A066540 -id:A066540 - cp's OEIS frontend

A069567 Smaller of two consecutive primes which are anagrams of each other.

1913, 18379, 19013, 25013, 34613, 35617, 35879, 36979, 37379, 37813, 40013, 40213, 40639, 45613, 48091, 49279, 51613, 55313, 56179, 56713, 58613, 63079, 63179, 64091, 65479, 66413, 74779, 75913, 76213, 76579, 76679, 85313, 88379, 90379, 90679, 93113, 94379, 96079

Offset: 1

Amarnath Murthy, Mar 24 2002

Smaller members of Ormiston prime pairs.
Given the n-th prime, it is occasionally possible to form the (n+1)th prime using the same digits in a different order. Such a pair is called an Ormiston pair.
Ormiston pairs occur rarely but randomly. It is thought that there are infinitely many but this has not been proved. They always differ by a multiple of 18. Ormiston triples also exist - see A075093.
"Anagram" means that both primes must not only use the same digits but must use each digit the same number of times.  [From Harvey P. Dale, Mar 06 2012]
Dickson's conjecture would imply that the sequence is infinite, e.g. that there are infinitely many k for which 1913+3972900*k and 1931+3972900*k form an Ormiston pair. - Robert Israel, Feb 23 2017

			1913 and 1931 are two successive primes.
Although 179 and 197 are composed of the same digits, they do not form an Ormiston pair as several other primes intervene (i.e. 181, 191, 193).

Andy Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp. 12-13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Ormiston Tuples
Andy Edwards, Ormiston Pairs [Archive machine link]; local copy [with permission]
Eric Weisstein's World of Mathematics, Rearrangement Prime Pair.

Cf. A072274, A075093, A161160, A066540.

Cf. A000040, A028906.

import Data.List (sort)
a069567 n = a069567_list !! (n-1)
a069567_list = f a000040_list where
   f (p:ps@(p':_)) = if sort (show p) == sort (show p')
                     then p : f ps else f ps
-- Reinhard Zumkeller, Apr 03 2015

N:= 10^6: # to get all terms <= N
R:= NULL: p:= 3: q:= 5:
while p <= N do
  p:= q;
  q:= nextprime(q);
  if q-p mod 18 = 0 and sort(convert(p,base,10)) = sort(convert(q,base,10)) then
    R:= R, p
  fi
od:
R; # Robert Israel, Feb 23 2017

Prime[ Select[ Range[10^4], Sort[ IntegerDigits[ Prime[ # ]]] == Sort[ IntegerDigits[ Prime[ # + 1]]] & ]]
a = {1}; b = {2}; Do[b = Sort[ IntegerDigits[ Prime[n]]]; If[a == b, Print[ Prime[n - 1], ", ", Prime[n]]]; a = b, {n, 1, 10^4}]
Transpose[Select[Partition[Prime[Range[8600]],2,1],Sort[IntegerDigits[ First[#]]] == Sort[ IntegerDigits[Last[#]]]&]][[1]] (* Harvey P. Dale, Mar 06 2012 *)

is(n)=isprime(n)&&vecsort(Vec(Str(n)))==vecsort(Vec(Str(nextprime(n+1)))) \\ Charles R Greathouse IV, Aug 09 2011

p=2;forprime(q=3,1e5,if((q-p)%18==0&&vecsort(Vec(Str(p)))==vecsort(Vec(Str(q))),print1(p", "));p=q) \\ Charles R Greathouse IV, Aug 09 2011, minor edits by M. F. Hasler, Oct 11 2012

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    p, hp, q, hq = 2, "2", 3, "3"
    while True:
        if hp == hq: yield p
        p, q = q, nextprime(q)
        hp, hq = hq, "".join(sorted(str(q)))
print(list(islice(agen(), 38))) # Michael S. Branicky, Feb 19 2024

Comments and references from Andy Edwards (AndynGen(AT)aol.com), Jul 09 2002
Edited by Robert G. Wilson v, Jul 15 2002 and Aug 29 2002
Minor edits by Ray Chandler, Jul 16 2009

A075093 Smallest member of Ormiston prime triple.

