A121805 -id:A121805 - cp's OEIS frontend

A330128 a(n) is the number of terms in the analog of A121805 but starting with n, or -1 if that sequence is infinite.

2137453, 194697747222394, 2, 199900, 19706, 209534289952018960, 15, 198104936410, 19511030, 20573, 20572, 2137452, 20534, 19238, 2, 2089707, 20670629294, 1, 21482, 19278442756937613, 2074, 19278442756937612, 20571, 194697747222393, 193, 197062, 1, 197, 2061823

Offset: 1

Giovanni Resta, Dec 02 2019

The final terms of the corresponding sequences are given in A330129.

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

Cf. A330129 (corresponding last term), A121805, A139284, A366492.

For record high points see A367364 and A367365.

nxt[x_] := Block[{p=1, n=x}, While[n >= 10, n = Floor[n/10]; p *= 10]; p (n + 1)]; a[n_] := Block[{nT=1, nX=n, w1, w2, w3, x, it, stp, oX}, stp = 100; w1 = w2 = w3 = 0; While[True, oX = nX; nT++; x = 10*Mod[oX, 10]; nX = SelectFirst[Range[9], IntegerDigits[oX + x + #][[1]] == # &, 0]; If[nX == 0, Break[], nX = nX + oX + x]; If[nT == stp, stp += 100; w1=w2; w2=w3; w3=nX; If[w3 + w1 == 2 w2 && Mod[w3 - w2, 100] == 0, it = Floor[(nxt[nX] - nX - 1)/(w3 - w2)]; nT += it*100; nX += (w3 - w2)*it; w3=nX; stp += it*100]]]; nT - 1]; Array[a, 30]

def nxt(x):
    p, n = 1, x
    while n >= 10:
        n //= 10
        p *= 10
    return p * (n + 1)
def a(n):
    nT, nX, w1, w2, w3, stp  = 1, n, 0, 0, 0, 100
    while True:
        oX = nX
        nT += 1
        x = 10*(oX%10)
        nX = next((y for y in range(1, 10) if str(oX+x+y)[0] == str(y)), 0)
        if nX == 0: break
        else: nX += oX + x
        if nT == stp:
            stp += 100
            w1, w2, w3 = w2, w3, nX
            if w3 + w1 == 2*w2 and (w3 - w2)%100 == 0:
                it = (nxt(nX) - nX - 1)//(w3 - w2)
                nT += it*100
                nX += (w3 - w2)*it
                w3 = nX
                stp += it*100
    return nT - 1
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Nov 18 2023 after Giovanni Resta

Escape clause added to definition by N. J. A. Sloane, Nov 14 2023

A330129 a(n) is the last term of the analogous sequence to A121805, but with initial term n, or -1 if that sequence is infinite.

Original entry on oeis.org

99999945, 9999999999999918, 36, 9999945, 999945, 9999999999999999936, 936, 9999999999972, 999999936, 999936, 999936, 99999945, 999954, 999918, 72, 99999918, 999999999927, 18, 999981, 999999999999999963, 99981, 999999999999999963, 999936, 9999999999999918, 9963

Offset: 1

Giovanni Resta, Dec 02 2019

The numbers of terms of the corresponding sequences are in A330128.

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

Cf. A330128, A121805, A139284, A366492.

nxt[x_] := Block[{p=1, n=x}, While[n >= 10, n = Floor[n/10]; p *= 10]; p (n + 1)]; a[n_] := Block[{nT=1, nX=n, w1, w2, w3, x, it, stp, oX}, stp = 100; w1 = w2 = w3 = 0; While[True, oX = nX; nT++; x = 10*Mod[oX, 10]; nX = SelectFirst[Range[9], IntegerDigits[oX + x + #][[1]] == # &, 0]; If[nX == 0, Break[], nX = nX + oX + x]; If[nT == stp, stp += 100; w1=w2; w2=w3; w3=nX; If[w3 + w1 == 2 w2 && Mod[w3 - w2, 100] == 0, it = Floor[(nxt[nX] - nX - 1)/(w3 - w2)]; nT += it*100; nX += (w3 - w2)*it; w3=nX; stp += it*100]]]; oX]; Array[a, 30]

