A048398 -id:A048398 - cp's OEIS frontend

A182781 Number of n-digit terms in A048398.

4, 4, 2, 1, 12, 20, 35, 28, 80, 114, 211, 228, 736, 1214, 2101, 2536, 7799, 13830, 22107, 27265, 82611, 144324, 259260, 354029, 901774, 1651718, 2913981, 3913728, 11048656, 19782855, 33483206, 49533124

Offset: 1

Zak Seidov, Feb 01 2011

Also, number of n-digit primes in A033075.

Appears to be strictly increasing for n >= 8. - Chai Wah Wu, May 31 2017

Cf. A033075, A048398.

A182781aux := proc(Lhig,n) local lsb,a ; if n = 0 then if isprime(Lhig) then    1; else 0; end if; else a := 0 ; lsb := Lhig mod 10 ; if lsb > 0 then a := a + procname(10*Lhig+lsb-1,n-1) ; end if; if lsb < 9 then a := a + procname(10*Lhig+lsb+1,n-1) ; end if; a; end if; end proc:
A182781 := proc(n) if n = 1 then 4; else a := 0 ; for l from 1 to 9 do a := a + A182781aux(l,n-1) ; end do: a ; end if; end proc: # R. J. Mathar, Feb 01 2011

a(22)-a(24) from Chai Wah Wu, May 31 2017

a(25)-a(32) from Chai Wah Wu, Jun 05 2017

A185893 Terms in A048398 ending with 1.

Original entry on oeis.org

101, 12101, 32321, 210101, 432121, 1012321, 1212121, 3210101, 3210121, 3212101, 3232321, 3432101, 5432321, 5434321, 21212101, 21232121, 23210101, 23232101, 43432321, 45434321, 101012321, 101210101, 101232121, 121232101, 123210121, 123232121, 321012121

Offset: 1

Zak Seidov, Feb 05 2011

Among first 114956 terms, number of n-digit terms are: 0, 0, 1, 0, 2, 2, 9, 6, 22, 23, 32, 57, 166, 246, 382, 572, 1770, 2936, 3956, 6183, 18132, 30818, 49641.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A048398, A182781.

Select[Prime[Range[100000]], IntegerDigits[#][[-1]] == 1 && Union[Abs[Differences[IntegerDigits[#]]]] == {1} &]

A033075 Positive numbers all of whose pairs of consecutive decimal digits differ by 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876

Offset: 1

Clark Kimberling

Number of n-digit terms: 9, 17, 32, 61, 116, 222, 424 (= A090994).

Also called 10-esthetic numbers (where in general a q-esthetic number has the property that the consecutive digits of its base-q representation differ by 1, see "Esthetic numbers" by J. M. De Koninck and N. Doyon). - Narad Rampersad, Aug 03 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. M. De Koninck and N. Doyon, Esthetic numbers, Annales des Sciences mathématiques du Québec, 33 (2009), 155-164.

Cf. A090994, A048398 (primes), A048411 (squares), A207954 (palindromes).

-- import Data.Set (fromList, deleteFindMin, insert)
a033075 n = a033075_list !! (n-1)
a033075_list = f (fromList [1..9]) where
   f s | d == 0    = m : f (insert (10*m+1) s')
       | d == 9    = m : f (insert (10*m+8) s')
       | otherwise = m : f (insert (10*m+d-1) (insert (10*m+d+1) s'))
       where (m,s') = deleteFindMin s
             d = mod m 10
-- Reinhard Zumkeller, Feb 21 2012

Join[Range[9],Select[Range[2000],Union[Abs[Differences[IntegerDigits[#]]]]=={1}&]] (* Harvey P. Dale, Dec 28 2011 *)

diff(v)=vector(#v-1,i,v[i+1]-v[i])
is(n)=if(n>9, Set(abs(diff(digits(n))))==[1], n>0) \\ Charles R Greathouse IV, Mar 11 2014

def ok(n):
    s = str(n)
    return all(abs(int(s[i]) - int(s[i+1])) == 1 for i in range(len(s)-1))
print(list(filter(ok, range(1, 877)))) # Michael S. Branicky, Aug 22 2021

# faster version for initial segment of sequence
def gen(d, s=None): # generate remaining d digits, from start digit s
    if d == 0:
        yield tuple()
        return
    if s == None:
        yield from [(i, ) + g for i in range(1, 10) for g in gen(d-1, s=i)]
    else:
        if s > 0:
            yield from [(s-1, ) + g for g in gen(d-1, s=s-1)]
        if s < 9:
            yield from [(s+1, ) + g for g in gen(d-1, s=s+1)]
def agentod(digits):
    for d in range(1, digits+1):
        yield from [int("".join(map(str, g))) for g in gen(d, s=None)]
print(list(agentod(11))) # Michael S. Branicky, Aug 22 2021

a(n) >> n^3.53267..., where the exponent is log 10/log k and k is the largest root of x^5 - x^4 - 4x^3 + 3x^2 + 3x - 1. - Charles R Greathouse IV, Mar 11 2014