Original entry on oeis.org

11117123, 12980783, 14964017, 32638213, 32964341, 33539783, 35868013, 44058013, 46103237, 48015013, 50324237, 52402783, 58005239, 60601237, 61395239, 74699789, 76012879, 78163123, 80905879, 81966341, 82324237

Offset: 1

Robert G. Wilson v, Aug 31 2002

A subsequence of A069567 which is in turn a subsequence of A072274. More precisely, this A075093 lists exactly the terms which precede repeated elements in A072274, since an Ormiston triple (p,q,r), i.e., three consecutive primes whose decimal representations are anagrams of each other, corresponds to two pairs (p,q) and (q,r), so that p=a(n) implies p=A072274(2k-1)=A069567(k) for some k, and q=A072274(2k)=A072274(2k+1)=A069567(k+1) and r=A072274(2k+2). - M. F. Hasler, Oct 11 2012

			The first Ormiston triples are 11117123, 11117213 and 11117321.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Cf. A069567 - smaller member of an Ormiston prime pair.

Cf. A072274, A161160, A066540. - M. F. Hasler, Oct 11 2012

a075093 n = a075093_list !! (n-1)
a075093_list = f a000040_list where
   f (p:ps@(q:r:_)) =
     if sort (show p) == sort (show q) && sort (show q) == sort (show r)
        then p : f ps else f ps
-- Reinhard Zumkeller, Mar 09 2012

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = 0; b = 1; Do[c = NextPrim[b]; If[ Sort[ IntegerDigits[a]] == Sort[ IntegerDigits[b]] == Sort[ IntegerDigits[c]], Print[a]]; a = b; b = c, {n, 1, 10^7}]
op3Q[{a_,b_,c_}]:=Sort[IntegerDigits[a]]==Sort[IntegerDigits[b]] == Sort[ IntegerDigits[ c]]; Transpose[Select[Partition[Prime[ Range[ 5000000]],3,1], op3Q]][[1]] (* Harvey P. Dale, Jun 16 2014 *)

is_A075093(n)={isprime(n) & vecsort(Vec(Str(n)))==vecsort(Vec(Str(n=nextprime(n+1)))) & vecsort(Vec(Str(n)))==vecsort(Vec(Str(nextprime(n+1))))} \\ M. F. Hasler, Oct 11 2012

from sympy import nextprime
from itertools import islice
def hash(n): return "".join(sorted(str(n)))
def agen(start=2): # generator of terms
    p = nextprime(start-1); q=nextprime(p); r=nextprime(q)
    hp, hq, hr = list(map(hash, [p, q, r]))
    while True:
        if hp == hq == hr: yield p
        p, q, r = q, r, nextprime(r)
        hp, hq, hr = hq, hr, hash(r)
print(list(islice(agen(), 3))) # Michael S. Branicky, Feb 19 2024

A209396 Each entry is the first of three consecutive primes with equal digital sum.

Original entry on oeis.org

22193, 25373, 69539, 107509, 111373, 167917, 200807, 202291, 208591, 217253, 221873, 236573, 238573, 250073, 250307, 274591, 290539, 355573, 373073, 382373, 404273, 407083, 415391, 417383, 439009, 441193, 447907, 515173, 542837, 581873, 582083, 591673

Offset: 1

Antonio Roldán, Mar 13 2012

Subsequence of A066540.

The differences between the three primes of the triple are multiples of 18.

			200807 is in the sequence because 200807, 200843, 200861 are consecutive primes and sum_of_digits(200807)= sum_of_digits(200843)= sum_of_digits(200861)=17

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

Cf. A066540, A209663, A210629.

Select[Prime[Range[100000]], Total[IntegerDigits[#]] == Total[IntegerDigits[NextPrime[#, 1]]] == Total[IntegerDigits[NextPrime[#, 2]]] &] (* T. D. Noe, Mar 13 2012 *)
Transpose[Select[Partition[Prime[Range[50000]],3,1],Differences[ Total/@ (IntegerDigits/@#)]=={0,0}&]][[1]] (* Harvey P. Dale, Jul 22 2016 *)

A210629 Each entry is the first of four consecutive primes with equal digital sum.