def nxt(x):
    p, n = 1, x
    while n >= 10:
        n //= 10
        p *= 10
    return p * (n + 1)
def a(n):
    nT, nX, w1, w2, w3, stp  = 1, n, 0, 0, 0, 100
    while True:
        oX = nX
        nT += 1
        x = 10*(oX%10)
        nX = next((y for y in range(1, 10) if str(oX+x+y)[0] == str(y)), 0)
        if nX == 0: break
        else: nX += oX + x
        if nT == stp:
            stp += 100
            w1, w2, w3 = w2, w3, nX
            if w3 + w1 == 2*w2 and (w3 - w2)%100 == 0:
                it = (nxt(nX) - nX - 1)//(w3 - w2)
                nT += it*100
                nX += (w3 - w2)*it
                w3 = nX
                stp += it*100
    return oX
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Nov 18 2023 after Giovanni Resta

Escape clause added to definition by N. J. A. Sloane, Nov 14 2023

A139284 Analog of A121805, but starting with 2.

Original entry on oeis.org

2, 24, 71, 89, 180, 181, 192, 214, 256, 319, 413, 447, 522, 547, 623, 659, 756, 824, 872, 901, 920, 929, 1020, 1021, 1032, 1053, 1084, 1125, 1176, 1237, 1308, 1389, 1480, 1481, 1492, 1513, 1544, 1585, 1636, 1697, 1768, 1849, 1940, 1941, 1952, 1973, 2005

Offset: 1

N. J. A. Sloane (based on Angelini's article), Jun 08 2008

It appears that this sequence and A121805 have no terms in common. Furthermore, this sequence exists for at least 1551000000 terms. - Jacques ALARDET, Jul 22 2008

The last term of the sequence is a(194697747222394) = 9999999999999918. - Giovanni Resta, Nov 30 2019

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 980. The "Commas" sequence, The Prime Puzzles and Problems Connection.

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

a:= proc(n) option remember; local k, t, y; if n=1 then 2 else k:= a(n-1); for y from 0 to 9 do t:= k +10* irem (k, 10) +y; if convert (t, base, 10)[ -1]=y then return t fi od; NULL fi end: seq(a(n), n=1..80); # Alois P. Heinz, Aug 13 2009

a[1] = 2; a[n_] := a[n] = For[x=Mod[a[n-1], 10]; y=0, y <= 9, y++, an = a[n-1] + 10*x + y; If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 80] (* Jean-François Alcover, Nov 25 2014 *)

from itertools import islice
def agen(): # generator of terms
    an, y = 2, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 47))) # Michael S. Branicky, Apr 08 2022

More terms from Alois P. Heinz, Aug 13 2009

A366492 Analog of A121805, but starting with 4.

Original entry on oeis.org

4, 48, 129, 221, 233, 265, 318, 402, 426, 490, 494, 539, 635, 691, 708, 795, 853, 891, 910, 919, 1010, 1011, 1022, 1043, 1074, 1115, 1166, 1227, 1298, 1379, 1470, 1471, 1482, 1503, 1534, 1575, 1626, 1687, 1758, 1839, 1930, 1931, 1942, 1963, 1994, 2036, 2098, 2180, 2182

Offset: 1

N. J. A. Sloane, Nov 14 2023

If instead we start with 3, the sequence is the two-term sequence [3, 36].

The present sequence is finite, with last term a(199900) = 9999945.

