A121805 -id:A121805 - cp's OEIS frontend

A367600 Numbers that are not the comma-successor of any number.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 49, 50, 51, 52, 53, 54, 60, 62, 63, 64, 65, 70, 74, 75, 76, 80, 86, 87, 90, 98, 200, 300, 400, 500, 600, 700, 800, 900, 2000, 3000, 4000, 5000, 6000, 7000

Offset: 1

Giovanni Resta, Nov 23 2023

These are the positive integers that do not appear in A367338.

All terms > 98 are of the form c*10^i for i >= 2 and 2 <= c <= 9; see proof in links. - Michael S. Branicky, Nov 28 2023

Michael S. Branicky, Table of n, a(n) for n = 1..110 (all terms < 10^9)
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Michael S. Branicky, Comma-Predecessor Theorem

Cf. A121805, A337338, A367614.

from itertools import count, islice
def A367338(n):
    nn = n + 10*(n%10)
    return next((nn+y for y in range(1, 10) if str(nn+y)[0] == str(y)), -1)
def agen():
    A367338_set = set()
    for n in count(1):
        A367338_set.add(A367338(n))
        if n not in A367338_set:
            yield n
        # A367338_set.discard(n-100) # uncomment if memory is an issue
print(list(islice(agen(), 86))) # Michael S. Branicky, Nov 28 2023

A367603 Indices of records in A367601.

Original entry on oeis.org

1, 2, 6, 40, 4000, 20000, 80000000000, 40000000000000000000000000000, 30000000000000000000000000000000, 3000000000000000000000000000000000000000, 6000000000000000000000000000000000000000

Offset: 1

Michael S. Branicky, Dec 06 2023

Michael S. Branicky, Table of n, a(n) for n = 1..133
Michael S. Branicky, Table of indices and record highs in A367601

Cf. A121805, A330128, A330129, A367364, A367365, A367602.

A367616 a(n) is the unique k such that n is a comma-child of k, or -1 if k does not exist.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 20, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, 30, 21, 12, 3, -1, -1, -1, -1, -1, -1, -1, 40, 31, 22, 13, 4, -1, -1, -1, -1, -1, -1, 50, 41, 32, 23, 14, 14, 5, -1, -1, -1, -1, 60, 51, 42, 33, 33, 24, 15, 6, -1, -1, -1, 70, 61, 52, 52, 43, 34, 25, 16, 7

Offset: 1

N. J. A. Sloane, Dec 18 2023

Similar to A367614, but here we give the k such that n is a comma-child of k, whereas in A367614 n has to be a comma-successor of k. See A367338 for definitions.
The first difference between A367614 and the present sequence arises because 14 has one comma-successor, 59, but has two comma-children, 59 and 60. So A367614(59) = 14, A367614(60) = -1, while in the present sequence we have a(59) = a(60) = 14.
There are similar differences at n = 69 and 70, because both are comma-children of 33, and at many other places.

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

Cf. A121805, A367338, A367346, A367614.

def a(n):
    y = int(str(n)[0])
    x = (n-y)%10
    k = n - y - 10*x
    return k if k > 0 else -1
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Dec 18 2023

A134647 Let a(1) = 0, a(2) = 1; and a(n) = a(n-1) + [the two-digit integer split by the comma which separates a(n-1) and a(n-2)].

Original entry on oeis.org

0, 1, 2, 14, 35, 78, 135, 216, 268, 330, 413, 417, 451, 525, 540, 595, 600, 656, 662, 728, 755, 842, 900, 929, 938, 1037, 1118, 1189, 1270, 1361, 1362, 1373, 1394, 1425, 1466, 1517, 1578, 1649, 1730, 1821, 1822, 1833, 1854, 1885, 1926, 1977

Offset: 1

N. J. A. Sloane, Oct 17 2009, based on a posting to the Sequence Fans Mailing List by Eric Angelini, Oct 15 2009

			135 = 78 + 57, since the comma in 35,78 splits the number 57.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (First 1000 terms from _Harvey P. Dale_)

Cf. A121805, A367357 (first differences).

