Jean-Marc Falcoz - cp's OEIS frontend

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

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

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

A375244 List of triples {w;x;y} where «w» is the w-th «pyramid», "x" = the number of elements in the «pyramid» that are not erased (before the end-level erasure); "y" is the number of steps in the pyramid until the iteration stops. See the Comments section for more details.

Original entry on oeis.org

1, 13, 12, 2, 12, 11, 3, 12, 10, 4, 11, 10, 5, 2, 4, 6, 11, 9, 7, 127, 79, 8, 10, 9, 9, 9, 7, 10, 1, 3, 11, 1, 2, 12, 10, 8, 13, 10, 8, 14, 126, 78, 15, 121, 77, 16, 9, 8, 17, 119, 75, 18, 8, 6, 19, 118, 74, 20, 1, 3, 21, 136, 76, 22, 1, 2, 23, 134, 75, 24, 120, 74, 25, 9, 9, 26, 121, 74, 27, 7, 8, 28, 116, 73

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 07 2024

Start the top of a "pyramid" with an integer w.
Form the lower level by adding w to each digit of w.
Erase any term having one or more duplicates, as well as its duplicates.
Iterate.
All "pyramids" will be blocked at some point, because their lowest level will end up completely erased.

			We start the first pyramid with w = 1:
.
                                 1
                                 2
                                 4
                                 8
                                16
                               17.22
                            18.24.24.24
             (a triple erasure, 22 will be erased later)
                               19.26
                            20.28.28.32
             (a double erasure, 20 will be erased later)
                            22.20.35.34
              (we erase now the 22-pair and the 20-pair)
                            38.40.37.38
      (a double erasure again, 40 will be erased at the next step)
                            44.40.40.44
.
The iteration stops there. w = 1, x = 13 as 13 terms were not erased in the blocked pyramid, y = 12 as the now blocked-pyramid has 12 levels.
Those numbers form the first triple of the sequence {1;13;12}.

Eric Angelini, This is not a fractal pipe, Personal blog of the author.

Cf. A351330.

A375243 Infinite variant of A375232.

Original entry on oeis.org

0, 10, 20, 100, 30, 102, 40, 101, 203, 105, 60, 1024, 300, 107, 200, 150, 304, 1026, 80, 109, 230, 10457, 0, 120, 306, 110, 204, 1058, 303, 10279, 0, 1046, 302, 501, 0, 201, 3048, 170, 206, 1059, 330, 1042, 0, 1000, 320, 105678, 400, 210, 3000, 190, 202, 1045, 360, 1027, 800, 1001, 2034, 510, 0, 10269

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 07 2024

Instead of stopping the sequence when no integer is available, we extend it with 0 and go on (0 being the only term allowed to be repeated whenever nothing else works). This method seems to work ad infinitum.
Around 90% of the terms are not equal to 0.
For the first 1000 terms, the largest chunk between two successive 0 is the 24-integer long serie [101101, 2630, 8015, 40044, 10122, 333330, 1907, 200222, 10456, 10200, 80008, 101110, 3204, 5107, 6660, 9012, 1400, 202000, 8051, 10276, 40400, 101111, 20223, 5109].

			The finite sequence A375232 ends with 80, 109, 230, 10457. If we extend it with a(23) = 0, we can compute a(24) = 120, a(25) = 306 then 110, 204, 1058, 303 and 10279. No more integers are available at that stage. But, again, we can extend the sequence with a(31) = 0, then a(32) = 1046 and 302, 501, 0, 201, 3048, 170, etc.
A repeated single 0 is counted as a term of the sequence.

Cf. A284516, A375232.

A375232 Two terms that contain the digit "d" are always separated by "d" terms that do not contain the digit "d". This is the lexicographically earliest sequence of distinct nonnegative integers with this property.

Original entry on oeis.org

0, 10, 20, 100, 30, 102, 40, 101, 203, 105, 60, 1024, 300, 107, 200, 150, 304, 1026, 80, 109, 230, 10457

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 06 2024

The sequence is finite, there is no 23rd term.