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

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

from itertools import islice
def agen(start=4): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Nov 18 2023

A367341 Numbers without comma-successors: these are the numbers k such that if the commas sequence A121805 is started at k instead of 1, there is no second term.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 72, 81, 918, 927, 936, 945, 954, 963, 972, 981, 9918, 9927, 9936, 9945, 9954, 9963, 9972, 9981, 99918, 99927, 99936, 99945, 99954, 99963, 99972, 99981, 999918, 999927, 999936, 999945, 999954, 999963, 999972, 999981, 9999918, 9999927, 9999936, 9999945, 9999954, 9999963, 9999972, 9999981

Offset: 1

N. J. A. Sloane, Nov 15 2023

Comment from N. J. A. Sloane, Nov 19 2023 (Start)
Theorem. This sequence consists precisely of the decimal numbers of the form
  99...9xy = 100*(10^i-1) + 9*x + 9,
with i >= 0 copies of 9, and 1 <= x <= 8.
(See link for proof.) This was stated without proof by David W. Wilson in 2007 (see the Angelini link), and was conjectured (in a slightly less precise form) by Ivan N. Ianakiev, Nov 16 2023.
This implies that the conjecture below is true, as well as the conjecture in A367342.
All terms are multiples of 9, and A367342 gives a(n)/9.
(End)
Numbers k such that A367338(k) = A367339(k) = -1.
By definition, A330129 is a subsequence.

N. J. A. Sloane, Table of n, a(n) for n = 1..408
Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
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

Cf. A121805, A367338, A367339, A367340, A367342.

for i from 0 to 4 do t1:=100*(10^i-1);
 for x from 1 to 8 do lprint(t1+9*x+9);
od: od:

fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]}, Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]] =={};
Select[Range[10^5],fQ[#]&] (* Ivan N. Ianakiev, Nov 16 2023 *)

from itertools import islice
def ok(n):
    an, y = n, 1
    while y < 10:
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
        if y < 10:
            return False
    return True
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Nov 15 2023

The first eight terms are given by a(i) = 9*(i+1), for 1 <= i <= 8; thereafter, each successive block of eight terms is obtained by prefixing the terms of the previous block by 9. - Michael S. Branicky, Nov 15 2023 [This follows from the theorem above. - N. J. A. Sloane, Nov 19 2023]

a(33) and beyond from Michael S. Branicky, Nov 15 2023

A367355 Comma sequence, analogous to A121805, starting at 1, but the calculations are done in base 3.

Original entry on oeis.org

1, 5, 12, 13, 17, 25, 29, 36, 37, 41, 48, 49, 53, 61, 66, 68, 76

Offset: 1

N. J. A. Sloane, Nov 18 2023

Cf. A121805, A367343, A367344, A367356.

A367605 and A367605 give the length and last term for the analogous sequence in base b.

# Comma Successor in Maple in base "bas", from N. J. A. Sloane, Dec 06 2023
bas := 3;
Ldigit:=proc(n) local v; v:=convert(n, base, bas); v[-1]; end; # Returns leading digit
# Return comma-successor to a or -1 if no successor exists
commsucc := proc(a) local f,i,d;
f := (a mod bas);
d:=bas*f;
for i from 1 to bas-1 do
d := d+1;
if Ldigit(a+d) = i then return(a+d); fi;
od:
return(-1);
end;
a:=[1]; s:=1; for n from 1 to 16 do s:=commsucc(s); a:=[op(a),s]; od: a;

A367337 Analog of A121805, but starting with 5.

Original entry on oeis.org

5, 61, 78, 159, 251, 263, 295, 348, 432, 456, 521, 536, 602, 628, 715, 772, 799, 897, 976, 1037, 1108, 1189, 1280, 1281, 1292, 1313, 1344, 1385, 1436, 1497, 1568, 1649, 1740, 1741, 1752, 1773, 1804, 1845, 1896, 1957, 2029, 2121, 2133, 2165, 2217, 2289, 2381, 2393, 2425

Offset: 1

N. J. A. Sloane, Nov 14 2023

This sequence is finite, with last term a(19706) = 999945.

