A182177 -id:A182177 - cp's OEIS frontend

A182178 Beginning with 1, smallest positive integer not yet in the sequence such that two adjacent digits of the sequence (also ignoring commas between terms) sum to a prime.

1, 2, 3, 4, 7, 6, 5, 8, 9, 20, 21, 11, 12, 14, 16, 50, 23, 25, 29, 41, 43, 47, 49, 83, 85, 61, 65, 67, 411, 111, 112, 30, 32, 34, 38, 52, 56, 58, 92, 94, 70, 74, 76, 114, 98, 302, 116, 120, 202, 121, 123, 89, 203, 205, 207, 412, 125, 211, 129, 212, 141, 143

Offset: 1

Jim Nastos and Eric Angelini, Apr 16 2012

See A219110 for the numbers which do not occur in this sequence. See A219250 for the analog when "sum" is replaced with "absolute difference", and A219248-A219251 for related sequences. - M. F. Hasler, Apr 11 2013

			20 follows 9 since 9+2 and 2+0 is prime, and no number less than 20 (not already in the sequence) satisfies the stated property.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A182175, A182177.

a[1] = 1; a[n_] := a[n] = For[id = IntegerDigits[a[n-1]]; k = 1, True, k++, If[FreeQ[Array[a, n-1], k], dd = Join[id, IntegerDigits[k]]; If[And @@ PrimeQ /@ Plus @@@ Transpose[{Most[dd], Rest[dd]}], Return[k]]]]; Array[a, 62] (* Jean-François Alcover, Apr 17 2013 *)

A182178_vec={(n, a=[1], u=0)->while(#aM. F. Hasler, Apr 11 2013

A219248 Numbers such that the absolute difference of any two adjacent (decimal) digits is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 18, 20, 24, 25, 27, 29, 30, 31, 35, 36, 38, 41, 42, 46, 47, 49, 50, 52, 53, 57, 58, 61, 63, 64, 68, 69, 70, 72, 74, 75, 79, 81, 83, 85, 86, 92, 94, 96, 97, 130, 131, 135, 136, 138, 141, 142, 146, 147, 149, 161, 163, 164

Offset: 1

M. F. Hasler, Apr 12 2013

Numbers which may (and do) occur in A219250 and A219249 (union {0}).

This is to A219250 and A219249 what A182175 is to A182177 and A182178.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
E. Angelini, Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

Select[Range[0,200],And@@PrimeQ[Abs[Differences[IntegerDigits[#]]]]&] (* Harvey P. Dale, Jun 06 2014 *)

is_A219248(n)={!for(i=2,#n=digits(n),isprime(abs(n[i-1]-n[i]))||return)}

def ok(n):
    d = list(map(int, str(n)))
    return all(abs(d[i]-d[i-1]) in {2,3,5,7} for i in range(1, len(d)))
print([k for k in range(164) if ok(k)]) # Michael S. Branicky, Sep 11 2024

from itertools import count, islice
def A219248gen(seed=None): # generator of terms
    nxt = {None:"123456789", "0":"2357", "1":"3468", "2":"04579",
        "3":"01568", "4":"12679", "5":"02378", "6":"13489",
        "7":"02459", "8":"1356", "9":"2467"}
    def bgen(d, seed=None):
        if d == 0: yield tuple(); return
        yield from ((i,)+t for i in nxt[seed] for t in bgen(d-1, seed=i))
    yield 0
    for d in count(1):
        yield from (int("".join(t)) for t in bgen(d, seed=seed))
print(list(islice(A219248gen(), 65))) # Michael S. Branicky, Sep 11 2024

A219250 Lexicographically earliest sequence of nonnegative integers such that the absolute difference of any two adjacent digits is prime.

Original entry on oeis.org

0, 2, 4, 1, 3, 5, 7, 9, 6, 8, 13, 14, 16, 18, 30, 20, 24, 25, 27, 29, 41, 31, 35, 36, 38, 50, 52, 42, 46, 47, 49, 61, 63, 53, 57, 58, 64, 68, 69, 70, 72, 74, 75, 79, 202, 92, 94, 96, 81, 83, 85, 86, 97, 203, 130, 205, 207, 241, 302, 413, 131, 303, 135, 242, 414, 136, 138, 141, 305, 246, 142, 416, 146, 147, 247, 249

Offset: 1

M. F. Hasler, Apr 11 2013

See A219249 for the version allowing only positive integers, i.e., starting with a(1)=1.
See A219248 (= range of A219250) for the numbers which occur in this sequence, and A219251 for the complement.
A182177 is the analog of this sequence for replacing "absolute difference" by "sum", A182178 is the same analog for A219249; A182175 is the analog of A219248 and A219110 corresponds to A219251.