			As we start the sequence with a(1) = 0, the digit 0 must be present in every term of the sequence.
We extend it now with a(2) = 10 as 10 is the smallest integer not present that contains the digit 0.
The next term will be a(3) = 20 as 20 is the smallest integer not present that contains the digit 0.
The next term will be a(4) = 100 as 100 is the smallest integer not present that contains both the digits 0 and 1.
The next term will be a(5) = 30 as 30 is the smallest integer not present that contains the digit 0.
The next term will be a(6) = 102 as 102 is the smallest integer not present that contains the digits 0, 1 and 2.
The next term will be a(7) = 40 as 40 is the smallest integer not present that contains the digit 0.
The next term will be a(8) = 101 as 101 is the smallest integer not present that contains both the digits 0 and 1.
Etc.

Eric Angelini, Digits and gaps, personal blog of the author.

Cf. A284516.

a(14) and successive terms computed by Michael S. Branicky.

A373356 Numbers which have no imbalance when they are written in French and cut at the right place (see the definition of the imbalance in the Comments section).

Original entry on oeis.org

8, 34, 52, 77, 82, 108, 121, 127, 130, 139, 172, 186, 232, 274, 287, 291, 313, 314, 319, 333, 341, 347, 355, 362, 394, 409, 426, 434, 471, 489, 535, 569, 576, 598, 602, 636, 674, 688, 731, 737, 743, 763, 807, 811, 838, 843, 854, 896, 921, 927, 934, 939, 986, 1094, 1162, 1255, 1292, 1306

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jun 02 2024

Take a number. Write this number in capital letters. Remove any spaces or hyphens to form a single block of contiguous letters. Cut this block into two parts using a vertical line which is tangent to at least one of the letters. The word ONE, for example, admits four cuts, the first before the O, the second between the O and the N, the third between the N and the E, the fourth after the E. These respective cuts produce the four pairs of subsets of following letters: —/ONE, O/NE, ON/E, ONE/—. A weight is now assigned to each subset. To do this, we replace each letter in the subset by its rank in the alphabet. Then we add up these ranks within the subset. The weights of the four successive pairs above are therefore {0;34}, {15;19}, {29;5}, {34;0}. The absolute difference of the two numbers that form such a pair is called imbalance. The successive imbalances in the example here are 34, 4, 24 and 34.

			a(1) = 8 = HUIT is balanced when cut between U and I as H+U = 29 = I+T;
a(2) = 34 = TRENTEQUATRE is balanced when cut between E and Q as T+R+E+N+T+E = 82 = Q+U+A+T+R+E;
a(3) = 52 = CINQUANTEDEUX is balanced when cut between N and T as C+I+N+Q+U+A+N = 79 = T+E+D+E+U+X; etc.
The cuts in the first integers of the sequence below are made using the equal (=) sign:
HU=IT (29=29)
TRENTE=QUATRE (82=82)
CINQUAN=TEDEUX (79=79)
SOIXANT=EDIXSEPT (102=102)
QUATREV=INGTDEUX (104=104)
CENTH=UIT (50=50)
CENTVIN=GTETUN (87=87)
CENTVIN=GTSEPT (87=87)
CENTT=RENTE (62=62)
CENTTRE=NTENEUF (85=85)
CENTSOIXA=NTEDOUZE (110=110)
CENTQUATRE=VINGTSIX (124=124)
DEUXCENTT=RENTEDEUX (116=116)
DEUXCENTSOIX=ANTEQUATORZE (163=163)
DEUXCENTQUAT=REVINGTSEPT (155=155)
DEUXCENTQUAT=REVINGTONZE (155=155)
TROISCEN=TTREIZE (103=103)
TROISCENT=QUATORZE (123=123)
TROISCEN=TDIXNEUF (103=103)
TROISCENTT=RENTETROIS (143=143)
TROISCENTQ=UARANTEETUN (140=140)
TROISCENTQ=UARANTESEPT (140=140)
TROISCENTCI=NQUANTECINQ (135=135)
TROISCENTS=OIXANTEDEUX (142=142)
TROISCENTQUATR=EVINGTQUATORZE (200=200)
QUATREC=ENTNEUF (85=85)
QUATRECENT=VINGTSIX (124=124)
QUATRECENTT=RENTEQUATRE (144=144)
QUATRECENTSO=IXANTEETONZE (158=158)
QUATRECENTQU=ATREVINGTNEUF (162=162)
CINQCENTT=RENTECINQ (105=105)
CINQCENTSO=IXANTENEUF (119=119)
CINQCENTSOI=XANTESEIZE (128=128)
CINQCENTQUATRE=VINGTDIXHUIT (167=167)
SIXCEN=TDEUX (74=74)
...

