Eric Angelini - cp's OEIS frontend

A376297 Smallest terms which are the concatenation of two unambiguous distinct integers > 0.

12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 120, 130, 140, 150, 160, 170, 180, 190, 202, 203, 204, 205, 206, 207, 208, 209, 210, 220, 222, 230, 240, 250, 260, 270, 280, 290, 330, 333, 340, 350, 360, 370, 380, 390

Offset: 1

Eric Angelini, Sep 19 2024

The sequence is infinite as it can always be extended with the smallest integer <1(n times 0)1> not yet present.

			10 is not in the sequence because 0 is not > 0.
11 is not in the sequence because 1 and 1 are not distinct.
21 is not in the sequence because 12 is already in it.
101 is in the sequence because it can be unambiguously split into the two distinct integers 10 and 1.
110 is not in the sequence because 101 is already in it.
111 is in the sequence because it can be split into the two distinct integers 1 and 11 (though in two ways).

A376128 The absolute difference of two successive terms is prime and the absolute difference of two successive digits is also prime.

Original entry on oeis.org

0, 2, 4, 1, 3, 5, 7, 9, 6, 8, 13, 16, 14, 25, 20, 27, 24, 29, 42, 47, 49, 46, 35, 30, 53, 50, 31, 36, 38, 57, 52, 41, 64, 61, 63, 58, 69, 72, 70, 75, 86, 81, 68, 135, 74, 79, 242, 469, 246, 83, 85, 252, 425, 202, 413, 130, 203, 136, 131, 358, 147, 94, 92, 97, 270, 241, 302, 429, 250, 207, 205, 258, 149

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Sep 11 2024

			Terms a(1) to a(15) are 0,2,4,1,3,5,7,9,6,8,13,16,14,25,20.
The successive absolute differences between two terms are the primes 2,2,3,2,2,2,2,3,2,5,3,2,11,5.
The successive absolute differences between two digits are the primes 2,2,3,2,2,2,2,3,2,7,2,2,5,5,3,2,3,3,2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Angelini, More digits, terms and primes, personal blog of the author.

Cf. A219248, A281878.

# uses A219248gen in A219248
from sympy import isprime
from itertools import count, islice
def c(an, k):
    return isprime(abs(an-k)) and isprime(abs(an%10-int(str(k)[0])))
def agen(): # generator of terms
    an, aset = 0, {0}
    while True:
        yield an
        an = next(k for k in A219248gen(seed=str(an%10)) if k not in aset and c(an, k))
        aset.add(an)
print(list(islice(agen(), 73))) # Michael S. Branicky, Sep 11 2024

A375614 Lexicographically earliest infinite sequence of distinct nonnegative pairs of terms that interpenetrate to produce a prime number.

Original entry on oeis.org

0, 11, 3, 17, 2, 23, 6, 13, 1, 21, 4, 19, 7, 27, 5, 33, 8, 39, 103, 10, 153, 20, 107, 12, 131, 15, 109, 16, 111, 26, 113, 24, 101, 30, 119, 14, 123, 25, 117, 29, 141, 22, 127, 18, 133, 31, 129, 28, 121, 34, 169, 36, 137, 32, 167, 38, 147, 35, 171, 43, 157, 37, 9, 41, 149, 44, 159, 55, 139, 46, 151, 45, 163, 48, 173, 42, 143, 51, 187, 49, 177, 52, 183, 50, 161, 54, 179, 47, 189

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 22 2024

The term a(n) must always be exactly one digit longer or shorter than the term a(n+1).

			Interpenetrate a(1) = 0 and a(2) = 11 to form 101 (a prime number);
interpenetrate a(2) = 11 and a(3) = 3 to form 131 (a prime number);
interpenetrate a(3) = 3 and a(4) = 17 to form 137 (a prime number);
interpenetrate a(4) = 17 and a(5) = 2 to form 127 (a prime number);
interpenetrate a(5) = 2 and a(6) = 23 to form 223 (a prime number);
interpenetrate a(6) = 23 and a(7) = 6 to form 263 (a prime number);
interpenetrate a(7) = 6 and a(8) = 13 to form 163 (a prime number);
interpenetrate a(8) = 13 and a(9) = 1 to form 113 (a prime number);
(...)
interpenetrate a(18) = 39 and a(19) = 103 to form 13093 (a prime number);
(...)
interpenetrate a(167) = 277 and a(168) = 1009 to form 1207079 (a prime number); etc.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Angelini, Combs and Pharaohs, personal blog of the author.

Cf. A213457, A000040.