# to compute the first M terms, from N. J. A. Sloane, Nov 20 2023
Ldigit:=proc(n) local v; v:=convert(n, base, 10); v[-1]; end;
M:=100;
s:=[0,1]; i:=0; j:=1;
for m from 2 to M do
k := j + 10*(i mod 10) + Ldigit(j); s:=[op(s),k]; i:=j; j:=k; od:
s;

a[1]=0;a[2]=1;
a[n_]:=a[n]=a[n-1]+10 Mod[a[n-2],10]+IntegerDigits[a[n-1]][[1]];
Table[a[k],{k,100}]
nxt[{a_,b_}]:={b,b+10IntegerDigits[a][[-1]]+First[IntegerDigits[b]]}; NestList[ nxt,{0,1},50][[All,1]] (* Harvey P. Dale, Jul 01 2020 *)

Corrected and extended with Mma code by Farideh Firoozbakht, Oct 15 2009

A260261 Concatenation of their first digits yields the difference of two subsequent terms, a(1)=1. This is the lexicographically first infinite sequence with this property.

Original entry on oeis.org

0, 1, 12, 24, 48, 97, 188, 199, 211, 233, 255, 277, 299, 322, 355, 388, 422, 466, 511, 566, 622, 688, 755, 833, 922, 1013, 1024, 1035, 1046, 1057, 1068, 1079, 1090, 1101, 1112, 1123, 1134, 1145, 1156, 1167, 1178, 1189, 1200, 1211, 1222, 1233, 1244, 1255, 1266, 1277

Offset: 0

Eric Angelini and M. F. Hasler, Jul 21 2015

At each step, the smallest possible choice must be made for the next term.
The initial term a(0) = 0 has been added since it can be thought to make sense, but it is maybe not without controversy ("leading zero", unambiguous continuation, ...) and should therefore be considered as purely conventional or be ignored completely.
In contrast to A121805, this sequence is infinite.

			a(1)=1 cannot be followed by a(2) = 2, 3, ..., 10 or 11 because this would yield concatenated first digits equal to 12, 13, ..., 19, 11 and 11, all different from the gap a(2) - a(1). But a(2) = 12 does produce a gap of 11 equal to the first two digits concatenated.
Then must follow a term >= 12 + 11 = 23 since the concatenation of the first two digits will be at least 11, but 23 would yield 12 for the concatenation, not equal to the gap. For a(3) = 24 the same concatenation is equal to the gap.
After a(25)=1013, and also after a(282)=10044, there follow many gaps of 11, then, e.g. after a(1187)=19999, one gap of 12 followed by about 450 gaps of 22, then one gap of 23 and about 300 gaps of 33, etc. After about 100 gaps of 99 the sequence reaches a(2850) = 99911, followed by a gap of 91, and then again more than 9000 gaps of 11.
This behavior will continue for each order of magnitude, so it is easily seen that the sequence is infinite.

M. F. Hasler, Table of n, a(n) for n = 0..10000
E. Angelini, A story of leftmost digit (corrected), Jul 21 2015

Cf. A121805 (gap equals concatenation of the last and first digits, respectively).

a=1;for(n=1,100,print1(a",");a+=10*digits(a)[1];for(k=1,9,digits(a+k)[1]==k&&(a+=k)&&next(2));error) \\ The error never occurs.

a(10^7)=178567225.

A367347 a(1) = 1; thereafter, a(n+1) = a(n) + x + 10*y, where x and y are respectively the first and last digits of a(n).