N. J. A. Sloane, Table of n, a(n) for n = 1..19706

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

from itertools import islice
def agen(start=5): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(agen())) # Michael S. Branicky, Nov 18 2023

A367350 Analog of A121805, but starting with 6.

Original entry on oeis.org

6, 73, 104, 145, 196, 258, 341, 354, 397, 471, 485, 540, 545, 601, 617, 693, 730, 737, 815, 873, 912, 941, 960, 969, 1060, 1061, 1072, 1093, 1124, 1165, 1216, 1277, 1348, 1429, 1520, 1521, 1532, 1553, 1584, 1625, 1676, 1737, 1808, 1889, 1980, 1981, 1992, 2014, 2056, 2118

Offset: 1

N. J. A. Sloane, Nov 17 2023

Contains 209534289952018960 terms, the last term being a(209534289952018960) = 9999999999999999936.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

b = 10; m = b - 1; a[1] = 6; a[n_] := a[n] = For[r = Mod[a[n - 1], b]; y = 0, y <= m, y++, If[y == IntegerDigits[#, b][[1]], Return[#]] &[a[n - 1] + b r + y]]; Array[a, 45] (* Michael De Vlieger, Nov 18 2023, after Jean-François Alcover at A121805 *)

from itertools import count, islice
def agen(start=6): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Nov 18 2023

A367351 Analog of A121805, but starting with 7.

Original entry on oeis.org

7, 85, 136, 197, 269, 362, 385, 439, 534, 579, 675, 732, 759, 857, 936

Offset: 1

N. J. A. Sloane, Nov 17 2023

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

b = 10; m = b - 1; a[1] = 7; a[n_] := a[n] = For[r = Mod[a[n - 1], b]; y = 0, y <= m, y++, If[y == IntegerDigits[#, b][[1]], Return[#]] &[a[n - 1] + b r + y]]; TakeWhile[Array[a, 45], IntegerQ] (* Michael De Vlieger, Nov 18 2023, after Jean-François Alcover at A121805 *)

def agen(start=7): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(agen())) # Michael S. Branicky, Nov 18 2023

A367352 Analog of A121805, but starting with 8.

Original entry on oeis.org

8, 97, 168, 250, 252, 274, 317, 390, 393, 427, 502, 527, 603, 639, 736, 804, 852, 880, 888, 977, 1048, 1129, 1220, 1221, 1232, 1253, 1284, 1325, 1376, 1437, 1508, 1589, 1680, 1681, 1692, 1713, 1744, 1785, 1836, 1897, 1968, 2050, 2052, 2074, 2116, 2178, 2260, 2262, 2284

Offset: 1

N. J. A. Sloane, Nov 17 2023

Contains 198104936410 terms, the last being 9999999999972.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129.

b = 10; m = b - 1; a[1] = 8; a[n_] := a[n] = For[r = Mod[a[n - 1], b]; y = 0, y <= m, y++, If[y == IntegerDigits[#, b][[1]], Return[#]] &[a[n - 1] + b r + y]]; Array[a, 45] (* Michael De Vlieger, Nov 18 2023, after Jean-François Alcover at A121805 *)

from itertools import islice
def agen(start=8): # generator of terms
    an, y = start, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Nov 18 2023

cp's OEIS Frontend

A330128 a(n) is the number of terms in the analog of A121805 but starting with n, or -1 if that sequence is infinite.

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A330129 a(n) is the last term of the analogous sequence to A121805, but with initial term n, or -1 if that sequence is infinite.

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A139284 Analog of A121805, but starting with 2.

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A366492 Analog of A121805, but starting with 4.

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A367341 Numbers without comma-successors: these are the numbers k such that if the commas sequence A121805 is started at k instead of 1, there is no second term.

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A367355 Comma sequence, analogous to A121805, starting at 1, but the calculations are done in base 3.

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A367337 Analog of A121805, but starting with 5.

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A367350 Analog of A121805, but starting with 6.

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