A006055 Primes with consecutive (ascending) digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 67, 89, 4567, 78901, 678901, 23456789, 45678901, 9012345678901, 789012345678901, 56789012345678901234567890123, 90123456789012345678901234567, 678901234567890123456789012345678901

Offset: 1

N. J. A. Sloane, Richard C. Schroeppel

J. S. Madachy, Consecutive-digit primes - again, J. Rec. Math., 5 (No. 4, 1972), 253-254.
Thomas E. Moore, A Note on the Distribution of Primes in Arithmetic Progressions, J. Rec. Math., 5 (1972), 253-254.
R. C. Schroeppel, personal communication, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Zwillinger, Consecutive-Digit Primes - In Different Bases, J. Rec. Math., 10 (1972), 32-33.

Paul Tek, Table of n, a(n) for n = 1..36
J. S. Madachy, Consecutive-digit primes - again, J. Rec. Math., 5 (No. 4, 1972), 253-254. (Annotated scanned copy with letter to N. J. A. Sloane)
R. Schroeppel, Email to N. J. A. Sloane, Jun 1991
R. Schroeppel, Email to N. J. A. Sloane, Jun. 1991
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A052016, A052017, A048398, A120804, A120805.

f[n_] := Block[{u = Range@n, t = Table[1, {n}]}, Select[ Drop[ Union@ Flatten@ Table[ FromDigits[ Mod[u + i*t, 10]], {i, 10}], 2], PrimeQ@# &]]; Array[f, 35] // Flatten (* Robert G. Wilson v, Jul 05 2006 *)

from sympy import isprime
from itertools import count, islice
def bgen(): yield from (int("".join(str((s0+i)%10) for i in range(d))) for d in count(1) for s0 in range(1, 10))
def agen(): yield from filter(isprime, bgen())
print(list(islice(agen(), 18))) # Michael S. Branicky, May 26 2022

a(17) from Robert G. Wilson v, Jul 05 2006

Entry revised by N. J. A. Sloane, Feb 07 2007

A048405 Primes with consecutive digits that differ exactly by 8.

Original entry on oeis.org

2, 3, 5, 7, 19, 191, 919, 919191919, 91919191919, 91919191919191919, 91919191919191919191919, 191919191919191919191919191919191

Offset: 1

Patrick De Geest, Apr 15 1999

The next term (a(13)) has 133 digits. - Harvey P. Dale, Jan 04 2023

			2 is a term since all its consecutive digits differ by 5 (there aren't any).
19 is a term because 1 and 9 differ by 8.
23 is not a term because its consecutive digits differ only by 1.

Sean A. Irvine, Table of n, a(n) for n = 1..13

Cf. A048398, A048399, A048400, A048401, A048402, A048403, A048404, A048410.

Module[{nn=500,nine,one},one=Select[Table[FromDigits[PadRight[{},n,{1,9}]],{n,nn}],PrimeQ];nine=Select[Table[FromDigits[PadRight[{},n,{9,1}]],{n,nn}],PrimeQ];Sort[Join[{2,3,5,7},nine,one]]] (* Harvey P. Dale, Jan 04 2023 *)

Offset corrected by Sean A. Irvine, Jun 16 2021

A048399 Primes with consecutive digits that differ exactly by 2.

Original entry on oeis.org

2, 3, 5, 7, 13, 31, 53, 79, 97, 131, 313, 353, 757, 797, 31357, 35353, 35753, 35797, 75353, 75797, 79757, 97579, 131357, 135353, 135757, 353531, 531353, 535757, 575753, 579757, 757579, 797579, 975313, 975797, 979757, 1313579, 3131353

Offset: 1

Patrick De Geest, Apr 15 1999

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A033088, A048398, A048400, A048401, A048402, A048403, A048404, A048405.

Join[{2,3,5,7},Select[Prime[Range[230000]],Union[Abs[ Differences[ IntegerDigits[ #]]]]=={2}&]] (* Harvey P. Dale, Nov 03 2013 *)

A048400 Primes with consecutive digits that differ exactly by 3.

Original entry on oeis.org

2, 3, 5, 7, 41, 47, 14741, 14747, 74747, 1414741, 1474141, 7414741, 4141414747, 4147414147, 14141414141, 14141414741, 14141474741, 14141474747, 14147414741, 14147474141, 14147474147, 14741414747, 74141414147, 74141414741, 74147414741, 74741474741, 74747414141

Offset: 1

Patrick De Geest, Apr 15 1999

All terms with more than a single digit must comprise only the digits 1, 4, and 7, because no number comprising the digits 2, 5, and 8 or the digits 3, 6, and 9 can be prime. - Harvey P. Dale, Mar 01 2023

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A033081, A048398, A048399, A048401, A048402, A048403, A048404, A048405.