Original entry on oeis.org

1, 12, 33, 66, 132, 153, 184, 225, 277, 349, 442, 466, 530, 535, 590, 595, 650, 656, 722, 749, 846, 914, 963, 1002, 1023, 1054, 1095, 1146, 1207, 1278, 1359, 1450, 1451, 1462, 1483, 1514, 1555, 1606, 1667, 1738, 1819, 1910, 1911, 1922, 1943, 1974, 2015, 2067, 2139, 2231, 2243, 2275, 2327, 2399

Offset: 1

N. J. A. Sloane, Nov 16 2023

This sequence and A367348 are simple models for the Commas sequence A121805.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A121805, A367348.

NestList[#+First[IntegerDigits[#]]+10Mod[#,10]&,1,100] (* Paolo Xausa, Nov 17 2023 *)

from itertools import islice
def agen(): # generator of terms
    an = 1
    while True:
        yield an
        s = str(an)
        an += int(s[0]) + 10*int(s[-1])
print(list(islice(agen(), 55))) # Michael S. Branicky, Nov 17 2023

a(888508) = A367348(837058) = 40000026, so a(888508+i) = A367348(837058+i) for i >= 0. - Michael S. Branicky, Nov 17 2023

A367348 a(1) = 1, a(2) = 11; thereafter, a(n+1) = a(n) + x + 10*y, where x and y are respectively the first and last digits of a(n).

Original entry on oeis.org

1, 11, 22, 44, 88, 176, 237, 309, 402, 426, 490, 494, 538, 623, 659, 755, 812, 840, 848, 936, 1005, 1056, 1117, 1188, 1269, 1360, 1361, 1372, 1393, 1424, 1465, 1516, 1577, 1648, 1729, 1820, 1821, 1832, 1853, 1884, 1925, 1976, 2037, 2109, 2201, 2213, 2245, 2297, 2369

Offset: 1

N. J. A. Sloane, Nov 16 2023

This sequence and A367347 are simple models for the Commas sequence A121805.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A121805, A367347.

Join[{1},NestList[#+First[IntegerDigits[#]]+10Mod[#,10]&,11,100]] (* Paolo Xausa, Nov 17 2023 *)

from itertools import islice
def agen(): # generator of terms
    yield 1
    an = 11
    while True:
        yield an
        s = str(an)
        an += int(s[0]) + 10*int(s[-1])
print(list(islice(agen(), 55))) # Michael S. Branicky, Nov 17 2023

a(837058) = A367347(888508) = 40000026, so a(837058+i) = A367347(888508+i) for i >= 0. - Michael S. Branicky, Nov 17 2023

A367349 List of numbers in the range 1 to 99 missing from A366487.

Original entry on oeis.org

4, 10, 15, 20, 24, 30, 40, 44, 50, 60, 64, 65, 70, 80, 84, 90

Offset: 1

N. J. A. Sloane, Nov 17 2023

These numbers do not occur as differences between successive terms in A121805.

Carlos Rivera, Puzzle 980. The "Commas" sequence, The Prime Puzzles and Problems Connection.

Cf. A121805, A366487.

a[1] = 1; a[n_] := a[n] = For[r = Mod[a[n - 1], 10]; y = 0, y <= 9, y++, If[y == IntegerDigits[#][[1]], Return[#]] &[a[n - 1] + 10 r + y]]; Complement[Range[99], Union@ Differences@ Array[a, 10^6]] (* Michael De Vlieger, Nov 18 2023, after Jean-François Alcover at A121805 *)

A367363 First term is 1; thereafter, if u and v are consecutive terms, with decimal expansions u = bc...ef, v = hi...jk, then v-u has decimal expansion efhi, formed by concatenating the last two digits of u and the first two digits of v. If there is a choice for v, pick the smallest.

Original entry on oeis.org

1, 112, 1325, 3863, 10173, 17490, 26516, 28144, 32576, 40216, 41857, 47604, 48052, 53305, 53858, 59717, 61478, 69347, 74121, 76297, 86083, 94477, 102187, 110898, 120710, 121722, 123934, 127346, 131959, 137872, 145086, 153701, 153816, 155431, 158546, 163162

Offset: 1

N. J. A. Sloane, Nov 23 2023

A generalization of A121805.

			a(1) = 1, so ef = "01" = 1. So v-u will be a four-digit number (with a leading zero in this case), say v-u = 0xyz, with v = 1 + xyz. This suggests that we try x=1 and y=1, v = 1 + xyz = 1 + 11*, where * = 1+z. The smallest choice is z = 0, giving "efhi" = "0111" = 111, and a(2) = 1 + 111 = 112 works.