Q:= proc(a,b) local La, Lb, i;
  La:= convert(a,base,10);
  Lb:= convert(b,base,10);
  add(La[i]*10^(2*i-2),i=1..nops(La)) + add(Lb[i]*10^(2*i-1),i=1..nops(Lb))
end proc:
f:= proc(n) local d,x;
  d:= 1+ilog10(n);
  if n::odd then
    for x from 10^(d-2) to 10^(d-1) - 1 do
      if not(member(x,S)) and isprime(Q(n,x)) then return x fi
    od
  fi;
  for x from 10^d+1 to 10^(d+1) - 1 by 2 do
    if not(member(x,S)) and isprime(Q(x,n)) then return x fi
  od;
FAIL
end proc:
R:= 0,11: S:= {0,11}: v:= 11:
for i from 2 to 100 do
  v:= f(v);
  R:= R,v;
  S:= S union {v};
od:
R; # Robert Israel, Aug 22 2024

from sympy import isprime
from itertools import islice
def ip(s, t): return int("".join(x+v for x, v in zip(s, t))+s[-1])
def agen(): # generator of terms
    seen, an, found = set(), 0, True
    while found:
        yield an
        seen.add(an)
        s = str(an)
        d, found = len(s), False
        if s[-1] in "1379" and d > 1:
            for k in range(10**(d-2), 10**(d-1)):
                if k not in seen and isprime(ip(s, str(k))):
                    an, found = k, True
                    break
        if not found:
            for k in range(10**d, 10**(d+1)):
                if k not in seen and isprime(ip(str(k), s)):
                    an, found = k, True
                    break
print(list(islice(agen(), 90))) # Michael S. Branicky, Aug 22 2024

A375477 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10, said chunks being "linked" (see the Comments section for an explanation).

Original entry on oeis.org

0, 1, 2, 3, 4, 40, 6, 60, 10, 12, 20, 5, 21, 11, 8, 80, 101, 13, 15, 50, 14, 41, 23, 30, 7, 70, 100, 1001, 16, 102, 22, 24, 42, 31, 17, 10001, 19, 91, 103, 33, 32, 104, 43, 111, 105, 110, 100001, 106, 201, 107, 1000001, 109, 901, 112, 51, 113, 120, 10000001, 114, 121

Offset: 1

Eric Angelini and Michael S. Branicky, Aug 17 2024

The 1st chunk with digitsum = 10 is (0, 1, 2, 3, 4), ending with a "4". The next chunk with digitsum = 10 must start with a "4" (this is the "link") and is thus (40, 6). As the next chunk with digitsum = 10 must start with a "6", we have (60, 10, 12). The next chunk with digitsum = 10 must start with a "2" and we have (20, 5, 21), etc.
No chunk is allowed to end with a zero.  Thus, no intermediate chunk digit sum can be 9, and anytime a chunk needs a digitsum of 2 to "complete", the next term must be of the form 10^k + 1 for k >= 1.
Infinite since there are an infinity of terms with digits sums <= 10.
a(5367) has 1001 digits.

Michael S. Branicky, Table of n, a(n) for n = 1..5366
Eric Angelini, Tile my sequence, personal blog of the author.

Cf. A375460.

from itertools import count, islice
def bgen(ds): # generator of terms with digital sum ds
    def A051885(n): return ((n%9)+1)*10**(n//9)-1 # due to Chai Wah Wu
    def A228915(n): # due to M. F. Hasler
        p = r = 0
        while True:
            d = n % 10
            if d < 9 and r: return (n+1)*10**p + A051885(r-1)
            n //= 10; r += d; p += 1
    k = A051885(ds)
    while True: yield k; k = A228915(k)
def agen(): # generator of terms
    an, ds_block, seen, link_set, min2 = 0, 0, set(), "123456789", 11
    while True:
        yield an
        seen.add(an)
        if ds_block == 8:
            while min2 in seen: min2 = 10*min2 - 9
            an, ds_an, link_an = min2, 2, "1"
        else:
            cand_ds = list(range(1, 9-ds_block)) + [10-ds_block]
            dsg = [0] + [bgen(i) for i in range(1, 11-ds_block)]
            dsi = [0] + [(next(dsg[i]), i) for i in range(1, 11-ds_block)]
            while True:
                k, ds_k = min(dsi[j] for j in cand_ds)
                if k not in seen:
                    sk, dst = str(k), ds_k + ds_block
                    if sk[0] in link_set:
                        if dst < 9 or (dst == 10 and k%10 != 0):
                            an, ds_an, link_an = k, ds_k, sk[-1]
                            break
                dsi[ds_k] = (next(dsg[ds_k]), ds_k)
        ds_block = ds_block + ds_an
        if ds_block == 10: ds_block, link_set = 0, link_an
        else: link_set = "123456789"
print(list(islice(agen(), 60)))