Original entry on oeis.org

1442173, 2288509, 2660183, 2805773, 3830891, 4137473, 4951073, 5216137, 5517173, 5521819, 5521891, 5914591, 6474119, 6518173, 7118519, 7570273, 8508473, 8584273, 8689573, 8912591, 9383053, 9958519, 10116373, 10204391, 11418193, 11878873, 11890873, 12948773, 13738163, 13873073, 14377157, 14436391, 14677573, 14732191

Offset: 1

Harvey P. Dale, Mar 25 2012

The differences between each of the 4-consecutive primes are multiples of 18. - Harvey P. Dale, Jul 22 2016

			2288509 is in the sequence because 2288509, 2288527, 2288563, and 2288581 are consecutive primes and the sum of the digits of each = 34

Cf. A066540, A209663, A209396.

Transpose[Select[Partition[Prime[Range[1000000]],4,1],Total[ IntegerDigits[#[[1]]]]==Total[IntegerDigits[#[[2]]]] == Total[ IntegerDigits[#[[3]]]]==Total[IntegerDigits[#[[4]]]]&]][[1]]
Transpose[Select[Partition[Prime[Range[10^6]],4,1], Differences[ Total/@ (IntegerDigits/@#)]=={0,0,0}&]][[1]] (* Harvey P. Dale, Jul 22 2016 *)

A209663 Numbers n with property that n is prime, n+18 is prime and both have equal sum of digits.

Original entry on oeis.org

5, 13, 19, 23, 29, 43, 53, 79, 109, 113, 139, 149, 163, 173, 179, 223, 233, 239, 263, 313, 349, 379, 439, 443, 449, 491, 503, 523, 569, 613, 643, 659, 673, 691, 709, 733, 739, 743, 769, 809, 839, 859, 863, 919, 929, 953, 1013, 1033, 1069, 1091, 1153, 1163

Offset: 1

Antonio Roldán, Mar 13 2012

Supersequence of A066540.

			613 is in the sequence because 613 is prime, 613+18 = 631 is also prime. Sum_of_digits(613) = 10 and Sum_of_digits(631) = 10.

Cf. A066540.

[p: p in PrimesUpTo(1200) | IsPrime(p+18) and  &+Intseq(p) eq &+Intseq(p+18)]; // Bruno Berselli, Mar 13 2012

A227931 Smallest sets of 5 consecutive primes with equal digital sum. The initial prime is listed.

Original entry on oeis.org

5521819, 33014273, 36183593, 39874273, 47143739, 82934191, 83640653, 86225437, 89121073, 99551093, 104663773, 108616619, 109514719, 117611519, 131616409, 142348637, 151942291, 168056137, 168066791, 172096037, 196415237, 197604227, 203519819, 204983507

Offset: 1

Shyam Sunder Gupta, Oct 06 2013

			5521819 is in the sequence because 5521819, 5521891, 5521927, 5521963 and 5521981 are consecutive primes and the sum of the digits of each = 31.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..445

Cf. A066540, A209396, A210629, A071613.

a = {}; m = 1; s = 1; Do[If[(y = Apply[Plus, IntegerDigits[x = Prime[n]]]) == s , m = m + 1; If[m == 6, AppendTo[a, Prime[n - 5]]], m = 1]; s = y, {n, 2, 100000000}];a
Select[Partition[Prime[Range[11340000]],5,1],Length[Union[Total/@(IntegerDigits/@ #)]] == 1&][[All,1]] (* Harvey P. Dale, Apr 14 2022 *)

A209875 Primes p such that p and p+18 are consecutive primes with equal digital sum.

Original entry on oeis.org

523, 1069, 1259, 1759, 1913, 2503, 3803, 4159, 4373, 4423, 4463, 4603, 4703, 4733, 5059, 5209, 6229, 6529, 6619, 7159, 7433, 7459, 8191, 9109, 9749, 9949, 10691, 10753, 12619, 12763, 12923, 13763, 14033, 14303, 14369, 15859, 15973, 16529, 16673, 16903, 17239, 17359

Offset: 1

Antonio Roldán, Mar 14 2012

Subsequence of A066540 and A209663 (A066540 contains some consecutive primes with differences greater than 18; A209663 allows nonconsecutive primes).