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

Cf. A121805.

from itertools import islice
def agen(): # generator of terms
    an, y = 1, 1
    while y:
        yield an
        an, y = an + 100*(an%100), 1
        y = next((y for y in range(1, 100) if str(an+y)[:2] == str(y)), 0)
        an += y
print(list(islice(agen(), 36))) # Michael S. Branicky, Nov 23 2023

More terms from Michael S. Branicky, Nov 23 2023

A367504 a(1) = 2; for n > 1, a(n) = a(n-1) + 2*gpf(a(n-1)), where gpf(k) = A006530(k) = greatest prime dividing k.

Original entry on oeis.org

2, 6, 12, 18, 24, 30, 40, 50, 60, 70, 84, 98, 112, 126, 140, 154, 176, 198, 220, 242, 264, 286, 312, 338, 364, 390, 416, 442, 476, 510, 544, 578, 612, 646, 684, 722, 760, 798, 836, 874, 920, 966, 1012, 1058, 1104, 1150, 1196, 1242, 1288, 1334, 1392, 1450, 1508, 1566, 1624, 1682, 1740, 1798

Offset: 1

Scott R. Shannon, Nov 21 2023

nonn

Conjecture: with the requirement that the prime factorization of each term is written so that the primes are ordered from smallest to largest, the sequence is the lexicographically earliest infinite sequence of distinct positive numbers such that gpf(a(n-1)) * lpf(a(n)) = |a(n) - a(n-1)|, where gpf(k) = A006530(k) = greatest prime factor of k and lpf(k) = A020639(k) = least prime factor of k. In this way the sequence is the ordered prime factorization version of the 'Commas sequence', A121805. One can show that, for such a sequence to be infinite, no odd number can appear. Although for many terms a lower even number can be chosen for the following term, which can lead to even lower numbers for further terms, it is conjectured all such choices will ultimately halt the sequence as a number is eventually reached for which no unused next number exists which follows the required rule for the difference between the terms. Therefore all terms must be larger than the previous, and the earliest such infinite sequence is the given sequence.

See A367465 for the sequence when the requirement that the primes in the factorization of each term must be in order is removed.

			a(7) = 40 as a(6) = 30 = 2*3*5, thus A006530(30) = 5 and a(7) = a(6) + 2*5 = 30 + 2*5 = 40.

Cf. A027746, A006530, A020639, A367465, A121805, A123581, A180107, A036441, A076271.

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A367600 Numbers that are not the comma-successor of any number.

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A367603 Indices of records in A367601.

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A367616 a(n) is the unique k such that n is a comma-child of k, or -1 if k does not exist.

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A134647 Let a(1) = 0, a(2) = 1; and a(n) = a(n-1) + [the two-digit integer split by the comma which separates a(n-1) and a(n-2)].

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A260261 Concatenation of their first digits yields the difference of two subsequent terms, a(1)=1. This is the lexicographically first infinite sequence with this property.

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A367347 a(1) = 1; thereafter, a(n+1) = a(n) + x + 10*y, where x and y are respectively the first and last digits of a(n).

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A367348 a(1) = 1, a(2) = 11; thereafter, a(n+1) = a(n) + x + 10*y, where x and y are respectively the first and last digits of a(n).

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A367349 List of numbers in the range 1 to 99 missing from A366487.

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A367363 First term is 1; thereafter, if u and v are consecutive terms, with decimal expansions u = bc...ef, v = hi...jk, then v-u has decimal expansion efhi, formed by concatenating the last two digits of u and the first two digits of v. If there is a choice for v, pick the smallest.

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