A048405 -id:A048405 - cp's OEIS frontend

A048398 Primes with consecutive digits that differ exactly by 1.

2, 3, 5, 7, 23, 43, 67, 89, 101, 787, 4567, 12101, 12323, 12343, 32321, 32323, 34543, 54323, 56543, 56767, 76543, 78787, 78989, 210101, 212123, 234323, 234343, 432121, 432323, 432343, 434323, 454543, 456767, 654323, 654343, 678767, 678989

Offset: 1

Patrick De Geest, Apr 15 1999

Or, primes in A033075. - Zak Seidov, Feb 01 2011

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 67, p. 23, Ellipses, Paris 2008.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..1500 from Zak Seidov)

Cf. A033075, A048399-A048405, A052016, A052017, A006055.

Cf. A010051; intersection of A033075 and A000040.

a048398 n = a048398_list !! (n-1)
a048398_list = filter ((== 1) . a010051') a033075_list
-- Reinhard Zumkeller, Feb 21 2012, Nov 04 2010
(Python 3.2 or higher)
from itertools import product, accumulate
from sympy import isprime
A048398_list = [2,3,5,7]
for l in range(1,17):
    for d in [1,3,7,9]:
        dlist = [d]*l
        for elist in product([-1,1],repeat=l):
            flist = [str(d+e) for d,e in zip(dlist,accumulate(elist)) if 0 <= d+e < 10]
            if len(flist) == l and flist[-1] != '0':
                n = 10*int(''.join(flist[::-1]))+d
                if isprime(n):
                    A048398_list.append(n)
A048398_list = sorted(A048398_list) # Chai Wah Wu, May 31 2017

Select[Prime[Range[10000]], # < 10 || Union[Abs[Differences[IntegerDigits[#]]]] == {1} &]

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

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 *)

A048404 Primes with consecutive digits that differ exactly by 7.

Original entry on oeis.org

2, 3, 5, 7, 29, 181, 929, 18181, 929292929, 18181818181818181818181818181818181818181818181818181818181818181818181818181

Offset: 1

Patrick De Geest, Apr 15 1999

The next term (a(11)) has 163 digits. - Harvey P. Dale, Mar 23 2023

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

Cf. A048398, A048399, A048400, A048401, A048402, A048403, A048405, A048409.

Module[{s18,s81,s29,s92},s18=Select[Table[FromDigits[PadRight[{},n,{1,8}]],{n,1,181,2}],PrimeQ]; s81=Select[Table[FromDigits[PadRight[{},n,{8,1}]],{n,2,182,2}],PrimeQ];s29 = Select[ Table[FromDigits[PadRight[{},n,{2,9}]],{n,2,182,2}],PrimeQ]; s92 =Select[Table[ FromDigits[ PadRight[{},n,{9,2}]],{n,1,183,2}],PrimeQ]; Join[{2,3,5,7},s18,s81,s29,s92]//Sort] (* Harvey P. Dale, Mar 23 2023 *)

A048403 Primes with consecutive digits that differ exactly by 6.

Original entry on oeis.org

2, 3, 5, 7, 17, 71, 1717171717171717171717171717171, 1717171717171717171717171717171717171

Offset: 1

Patrick De Geest, Apr 15 1999

From Andrew Howroyd, Aug 13 2024: (Start)
Terms with more than 1 digit have digits alternating between 1 and 7.
No more terms < 10^3000. (End)

Primes in A048408.

Cf. A048398, A048399, A048400, A048401, A048402, A048404, A048405.

upto(limit)={my(L=List([t|t<-[2,3,5],t<=limit]),m=1); while(mAndrew Howroyd, Aug 13 2024

Offset changed by Andrew Howroyd, Aug 13 2024

A089291 Prime worms (as defined below).

Original entry on oeis.org

101, 787, 12101, 32323, 34543, 78787, 1012321, 1212121, 3212123, 3212323, 3454343, 7654567, 7656787, 7676567, 7678787, 7876567, 7898767, 101012321, 101210101, 101232121, 121232101, 123210121, 123232121, 321234343, 323232323

Offset: 1

Enoch Haga, Dec 23 2003

By analogy, primes of this type are worms with a head and tail and body composed of the same digit which weaves in and out of the prime. In a(2)=12101 the worm is defined by the digit 1.

			a(2)=12101 because that number is prime with identical first and last digits. Then abs(1-2)=1; abs(2-1)=1; abs(1-0)=1; abs(0-1)=1; and sum of absolute values is 4, one less than the 5 digits in the prime.

The concept is due to Carlos Rivera in his Puzzle 246 (where he asks for the first pandigital prime worm and for the first pandigital titanic prime worm - solutions are at his site).

Carlos Rivera, Puzzle 246. The worms

This is a subset of A048398. Cf. A089315-A089317, A048398-A048405.

Primes whose first and last digits are identical and whose successive digit differences have a uniformly absolute value of 1. Thus the sum of absolute values is one less than the number of digits in the prime. No two adjacent digits are identical or differ by more than one.

Edited by Charles R Greathouse IV, Aug 02 2010

A089315 Prime worms [successive digit differences with absolute value of 3].

Original entry on oeis.org

14741, 74747, 1414741, 1474141, 14141414141, 14141414741, 14141474741, 14147414741, 14147474141, 74141414147, 1474741414141, 7474141474747, 7474741414747, 14141474141414141, 14147414747474741, 14147474147474741

Offset: 0

Enoch Haga, Dec 25 2003

One of a family of prime worms differing according to the uniform absolute value of successive digit pairs. Sequence checked to 10^9.

This is a subset of A048400. Cf. A089291, A089316-A089317, A048398-A048405.

			a(1)=74747 because the number is prime, has identical first and last digits and abs(7-4)=3; abs(4-7)=3; abs(7-4)=3 and abs(4-7)=3. In this number, the worm is 7.

Carlos Rivera's primepuzzles.net, Puzzle 246.

Carlos Rivera, Puzzle 246. The worm

Select prime numbers having the same first and last digits; if the uniform absolute value of successive digit differences is 3, add to sequence.

More terms from David Wasserman, Sep 09 2005

A089317 Prime worms [successive digit differences with absolute value of 4].

Original entry on oeis.org

151, 373, 95959, 9515959, 159595151, 159595951, 15151595951, 15951595151, 95951515159, 1515159515951, 1515959515951, 1515959595151, 1595159515151, 1595159595151, 9515151515159, 9515159515159, 9515159595959, 9595159515959

Offset: 0

Enoch Haga, Dec 25 2003

			a(1)=373; first and last digits are 3; abs(3-7)=4; abs(7-3)=4; the worm is 3.

Carlos Rivera's primepuzzles.net, Puzzle 246

Carlos Rivera, Puzzle 246. The worm

This is a subsequence of A048401. Cf. A089291, A089315-A089316, A048398-A048405.

Select prime numbers having the same first and last digits; if the uniform absolute value of successive digit differences is 4, add to sequence.

More terms from David Wasserman, Sep 09 2005

cp's OEIS Frontend

A048398 Primes with consecutive digits that differ exactly by 1.

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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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A048402 Primes with consecutive digits that differ exactly by 5.

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A048404 Primes with consecutive digits that differ exactly by 7.

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A048403 Primes with consecutive digits that differ exactly by 6.

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A089291 Prime worms (as defined below).

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A089315 Prime worms [successive digit differences with absolute value of 3].

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