			19013 is in the sequence because 19013 is prime, 19013 + 18 = 19031 is the next prime, and sum_of_digits(19013) = sum_of_digits(19031) = 14.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..2999 from R. J. Mathar)

Cf. A066540, A209663.

{forprime(n=3, 20000, my(m=nextprime(n+1)); if(m-n==18 && sumdigits(n) == sumdigits(m), print1(n, ", ")))} \\ Antonio Roldán, Dec 21 2012

"Correction" of early 2012 undone by R. J. Mathar, Feb 20 2023

A217875 Primes which have the same (base 10) digital sum A007953 as the next smaller and next larger prime.

Original entry on oeis.org

22229, 25391, 69557, 107563, 111409, 167953, 200843, 202309, 208609, 217271, 221891, 236609, 238591, 250091, 250343, 274609, 290557, 355591, 373091, 382391, 404291, 407119, 415409, 417419, 439063, 441229, 447943, 515191, 542873, 581891, 582119, 591691, 614963

Offset: 1

M. F. Hasler, Oct 13 2012

M. F. Hasler, Table of n, a(n) for n = 1..733 (all terms below 10^7).
C. Caldwell (Ed.), 22229 (another Prime Pages' Curiosity).

A subsequence of A066540.

{q=r=0; forprime(p=1,default(primelimit),(p-q)%18==0 & (q-r)%18==0 & A007953(p)==A007953(q) & A007953(p)==A007953(r) & print1(q","); r=q; q=p)}

A227933 Smallest sets of 6 consecutive primes with equal digital sum. The initial prime is listed.

Original entry on oeis.org

354963229, 448024483, 467739719, 475313609, 525523709, 771943583, 790277219, 881160173, 901572019, 925569683, 1051470419, 1085896727, 1110999817, 1285560163, 1331768783, 1455016319, 1472310383, 1519074619, 1628600381, 1815368519, 1914032047, 1990306673

Offset: 1

Shyam Sunder Gupta, Oct 06 2013

			354963229 is in the sequence because 354963229, 354963283, 354963319, 354963337, 354963373 and 354963391 are consecutive primes and the sum of the digits of each = 43

Shyam Sunder Gupta, Table of n, a(n) for n = 1..46

Cf. A066540, A209396, A210629, A071613, A227931.

a = {}; m = 1; s = 1; Do[If[(y = Apply[Plus, IntegerDigits[x = Prime[n]]]) == s , m = m + 1; If[m == 6, AppendTo[a, Prime[n - 5]]], m = 1]; s = y, {n, 2, 200000000}];a

A230220 Smaller of two consecutive palindromic primes with equal digital sum.

Original entry on oeis.org

11, 16661, 31513, 74747, 75557, 1035301, 1074701, 1303031, 1363631, 1374731, 1557551, 1646461, 1714171, 1777771, 1909091, 3075703, 3127213, 3452543, 3627263, 3635363, 3646463, 3746473, 3784873, 3948493, 3983893, 7057507, 7302037, 7354537, 7365637, 7622267

Offset: 1

Shyam Sunder Gupta, Oct 11 2013

			11 is in the sequence because 11 and 101 are consecutive palindromic primes and the sum of the digits of each = 2.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..5000

Cf. A002385, A007953, A007605, A066540, A230199.

cp's OEIS Frontend

A069567 Smaller of two consecutive primes which are anagrams of each other.

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A075093 Smallest member of Ormiston prime triple.

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A209396 Each entry is the first of three consecutive primes with equal digital sum.

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A210629 Each entry is the first of four consecutive primes with equal digital sum.

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A209663 Numbers n with property that n is prime, n+18 is prime and both have equal sum of digits.

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A227931 Smallest sets of 5 consecutive primes with equal digital sum. The initial prime is listed.

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Mathematica

A209875 Primes p such that p and p+18 are consecutive primes with equal digital sum.

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A217875 Primes which have the same (base 10) digital sum A007953 as the next smaller and next larger prime.