A361337 -id:A361337 - cp's OEIS frontend

A361978 Complement of A361337.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 116

Offset: 1

Michael S. Branicky, Apr 02 2023

More than the usual number of terms are shown to distinguish the sequence from A034048.

Appears to be finite with 219 members, the largest being 3111.

Michael S. Branicky, Table of n, a(n) for n = 1..219

Cf. A034048, A011540, A361337, A277061.

Subsequence of A052382.

A361978=select( {is_A361978(n)=vecmin(digits(n))&& !for(p=1, logint(n, 10), is_A361978(vecprod(divrem(n, 10^p)))|| return)}, [1..10^5]) \\ Conjecturedly the full list: no terms between 3112 and 10^5. - M. F. Hasler, Apr 05 2023

def ok(n):
    if n < 10: return n != 0
    s = str(n)
    if "0" in s: return False
    return all(ok(int(s[:i])*int(s[i:])) for i in range(1, len(s)))
print([k for k in range(117) if ok(k)]) # Michael S. Branicky, Apr 02 2023

A361338 Number of different single-digit numbers that can be reached from n by any permissible sequence of split-and-multiply operations.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3

Offset: 0

N. J. A. Sloane, Apr 04 2023, following emails from Eric Angelini and Allan C. Wechsler

We always split an integer into two integers, then multiply them (and iterate). For example, 2023 can be split into 20 and 23 (producing 20*23 = 460), or split into 202 and 3 (producing 202*3 = 606). The split 2 and 023 is forbidden, as 023 is not an integer (but 460 can be split into 46 and 0 as 0 is an integer).
The sequence is the number of different single-digit numbers that can be obtained from n by any sequence of splits and multiplications.
a(n) can take on any value from 1 to 10, inclusive. There are many obvious questions. Clearly (by induction), a(10*k) = 1, but are there arbitrarily large n with a(n) = 1 that are not multiples of 10? If not, what is the largest such n? - Allan C. Wechsler, Apr 04 2023
All numbers of the form m = (c)(0^i)(d) where c in 1..9, d in 1..9, i > 0 and juxtaposition/exponentiation are concatenation and repeated concatenation, resp., have a(m) = 1 since they lead only to 0 or multiples of 10. - Michael S. Branicky, Apr 07 2023

			From 110 we can reach 11*0 = 0, or 1*10 = 10 -> 1*0 = 0, so we can only reach 0, and so a(110) = 1.
From 112 we can reach 11*2 = 22 -> 2*2 = 4, or 1*12 = 12 -> 1*2 = 2, so a(112) = 2.

Michael De Vlieger, Table of n, a(n) for n = 0..17999
Michael De Vlieger, Plot of digit d in sequence S(n) in black at (x, y) = (n, d) for n = 0..18000, d = 0..9, in blocks of 1000 arranged vertically from smallest at top to largest at bottom, 4X exaggeration. Digit 0 is most common, followed by {2, 4, 6, 8}, and then d = 5. Among reduced residues mod 10, d = 9 seems most common in this range.
Michael De Vlieger, Plot of digit d in sequence S(n) in black at (x, y) = (n, d) for n = 0..99999, d = 0..9, in rows of 1000 arranged vertically from smallest at top to largest at bottom, no exaggeration, no spacing between rows.

See A361337 for the numbers that reach 0, and A361339 for the smallest k such that a(k) = n.

A361340 a(n) = smallest number with the property that the split-and-multiply technique (see A361338) in base n can produce all n single-digit numbers.

Original entry on oeis.org

15, 23, 119, 167, 12049, 424, 735, 907, 17117, 1250, 307747, 2703, 49225, 9422, 57823, 5437, 2076131, 7747, 639987, 44960, 822799, 11537, 23809465, 24967, 1539917, 109346, 4643181, 26357, 5587832443, 37440, 1885949, 285085, 7782015, 265806, 1250473675, 66524, 8340541, 699890, 158607997, 85684

Offset: 2

N. J. A. Sloane, Apr 04 2023, based on an email from Zachary DeStefano

From Zachary DeStefano, May 17 2023: (Start)

