 4, 6, 8 or 9.

			11(1+1=2), 41(4+1=5), 61(6+1=7), 89(8+9=17), 101(1+0+1=2), 191(1+9+1=11)

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

Cf. A000040, A062088, A141468.

Select[FromDigits/@Select[Tuples[{0,1,4,6,8,9},4],PrimeQ[Total[#]]&],PrimeQ]//Flatten (* Harvey P. Dale, Oct 02 2021 *)

Entries checked by R. J. Mathar, May 19 2010

A253216 Smallest of four primes in arithmetic progression with common difference 6 and digit sum prime.

Original entry on oeis.org

1091, 15791, 30091, 369991, 421691, 501191, 661091, 1101091, 1539991, 2042591, 2210291, 2542091, 2811191, 3351191, 3512291, 3864691, 4411391, 4675591, 5960791, 5992291, 5998691, 6884191, 6918391, 7516891, 8608591, 8697791, 9297091, 9622891, 9646291, 12013091

Offset: 1

K. D. Bajpai, Dec 29 2014

			a (1) = 1091: 1091 + 6 = 1097; 1097 + 6 = 1103; 1103 + 6 = 1109; all four are prime. Their digit sums 1+0+9+1 = 11; 1+0+9+7 = 17; 1+1+0+3 = 5 and 1+1+0+9 = 11 are also prime.
a(2) = 15791: 15791 + 6 = 15797; 15797 + 6 = 15803; 15803 + 6 = 15809; all four are prime. Their digit sums 1+5+7+9+1 = 23, 1+5+7+9+7 = 29, 1+5+8+0+3 = 17 and 1+5+8+0+9 = 23 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2500

Cf. A000040, A033447, A062088, A253140.

A253216 = {}; Do[d = 6; k = Prime[n]; k1 = k + d; k2 = k + 2d; k3 = k + 3d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[Plus @@ IntegerDigits[k]] && PrimeQ[Plus @@ IntegerDigits[k1]] && PrimeQ[Plus @@ IntegerDigits[k2]] && PrimeQ[Plus @@ IntegerDigits[k3]], AppendTo[A253216, k]], {n, 1000000}]; A253216
prQ[{a_,b_,c_,d_}]:=AllTrue[{b,c,d},PrimeQ]&&AllTrue[Total/@ (IntegerDigits/@ {a,b,c,d}),PrimeQ]; Select[#+{0,6,12,18}& /@Prime[Range[800000]],prQ][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2018 *)

for( n=1, 10^6, k=prime(n); k1=k+6; k2=k+12; k3=k+18; if(isprime(k1)&isprime(k2)&isprime(k3) &isprime(eval(Str(sumdigits(k)))) &isprime(eval(Str(sumdigits(k1)))) &isprime(eval(Str(sumdigits(k2)))) &isprime(eval(Str(sumdigits(k3)))), print1(k,", ")))

Definition corrected by Harvey P. Dale, May 26 2018

A253232 Smallest of five consecutive primes in arithmetic progression with common difference 90 and equal digit sums.

Original entry on oeis.org

61, 83, 89, 593, 1399, 2063, 2287, 2351, 2441, 3491, 5081, 5171, 5479, 6599, 9497, 12073, 16561, 17569, 21377, 23099, 23189, 28573, 29063, 32143, 36293, 36497, 36587, 39569, 49279, 61291, 62383, 65449, 66373, 71167, 72379, 75347, 81457, 88591, 92377, 94261, 104369

Offset: 1

K. D. Bajpai, Dec 29 2014

90 is the smallest common difference (d) to get a set of five consecutive primes in arithmetic progression {p, p+d, p+2d, p+3d, p+4d} having digit sums equal; for p < prime(10^5).

			a(1) = 61: 61+90 = 151; 151+90 = 241; 241+90 = 331; 331+90 = 421; all five are prime. Their digit sums 6+1 = 1+5+1 = 2+4+1 = 3+3+1 = 4+2+1 = 7 are all equal.
a(2) = 83: 83+90 = 173; 173+90 = 263; 263+90 = 353; 353+90 = 443; all five are prime. Their digit sums 8+3 = 1+7+3 = 2+6+3 = 3+5+3 = 4+4+3 = 11 are all equal.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A033447, A062088, A253140.