M. F. Hasler in reply to E. Angelini, Re: Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

PARI
```
{(n,a=[0],u=0)->while(#a
				
```

A219110 Numbers for which at least one sum of two adjacent digits is not prime.

Original entry on oeis.org

10, 13, 15, 17, 18, 19, 22, 24, 26, 27, 28, 31, 33, 35, 36, 37, 39, 40, 42, 44, 45, 46, 48, 51, 53, 54, 55, 57, 59, 60, 62, 63, 64, 66, 68, 69, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105

Offset: 1

M. F. Hasler, Apr 11 2013

Different from A104211 where ("only") the sum of all digits is considered; of course exactly the two-digit terms coincide.

Numbers missing in A182177 and A182178. Otherwise said, complement of the range of A182177 (in the set of nonnegative integers) and of the range of A182178 (in the set of positive integers) and of A182175 in the set of integers > 9.

			102 is here because 1+0 is not prime (even though 0+2 is).

M. F. Hasler in reply to E. Angelini, Re: Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

Select[Range[10, 105], MemberQ[PrimeQ[Total /@ Partition[IntegerDigits[#], 2, 1]], False] &] (* T. D. Noe, Apr 16 2013 *)

is(n)=for(i=2,#n=digits(n),isprime(n[i-1]+n[i])||return(1))

A219249 Lexicographically earliest sequence of positive integers such that the absolute difference of any two adjacent digits is prime.

Original entry on oeis.org

1, 3, 5, 2, 4, 6, 8, 13, 14, 7, 9, 20, 24, 16, 18, 30, 25, 27, 29, 41, 31, 35, 36, 38, 50, 52, 42, 46, 47, 49, 61, 63, 53, 57, 58, 64, 68, 69, 70, 72, 74, 75, 79, 202, 92, 94, 96, 81, 83, 85, 86, 97, 203, 130, 205, 207, 241, 302, 413, 131, 303, 135, 242, 414, 136, 138, 141, 305, 246, 142, 416, 146, 147, 247, 249, 250

Offset: 1

Eric Angelini and M. F. Hasler, Apr 11 2013

See A219250 for the version allowing nonnegative integers, i.e., starting with a(1)=0.

See A219248 for the numbers which occur in this sequence, and A219251 for the complement.

E. Angelini, Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

Cf. A182175, A182177, A182178.

PARI
```
{A219249(n,a=[1],u=0)=while(#a
				
```

A219251 Numbers such that the absolute difference of a pair of adjacent decimal digits is not prime.

Original entry on oeis.org

10, 11, 12, 15, 17, 19, 21, 22, 23, 26, 28, 32, 33, 34, 37, 39, 40, 43, 44, 45, 48, 51, 54, 55, 56, 59, 60, 62, 65, 66, 67, 71, 73, 76, 77, 78, 80, 82, 84, 87, 88, 89, 90, 91, 93, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129

Offset: 1

M. F. Hasler, Apr 11 2013

Complement of A219248; numbers which do not occur in A219249 and A219250; analog of what is A219110 to A182177, A182178.

is_A219251(n)={for(i=2, #n=digits(n), isprime(abs(n[i-1]-n[i]))||return(1))}

cp's OEIS Frontend

A182178 Beginning with 1, smallest positive integer not yet in the sequence such that two adjacent digits of the sequence (also ignoring commas between terms) sum to a prime.

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A219248 Numbers such that the absolute difference of any two adjacent (decimal) digits is prime.

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A219250 Lexicographically earliest sequence of nonnegative integers such that the absolute difference of any two adjacent digits is prime.

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A219110 Numbers for which at least one sum of two adjacent digits is not prime.

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A219249 Lexicographically earliest sequence of positive integers such that the absolute difference of any two adjacent digits is prime.

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A219251 Numbers such that the absolute difference of a pair of adjacent decimal digits is not prime.

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