A375460 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 11, 20, 6, 12, 100, 7, 21, 8, 101, 9, 1000, 13, 14, 10000, 15, 22, 16, 30, 17, 110, 18, 100000, 19, 23, 31, 1000000, 24, 40, 25, 102, 26, 200, 27, 10000000, 28, 32, 41, 33, 103, 34, 111, 35, 1001, 36, 100000000, 37, 42, 112, 43, 120, 44, 1010, 45, 1000000000

Offset: 1

Eric Angelini, Aug 15 2024

The first integer that will never appear in the sequence is 29, as its digitsum exceeds 10.
From Michael S. Branicky, Aug 16 2024: (Start)
Infinite since A052224 is infinite (as are all sequences with digital sum 1..10).
a(6492) has 1001 digits. (End)

			The first chunk of integers with digitsum 10 is (0,1,2,3,4);
the next one is (5,10,11,20),
the next one is (6,12,100),
the next one is (7,21),
the next one is (8,101),
the next one is (9,1000),
the next one is (13,14,10000), etc.
The concatenation of the above chunks produce the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..6491

Cf. A166311, A051885, A228915.

Numbers with digital sum 1..10: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10).

from itertools import islice
def bgen(ds): # generator of terms with digital sum ds
    def A051885(n): return ((n%9)+1)*10**(n//9)-1 # due to Chai Wah Wu
    def A228915(n): # due to M. F. Hasler
        p = r = 0
        while True:
            d = n % 10
            if d < 9 and r: return (n+1)*10**p + A051885(r-1)
            n //= 10; r += d; p += 1
    k = A051885(ds)
    while True: yield k; k = A228915(k)
def agen(): # generator of terms
    an, ds_block = 0, 0
    dsg = [None] + [bgen(i) for i in range(1, 11)]
    dsi = [None] + [(next(dsg[i]), i) for i in range(1, 11)]
    while True:
        yield an
        an, ds_an = min(dsi[j] for j in range(1, 11-ds_block))
        ds_block = (ds_block + ds_an)%10
        dsi[ds_an] = (next(dsg[ds_an]), ds_an)
print(list(islice(agen(), 61))) # Michael S. Branicky, Aug 16 2024

a(46) and beyond from Michael S. Branicky, Aug 16 2024.

A375380 Lexicographically earliest sequence S of distinct nonnegative integers such that 9 out of the last 10 digits of S always sum to a square.

Original entry on oeis.org

1000000000, 0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11, 13, 12, 15, 14, 17, 16, 21, 18, 19, 20, 27, 22, 25, 23, 30, 24, 26, 31, 28, 29, 32, 34, 33, 36, 38, 35, 37, 40, 39, 41, 42, 43, 44, 47, 45, 50, 46, 51, 48, 52, 54, 55, 49, 57, 53, 56, 58, 59, 60, 64, 61, 62, 63

Offset: 1

Eric Angelini, Aug 14 2024

The smallest available 10-digit integer to start S with is a(1) = 1000000000.

The sequence is infinite as there are infinitely many integers whose 9 of the last 10 digits sum to a square.

			We start with a(1) = 1000000000:
S = 1000000000,
The last 10 digits of S are [1000000000]; we discard a 0 and the sum of the 9 remaining digits = 1 (a square). We try to extend S with 0:
S = 1000000000,0,
The last 10 digits of S are [0000000000]; we discard a 0 and the sum of the 9 remaining digits = 0 (a square). We try to extend S with 1:
S = 1000000000,0,1,
The last 10 digits of S are [0000000001]; we discard a 0 and the sum of the 9 remaining digits = 1 (a square). We try to extend S with 2:
S = 1000000000,0,1,2,
The last 10 digits of S are [0000000012]; we discard 2 and the sum of the 9 remaining digits = 1 (a square). We try to extend S with 3:
S = 1000000000,0,1,2,3,
The last 10 digits of S are [0000000123]; we discard 2 and the sum of the 9 remaining digits = 4 (a square). We try to extend S with 4:
S = 1000000000,0,1,2,3,4,
The last 10 digits of S are [0000001234]; we discard 1 and the sum of the 9 remaining digits = 9 (a square). We try to extend S with 5:
S = 1000000000,0,1,2,3,4,5,
The last 10 digits of S are [0000012345]; as no square can be reached, whatever we discard, we don’t extend S with 5. We try to extend S with 6:
S = 1000000000,0,1,2,3,4,6,
The last 10 digits of S are [0000012346]; we discard a 0 and the sum of the 9 remaining digits = 16 (a square).
Etc.