A253232 = {}; Do[d = 90; k = Prime[n]; k1 = k + d; k2 = k + 2 d; k3 = k + 3 d; k4 = k + 4 d; s = Plus @@ IntegerDigits[k]; s1 = Plus @@ IntegerDigits[k1]; s2 = Plus @@ IntegerDigits[k2]; s3 = Plus @@ IntegerDigits[k3]; s4 = Plus @@ IntegerDigits[k4]; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[k4] && s == s1 && s1 == s2 && s2 == s3 && s3 == s4, AppendTo[A253232, k]], {n, 50000}]; A253232
cd90Q[p_]:=Module[{q=p+90,r=p+180,s=p+270,t=p+360},AllTrue[{p,q,r,s,t},PrimeQ] && Length[Union[Total/@(IntegerDigits/@{p,q,r,s,t})]]==1]; Select[ Prime[ Range[ 10000]],cd90Q] (* Harvey P. Dale, May 13 2022 *)

A277607 Smallest of four consecutive primes in arithmetic progression with common difference 42 and all digit sums prime.

Original entry on oeis.org

5, 47, 157, 227, 317, 337, 557, 2027, 3037, 3217, 5147, 6047, 7457, 12527, 13757, 14657, 20357, 21017, 23747, 32057, 35027, 47417, 57047, 84137, 115727, 125627, 127247, 136337, 147137, 149027, 212057, 219937, 225257, 230017, 240047, 242357, 264137, 284117, 304127

Offset: 1

K. D. Bajpai, Oct 31 2016

			a(1) = 5: 5 + 42 = 47; 47 + 42 = 89; 89 + 42 = 131; all four are prime. Their digit sums 5, 4 + 7 = 11, 8 + 9 = 17 and 1 + 3 + 1 = 5 are also prime.
a(2) = 47: 47 + 42 = 89; 89 + 42 = 131; 131 + 42 = 173; all four are prime. Their digit sums  4 + 7 = 11, 8 + 9 = 17, 1 + 3 + 1 = 5 and 1 + 7 + 3 = 11 are also prime.

Cf. A000040, A033447, A062088, A253140.

A277607 = {}; Do[d = 42; k = Prime[n]; k1 = k + d; k2 = k + 2 d; k3 = k + 3 d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[Plus @@ IntegerDigits[k]] && PrimeQ[Plus @@ IntegerDigits[k1]] && PrimeQ[Plus @@ IntegerDigits[k2]] && PrimeQ[Plus @@ IntegerDigits[k3]], AppendTo[A25, k]], {n, 30000}]; A277607
FCPQ[n_] := Module[{a = n + 42, b = n + 84, c = n + 126}, AllTrue[{a, b, c}, PrimeQ] && AllTrue[Total /@ (IntegerDigits /@ {n, a, b, c}), PrimeQ]]; Select[Prime[Range[30000]], FCPQ]

A343834 Primes with digits in nondecreasing order, only primes, and with sum of digits also a prime.