There is a strong linear relationship between n^(n / phi(n)) and a(n) (see A000010 for phi(n)) which results from the final digit falling into subgroups of Z/nZ during split-and-multiply steps. This explains why a(n) is significantly smaller for prime n and significantly larger when n contains several small prime factors (ex. 2 * 3 * 5 = 30) (End)

			To reach the digits 0 though 9 in base 10 from 17117:
 171*17 -> 290*7  -> 203*0 -> 0
 1711*7 -> 1197*7 -> 837*9 -> 7*533 -> 373*1 -> 37*3  -> 1*11 -> 1*1 -> 1
 171*17 -> 2*907  -> 1*814 -> 8*14  -> 1*12  -> 1*2   -> 2
 1*7117 -> 711*7  -> 49*77 -> 377*3 -> 113*1 -> 1*13  -> 1*3  -> 3
 171*17 -> 2*907  -> 1*814 -> 8*14  -> 11*2  -> 2*2   -> 4
 1711*7 -> 1197*7 -> 837*9 -> 75*33 -> 247*5 -> 1*235 -> 23*5 -> 1*15 -> 1*5  -> 5
 17*117 -> 19*89  -> 169*1 -> 16*9  -> 1*44  -> 4*4   -> 1*6  -> 6
 1711*7 -> 1197*7 -> 837*9 -> 7*533 -> 37*31 -> 11*47 -> 51*7 -> 3*57 -> 17*1 -> 1*7 -> 7
 17*117 -> 1*989  -> 98*9  -> 88*2  -> 1*76  -> 7*6   -> 4*2  -> 8
 1*7117 -> 711*7  -> 49*77 -> 377*3 -> 113*1 -> 11*3  -> 3*3  -> 9

Zachary DeStefano and Tim Peters, Table of n, a(n) for n = 2..119
Michael S. Branicky, Python program
Michael De Vlieger, Plot of digit d in sequence S_n(m) in black at (x, y) = (m, d) for bases n = 2..12 as labeled, m = 1..1000, and digits d = 0..n-1, 12X exaggeration. This sequence shows the first occasion of a vertical black bar in base n.
Zachary DeStefano and Tim Peters, Known terms and bounds

Cf. A361337-A361349.

Table[Catch[Monitor[Do[(Set[c, Count[Union@Flatten[#], ?(# < b &)]]; If[c == b, Throw[i]]) &@ NestWhileList[Flatten@ Map[Function[w, Array[If[And[#[[-1, 1]] == 0, Length[#[[-1]]] > 1], Nothing, Times @@ Map[FromDigits[#, b] &, #]] &@ TakeDrop[w, #] &, Length[w] - 1]][IntegerDigits[#, b]] &, #] &, {i}, Length[#] > 0 &], {i, 0, Infinity}], {b, i, c}]], {b, 2, 6}] (* _Michael De Vlieger, Apr 04 2023, with Monitor to show progress *)

from itertools import count
from sympy.ntheory import digits
from functools import lru_cache
def fd(d, b): # from_digits
    return sum(di*b**i for i, di in enumerate(d[::-1]))
@lru_cache(maxsize=None)
def f(n, b):
    if n < b: return {n}
    s = digits(n, b)[1:]
    return {e for i in range(1, len(s)) if s[i]!=0 or i==len(s)-1 for e in f(fd(s[:i], b)*fd(s[i:], b), b)}
def a(n, printat=False):
    return next(k for k in count(1) if len(f(k, n))==n)
print([a(n) for n in range(2, 18)]) # Michael S. Branicky, Apr 04 2023

# see link for a version that is faster and uses less memory

a(21)-a(29) from Michael S. Branicky, Apr 04 2023

a(30)-a(41) from Zachary DeStefano, Apr 05 2023

A361339 a(n) is the smallest k such that A361338(k) = n.

Original entry on oeis.org

1, 112, 139, 219, 373, 719, 1133, 1919, 3377, 17117

Offset: 1

N. J. A. Sloane, Apr 04 2023, based on an email from Michael S. Branicky

			     a(n) n  {the digits reached}
       1  1  {1}
     112  2  {2, 4}
     139  3  {8, 4, 7}
     219  4  {8, 2, 4, 6}
     373  5  {1, 2, 4, 6, 8}
     719  6  {0, 2, 4, 6, 8, 9}
    1133  7  {0, 2, 4, 6, 7, 8, 9}
    1919  8  {0, 2, 4, 5, 6, 7, 8, 9}
    3377  9  {0, 2, 3, 4, 5, 6, 7, 8, 9}
   17117 10  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Cf. A361337-A361349.