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

Cf. A352000.

from itertools import count, islice, combinations
def f(k, last10):
    d = list(map(int, str(k)))
    if len(d) >= 10: return d[-10:]
    else: return last10[len(d):] + d
def c(k, last10):
    for pick9 in combinations(f(k, last10), 9):
        if sum(pick9) in {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}:
            return True
    return False
def agen(): # generator of terms
    an, seen, last10, mink = 1000000000, set(), [], 0
    while True:
        yield an
        seen.add(an)
        last10 = f(an, last10)
        an = next(k for k in count(mink) if k not in seen and c(k, last10))
        while mink in seen: mink += 1
print(list(islice(agen(), 66))) # Michael S. Branicky, Aug 14 2024

a(14) and beyond from Michael S. Branicky, Aug 14 2024

A375328 Terms as well as digits fit the prime/prime/nonprime pattern; this is the lexicographically earliest injective sequence with this property.

Original entry on oeis.org

2, 3, 0, 5, 7, 1, 23, 13, 20, 37, 17, 21, 53, 43, 24, 73, 47, 26, 229, 239, 22, 67, 29, 25, 83, 31, 27, 97, 59, 32, 127, 137, 4, 251, 271, 33, 157, 173, 6, 331, 359, 35, 433, 457, 8, 379, 521, 52, 653, 673, 9, 571, 739, 55, 677, 823, 12, 71, 751, 57, 827, 853, 15, 79, 2203, 28, 2207, 263, 30, 2213, 283, 34, 2243

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 13 2024

			a(4) = 5, a(5) = 7, a(6) = 1, a(7) = 23, a(8) = 13, a(9) = 20; we see that a(4), a(5), a(7) and a(8) are primes and that a(6) and a(9) are nonprimes. The digits involved fit the pattern prime/prime/nonprime too; they are 5,7,1,2,3,1,3,2 and 0.

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

Cf. A217555, A375326, A375327.

from sympy import isprime
from itertools import count, islice, product
def bgen(i): # generates terms with p/p/np, p/np/p, or np/p/p digits
    digs = ["014689", "2357"]
    for digits in count(1):
        patt = [digs[(i+j)%3 < 2] for j in range(digits)]
        yield from (int("".join(s)) for s in product(*patt) if digits==1 or s[0]!="0")
def agen(): # generator of terms
    seen, s = set(), 0
    for n in count(1):
        p = (n-1)%3 < 2
        an = next(k for k in bgen(s) if k not in seen and isprime(k)==p)
        yield an
        seen.add(an)
        s += len(str(an))
print(list(islice(agen(), 99))) # Michael S. Branicky, Aug 13 2024

A375327 Terms as well as digits fit the nonprime/nonprime/prime pattern; this is the lexicographically earliest injective sequence with this property.

Original entry on oeis.org

0, 1, 2, 4, 6, 3, 8, 9, 5, 10, 20, 13, 14, 21, 17, 16, 24, 43, 18, 26, 47, 40, 28, 67, 44, 30, 83, 46, 34, 97, 48, 36, 131, 12, 49, 7, 60, 38, 139, 15, 64, 29, 42, 66, 31, 45, 68, 59, 62, 69, 71, 63, 80, 79, 65, 81, 211, 39, 82, 11, 50, 85, 19, 51, 87, 41, 54, 92, 61, 56, 93, 89, 58, 95, 103, 84, 70, 151, 120

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 13 2024

			a(9) = 5, a(10) = 10, a(11) = 20, a(12) = 13, a(13) = 14, a(14) = 21 ; we see that a(9) and a(12) are primes and that a(10), a(11), a(13); and a(14) are nonprimes. The digits involved fit the pattern nonprime/nonprime/prime too; they are 5,1,0,2,0,1,3,1,4,2 and 1.

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

Cf. A217555, A375326, A375328.

from sympy import isprime
from itertools import count, islice, product
def bgen(i): # generates terms with np/np/p, np/p/np, or p/np/np digits
    digs = ["014689", "2357"]
    for digits in count(1):
        patt = [digs[(i+j)%3 == 2] for j in range(digits)]
        yield from (int("".join(s)) for s in product(*patt) if digits==1 or s[0]!="0")
def agen(): # generator of terms
    seen, s = set(), 0
    for n in count(1):
        p = (n-1)%3 == 2
        an = next(k for k in bgen(s) if k not in seen and isprime(k)==p)
        yield an
        seen.add(an)
        s += len(str(an))
print(list(islice(agen(), 99))) # Michael S. Branicky, Aug 13 2024

A375326 Terms as well as digits fit the nonprime/prime pattern; this is the lexicographically earliest injective sequence with this property.