Cf. A104059, A373321, A373317, A372222, A083967.

A373321 Numbers which are their own imbalance when written in French (see the definition of the imbalance in the Comments section).

Original entry on oeis.org

4, 16, 19, 22, 24, 25, 34, 41, 70, 97, 119, 128, 138, 152, 155, 170, 174, 222, 232, 258, 285, 296

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jun 01 2024

Take a number. Write this number in capital letters. Remove any spaces or hyphens to form a single block of contiguous letters. Cut this block into two parts using a vertical line which is tangent to at least one of the letters. The word ONE, for example, admits four cuts, the first before the O, the second between the O and the N, the third between the N and the E, the fourth after the E. These respective cuts produce the four pairs of subsets of following letters: —/ONE, O/NE, ON/E, ONE/—. A weight is now assigned to each subset. To do this, we replace each letter in the subset by its rank in the alphabet. Then we add up these ranks within the subset. The weights of the four successive pairs above are therefore {0;34}, {15;19}, {29;5}, {34;0}. The absolute difference of the two numbers that form such a pair is called imbalance. The successive imbalances in the example here are 34, 4, 24 and 34.

This sequence is conjectured to be finite and full.

			4 has an imbalance of 4 units when written in French and cut between A and T:
Q+U+A = 39 and T+R+E = 43. We have indeed 43 - 39 = 4.
16 has an imbalance of 16 units when written in French and cut between E and I:
S+E = 24 and I+Z+E = 40. We have indeed 40 - 24 = 16.
The cuts in the integers below are made using this sign ——:
QUA——TRE (43——39 = 4)
SE——IZE (24——40 = 16)
DIXN——EUF (51——32 = 19)
VING——TDEUX (52——74 = 22)
VINGTQ——UATRE (89——65 = 24)
VIN——GTCINQ (45——70 = 25)
TRENTEQ——UATRE (99——65 = 34)
QUARA——NTEETUN (58——99 = 41)
SOIXANTE——DIX (107——37 = 70)
QUATR——EVINGTDIXSEPT (77——174 = 97)
C——ENTDIXNEUF (3——122 = 119)
CEN——TVINGTHUIT (22——150 = 128)
CEN——TTRENTEHUIT (22——160 = 138)
CENTCINQUANTEDEU——X (176——24 = 152)
CENTCINQUANTECIN——Q (172——17 = 155)
CE——NTSOIXANTEDIX (8——178 = 170)
CENTSOIXANTEQUATO——RZE (223——49 = 174)
DEUXCENTVINGTDEUX—— (222——0 = 222)
DEUXCENTTRENTEDEUX—— (232——0 = 232)
DEUXCENTCINQUANTEHUIT—— (258——0 = 258)
D——EUXCENTQUATREVINGTCINQ (4——289 = 285)
DE——UXCENTQUATREVINGTSEIZE (9——305 = 296).
No integer has been found that could be cut in more than one way.

Cf. A104059, A373317 (English version), A083967.

cp's OEIS Frontend

User: Jean-Marc Falcoz

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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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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A375244 List of triples {w;x;y} where «w» is the w-th «pyramid», "x" = the number of elements in the «pyramid» that are not erased (before the end-level erasure); "y" is the number of steps in the pyramid until the iteration stops. See the Comments section for more details.

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A375243 Infinite variant of A375232.

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A375232 Two terms that contain the digit "d" are always separated by "d" terms that do not contain the digit "d". This is the lexicographically earliest sequence of distinct nonnegative integers with this property.

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A373356 Numbers which have no imbalance when they are written in French and cut at the right place (see the definition of the imbalance in the Comments section).

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