With[{s = Import["https://oeis.org/A361338/b361338.txt", "Data"][[All, -1]]}, {1}~Join~Table[-1 + FirstPosition[s, k][[1]], {k, 2, 10}]] (* Michael De Vlieger, Apr 04 2023, using bfile at A361338 *)

from itertools import count
def agen():
    n, adict = 1, dict()
    for k in count(1):
        v = A361338(k) # uses A361338() and reach1() in A361338
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
        if n == 11: return
print(list(agen())) # Michael S. Branicky, Apr 04 2023

A361342 Numbers k such that A361338(k) = 3.

Original entry on oeis.org

139, 148, 149, 167, 179, 187, 188, 189, 196, 197, 216, 217, 218, 226, 236, 238, 246, 247, 249, 256, 258, 261, 262, 263, 266, 269, 271, 276, 279, 283, 287, 288, 292, 294, 324, 329, 334, 337, 338, 339, 341, 342, 347, 349, 354, 362, 364, 368, 369, 372, 374, 376, 381, 382, 383, 386, 391, 392, 396, 399, 413, 416, 417, 423, 427, 428, 431, 436, 442, 443, 446

Offset: 1

N. J. A. Sloane, Apr 05 2023

{0, 6, 8} is by far the most frequent possible outcome for the numbers in this sequence (almost 60% of all cases up to 10^4, next most frequent being {0, 2, 6} and {0, 4, 6} and {0, 4, 8} in about 8% of the cases each). Up to 10^4, no term in this sequence can ever produce a 3. - M. F. Hasler, Apr 08 2023

Michael De Vlieger, Table of n, a(n) for n = 1..775

Cf. A361337-A361349.

-1 + Position[#, 3][[All, 1]] &@ Flatten@ Array[Map[Total, Transpose@ ImageData[ColorNegate@ Import["https://oeis.org/A361338/a361338_2.png", "PNG"], "Bit"][[10 # + 1 ;; 10 # + 10, 1 ;; 1000]]] &, 1, 0] (* Michael De Vlieger, Apr 06 2023, using image at A361338 *)

select( {is_A361342(n)=A361338(n)==3}, [1..456]) \\ M. F. Hasler, Apr 08 2023

A361343 Numbers k such that A361338(k) = 4.

Original entry on oeis.org

219, 257, 267, 274, 277, 278, 284, 286, 298, 299, 317, 319, 328, 344, 359, 363, 366, 377, 398, 418, 419, 433, 434, 437, 438, 449, 454, 464, 469, 471, 478, 482, 486, 492, 494, 527, 544, 547, 549, 576, 588, 616, 626, 633, 636, 639, 644, 657, 663, 673, 677, 681, 682, 694, 698, 699, 714, 717, 718, 727, 728, 733, 734, 736, 738, 762, 767, 773, 778, 792

Offset: 1

N. J. A. Sloane, Apr 05 2023

{0, 2, 4, 6} and {0, 2, 6, 8} are by far the most frequent possible outcome for these numbers. Up to 10^4, no number in this sequence ever produces a 1, and 1113 and 1311 are the only terms that can produce a 3, and {919, 1193, 1199, 1357, 1751, 1913, 2373} are the only terms that produce a 7. - M. F. Hasler, Apr 08 2023

Michael De Vlieger, Table of n, a(n) for n = 1..1037

Cf. A361337-A361349.

-1 + Position[#, 4][[All, 1]] &@ Flatten@ Array[Map[Total, Transpose@ ImageData[ColorNegate@ Import["https://oeis.org/A361338/a361338_2.png", "PNG"], "Bit"][[10 # + 1 ;; 10 # + 10, 1 ;; 1000]]] &, 1, 0] (* Michael De Vlieger, Apr 06 2023, using image at A361338 *)

select( {is_A361343(n)=A361338(n)==4}, [1..800]) \\ M. F. Hasler, Apr 08 2023

A361348 Numbers k such that A361338(k) = 9.