Original entry on oeis.org

0, 2, 1, 3, 4, 5, 6, 7, 8, 29, 20, 31, 21, 59, 24, 71, 26, 79, 28, 263, 9, 283, 12, 13, 15, 17, 42, 43, 45, 47, 62, 67, 63, 83, 65, 97, 82, 131, 30, 293, 85, 139, 34, 307, 87, 151, 36, 313, 92, 179, 38, 317, 93, 421, 39, 347, 95, 431, 50, 367, 120, 383, 121, 397, 124, 503, 126, 547, 128, 563, 129, 587, 130

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 12 2024

			a(9) = 8, a(10) = 29, a(11) = 20, a(12) = 31; we see that a(9) and a(11) are nonprimes and that a(10) and a(12) are primes. The digits involved fit the pattern nonprime/prime too; they are 8, 2, 9, 2, 0, 3, 1.

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

Cf. A217555.

from sympy import isprime
from itertools import count, islice, product
def bgen(i): # generates terms with prime/nonprime or nonprime/prime digits
    digs = ["014689", "2357"]
    for digits in count(1):
        patt = [digs[(i+j)&1] for j in range(digits)]
        yield from (int("".join(s)) for s in product(*patt) if s[0]!="0")
def agen(): # generator of terms
    seen, s, an = {0, 2}, 2, 2
    yield from [0, 2]
    for n in count(3):
        p = (n&1) == 0
        an = next(k for k in bgen(s) if k not in seen and isprime(k)==p)
        yield an
        seen.add(an)
        s += len(str(an))
print(list(islice(agen(), 99))) # Michael S. Branicky, Aug 12 2024

A375306 Terms as well as digits fit the even/odd/odd pattern; this is the lexicographically earliest injective sequence with this property.

Original entry on oeis.org

0, 1, 3, 2, 5, 7, 4, 9, 101, 10, 11, 21, 12, 13, 23, 14, 15, 25, 16, 17, 27, 18, 19, 29, 30, 31, 41, 32, 33, 43, 34, 35, 45, 36, 37, 47, 38, 39, 49, 50, 51, 61, 52, 53, 63, 54, 55, 65, 56, 57, 67, 58, 59, 69, 70, 71, 81, 72, 73, 83, 74, 75, 85, 76, 77, 87, 78, 79, 89, 90, 91, 211, 6, 93, 213, 8, 95, 215, 2110, 97

Offset: 1

Eric Angelini, Aug 11 2024

			a(7) = 4, a(8) = 9, a(9) = 101, a(10) = 10; terms follow the even/odd/odd pattern and so do their digits: 4/9/1 and 0/1/1.

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

Cf. A217555, A375305.

from itertools import count, islice, repeat
def c(s, i):
    return all(int(d)%2 == int((i+j)%3 > 0) for j, d in enumerate(s))
def agen(): # generator of terms
    seen, s = set(), 0
    for n in count(0):
        eo = int(n%3 > 0)
        an = next(k for k in count(eo, 2) if k not in seen and c(str(k), s))
        yield an
        seen.add(an)
        s += len(str(an))
print(list(islice(agen(), 99))) # Michael S. Branicky, Aug 11 2024

cp's OEIS Frontend

User: Eric Angelini

A376297 Smallest terms which are the concatenation of two unambiguous distinct integers > 0.

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A376128 The absolute difference of two successive terms is prime and the absolute difference of two successive digits is also prime.

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A375614 Lexicographically earliest infinite sequence of distinct nonnegative pairs of terms that interpenetrate to produce a prime number.

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A375477 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10, said chunks being "linked" (see the Comments section for an explanation).

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A375460 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10.

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A375380 Lexicographically earliest sequence S of distinct nonnegative integers such that 9 out of the last 10 digits of S always sum to a square.

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A375328 Terms as well as digits fit the prime/prime/nonprime pattern; this is the lexicographically earliest injective sequence with this property.

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A375327 Terms as well as digits fit the nonprime/nonprime/prime pattern; this is the lexicographically earliest injective sequence with this property.

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A375326 Terms as well as digits fit the nonprime/prime pattern; this is the lexicographically earliest injective sequence with this property.

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