Join[{2,3,5,7},Table[Select[FromDigits/@Tuples[{1,4,7},n],PrimeQ[#]&& Union[ Abs[ Differences[ IntegerDigits[ #]]]]=={3}&],{n,11}]//Flatten] (* Harvey P. Dale, Mar 01 2023 *)

More terms from Naohiro Nomoto, Jul 28 2001

More terms from Sean A. Irvine, Jun 15 2021

A048401 Primes with consecutive digits that differ exactly by 4.

Original entry on oeis.org

2, 3, 5, 7, 37, 59, 73, 151, 373, 15959, 95959, 515951, 595159, 595951, 9515959, 51515159, 159595151, 159595951, 5151515951, 5159515159, 5159515951, 5951515151, 5951515951, 5959515151, 5959595951, 15151595951, 15951515159

Offset: 1

Patrick De Geest, Apr 15 1999

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A048398, A048399, A048400, A048402, A048403, A048404, A048405, A048406.

pd[{a_,b_,c___}]:=Flatten[Table[Select[FromDigits/@Select[Tuples[ {a,b,c},n],Union[Abs[Differences[#]]]=={4}&],PrimeQ],{n,11}]]; Union[Join[{2,3,5,7},pd[{1,5,9}],pd[{3,7}]]] (* Harvey P. Dale, Aug 23 2011 *)

More terms from Naohiro Nomoto, Jul 28 2001

A134811 Giza primes.

Original entry on oeis.org

2, 3, 5, 7, 787, 34543, 345676543, 34567876543

Offset: 1

Omar E. Pol, Nov 30 2007

Giza numbers (A134810) that are prime numbers. For n > 4 the structure of digits represents a pyramid at Giza. Also the top of a mountain. This sequence has only these 8 terms in base 10. For more information, see A134810.

			Illustration using the final term of this sequence:
  . . . . . . . . . . .
  . . . . . 8 . . . . .
  . . . . 7 . 7 . . . .
  . . . 6 . . . 6 . . .
  . . 5 . . . . . 5 . .
  . 4 . . . . . . . 4 .
  3 . . . . . . . . . 3
  . . . . . . . . . . .
  . . . . . . . . . . .
  . . . . . . . . . . .

Chris K. Caldwell and G. L. Honaker, Jr; Prime Curios!, The Dictionary of Prime Number Trivia, CreateSpace (2009), p. 209.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 34567876543

Cf. A000040, A048398, A046479, A134810.

ups = Flatten[Table[Range[i, j - 1], {i, 1, 9}, {j, i + 1, 10}], 1];afull = Sort[  Map[ToExpression@StringJoin@Map[ToString, #[[;; -2]] ~Join~ Reverse[#]] &, ups]];Select[afull,PrimeQ] (* James C. McMahon, Apr 11 2025 *)

A000040 INTERSECT A134810. - Omar E. Pol, Mar 25 2011

Reference and link added by Omar E. Pol, Mar 25 2011

A048402 Primes with consecutive digits that differ exactly by 5.

Original entry on oeis.org

2, 3, 5, 7, 61, 83, 383, 727, 72727, 94949, 1616161, 383838383, 727272727, 383838383838383, 38383838383838383, 72727272727272727, 94949494949494949, 383838383838383838383

Offset: 1

Patrick De Geest, Apr 15 1999

Harvey P. Dale, Table of n, a(n) for n = 1..30

Cf. A048398, A048399, A048400, A048401, A048403, A048404, A048405, A048407.

Module[{nn=50,w1,w2},w1=Flatten[Table[Select[FromDigits/@Table[ PadRight[ {},n,{a,a+5}],{n,2,nn}],PrimeQ],{a,4}]];w2=Flatten[Table[Select[ FromDigits/@ Table[PadRight[{},n,{a+5,a}],{n,2,nn}],PrimeQ],{a,4}]];Join[ {2,3,5,7},w1,w2]//Union] (* Harvey P. Dale, Jan 09 2021 *)

cp's OEIS Frontend

A182781 Number of n-digit terms in A048398.

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A185893 Terms in A048398 ending with 1.

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A033075 Positive numbers all of whose pairs of consecutive decimal digits differ by 1.

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A006055 Primes with consecutive (ascending) digits.

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A048405 Primes with consecutive digits that differ exactly by 8.

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A048399 Primes with consecutive digits that differ exactly by 2.

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A048400 Primes with consecutive digits that differ exactly by 3.

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A048401 Primes with consecutive digits that differ exactly by 4.

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