Original entry on oeis.org

3377, 3713, 4779, 5319, 5919, 5981, 7947, 8159, 8839, 9531, 9591, 9883, 10599, 11307, 11339, 11977, 13371, 13377, 13397, 13713, 13969, 14779, 14791, 15319, 15789, 15919, 15933, 15981, 15997, 16393, 16997, 17149, 17183, 17667, 17717, 17733, 17771, 17941, 17947, 17963

Offset: 1

N. J. A. Sloane, Apr 05 2023

Cf. A361337-A361349.

select( {is_A361348(n)=A361338(n)==9}, [1..20000]) \\ M. F. Hasler, Apr 08 2023

A361344 Numbers k such that A361338(k) = 5.

Original entry on oeis.org

373, 387, 389, 393, 439, 479, 619, 627, 649, 661, 688, 696, 739, 742, 784, 788, 824, 827, 847, 862, 868, 878, 879, 886, 887, 889, 898, 914, 924, 929, 938, 943, 947, 966, 969, 973, 979, 982, 988, 996, 998, 1038, 1066, 1067, 1123, 1127, 1129, 1131, 1137, 1138, 1139, 1143, 1148, 1149, 1162, 1164, 1166, 1168, 1171, 1172, 1174, 1178, 1181, 1183

Offset: 1

N. J. A. Sloane, Apr 05 2023

Cf. A361337-A361349.

select( {is_A361344(n)=A361338(n)==5}, [1..1200]) \\ M. F. Hasler, Apr 08 2023

A361345 Numbers k such that A361338(k) = 6.

Original entry on oeis.org

719, 1117, 1119, 1147, 1157, 1159, 1167, 1177, 1187, 1189, 1197, 1253, 1317, 1327, 1359, 1373, 1389, 1391, 1453, 1499, 1579, 1599, 1657, 1713, 1717, 1731, 1737, 1747, 1757, 1767, 1769, 1773, 1797, 1799, 1921, 1927, 1933, 1939, 1961, 1963, 1971, 1981, 1991, 1993, 1997, 1999, 2117, 2119, 2157, 2191, 2199, 2217, 2237, 2331, 2457, 2517, 2519

Offset: 1

N. J. A. Sloane, Apr 05 2023

{0, 2, 4, 5, 6, 8} is by far the most frequent possible outcome for the terms in this sequence, in 669 out of the 790 cases up to 10^4. 1373 is the only term up to 10^4 that can produce a 1. - M. F. Hasler, Apr 08 2023

Cf. A361337-A361349.

select( {is_A361345(n)=A361338(n)==6}, [1..2555]) \\ M. F. Hasler, Apr 08 2023

A361346 Numbers k such that A361338(k) = 7.

Original entry on oeis.org

1133, 1339, 1387, 1519, 1597, 1719, 1917, 1969, 1973, 1979, 2139, 2177, 2357, 2359, 2577, 2599, 2771, 2939, 2959, 3117, 3119, 3137, 3149, 3169, 3189, 3191, 3437, 3571, 3573, 3589, 3591, 3639, 3657, 3731, 3773, 3797, 3859, 3917, 3929, 3979, 3991, 3999, 4179, 4191, 4359

Offset: 1

N. J. A. Sloane, Apr 05 2023

Cf. A361337-A361349.

select( {is_A361346(n)=A361338(n)==7}, [1..4444]) \\ M. F. Hasler, Apr 08 2023

cp's OEIS Frontend

A361978 Complement of A361337.

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A361338 Number of different single-digit numbers that can be reached from n by any permissible sequence of split-and-multiply operations.

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A361340 a(n) = smallest number with the property that the split-and-multiply technique (see A361338) in base n can produce all n single-digit numbers.

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A361339 a(n) is the smallest k such that A361338(k) = n.

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A361342 Numbers k such that A361338(k) = 3.

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A361343 Numbers k such that A361338(k) = 4.

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A361348 Numbers k such that A361338(k) = 9.

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A361344 Numbers k such that A361338(k) = 5.

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