Original entry on oeis.org

2, 3, 5, 7, 23, 223, 227, 337, 557, 577, 2333, 2357, 2377, 2557, 2777, 33377, 222337, 222557, 233357, 233777, 235577, 2222333, 2233337, 2235557, 3337777, 3355777, 5555777, 22222223, 22233577, 23333357, 23377777, 25577777, 222222227, 222222557, 222222577

Offset: 1

Mikk Heidemaa, May 01 2021

Intersection of A028864 and A062088.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A019546, A028864, A046704, A062088.

a[p_] := With[{dg = IntegerDigits@p}, PrimeQ@p && OrderedQ@dg && AllTrue[dg, PrimeQ] && PrimeQ@ Total@dg]; Cases[ Range[3*10^7], _?(a@# &)] (* or *)
upToDigitLen[k_] := Cases[ FromDigits@# & /@ Select[ Flatten[ Table[ Tuples[{2, 3, 5, 7}, {i}], {i, k}], 1], OrderedQ[#] &], _?(PrimeQ@# && PrimeQ@ Total@ IntegerDigits@# &)]; upToDigitLen[10]

from sympy import isprime
from sympy.utilities.iterables import multiset_combinations
def aupton(terms):
  n, digits, alst = 0, 1, []
  while len(alst) < terms:
    mcstr = "".join(d*digits for d in "2357")
    for mc in multiset_combinations(mcstr, digits):
      sd = sum(int(d) for d in mc)
      if not isprime(sd): continue
      t = int("".join(mc))
      if isprime(t): alst.append(t)
      if len(alst) == terms: break
    else: digits += 1
  return alst
print(aupton(35)) # Michael S. Branicky, May 01 2021

a(33) and beyond from Michael S. Branicky, May 01 2021

A360497 Maximal sequence of primes whose digits are primes and whose digit sum is also a term.

Original entry on oeis.org

2, 3, 5, 7, 23, 223, 2777, 7727, 27527, 33377, 33773, 35537, 35573, 35753, 37337, 52727, 55337, 55373, 55733, 73553, 75227, 75353, 75533, 222557, 222773, 223277, 225257, 225527, 233357, 235337, 235553, 253553, 253733, 277223, 322727, 323537, 332573, 335273

Offset: 1

Hongwei Jin, Feb 09 2023

The sequence is maximal in the sense that a nonempty set of primes cannot be added consistently.

			2 is a term because it is a prime with prime digits only and its digit sum 2 is also a term.
227 is not a term because the digit sum is 11 which is not a term because it has nonprime digits.
27527 is a term: it is a prime, each digit (2,5,7) is also a prime, and the sum of the digits (2+7+5+2+7 = 23) is also in the sequence.

A subsequence of A062088.

Cf. A000040, A007953.

R:= {2,3,5,7}: count:= 4:
S:= [2,3,5,7];
for d from 2 to 11 do
  S:= map(t -> (10*t+2,10*t+3,10*t+5,10*t+7), S);
  for x in S do
    if member(convert(convert(x,base,10),`+`),R) and isprime(x) then
       R:= R union {x}; count:= count+1;
    fi
  od;
od:
sort(convert(R,list)); # Robert Israel, Mar 02 2023

from sympy import isprime
seq = [2, 3, 5, 7]
for i in range(9, 10**6, 2):
    s = str(i)
    if set(s) <= set("2357") and sum(map(int, s)) in seq and isprime(i):
        seq.append(i)
print(seq)

cp's OEIS Frontend

A253140 Smallest of three consecutive primes in arithmetic progression with common difference 24 and digit sum prime.

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A110028 Primes with a prime number of digits, all of them prime, that add up to a prime.

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A070029 Primes with only prime digits and whose initial, all intermediate and final iterated sums of digits are primes.

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A117608 Let p be an element of A110028. Let L(p) be the sorted list of digits of p and let LL be the set of all L(p) with duplicates removed and ordered lexicographically. Then a(n) is the first element of A110028 such that L(a(n))=LL(n).

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Maple

A158205 Primes with every digit a nonprime and the sum of the digits a prime.

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Mathematica

Extensions

A253216 Smallest of four primes in arithmetic progression with common difference 6 and digit sum prime.

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Mathematica

PARI

Extensions

A253232 Smallest of five consecutive primes in arithmetic progression with common difference 90 and equal digit sums.

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Mathematica

A277607 Smallest of four consecutive primes in arithmetic progression with common difference 42 and all digit sums prime.

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