A361338 -id:A361338 - cp's OEIS frontend

A361349 Numbers k such that A361338(k) = 10.

17117, 17727, 17749, 18839, 19933, 21761, 22777, 29391, 30397, 31157, 31317, 31643, 33949, 34199, 35909, 39723, 39789, 39797, 39803, 39813, 42973, 44219, 44517, 46347, 47463, 49171, 49771, 49877, 53211, 53751, 57169, 57311, 57769, 57781, 57941, 57943, 57961, 59093

Offset: 1

N. J. A. Sloane, Apr 05 2023

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

Cf. A361338, A361341-A361348.

from itertools import count, islice
from math import prod
def A361338(n):
    c, d, m = {n}, set(), 0
    while True:
        c = set(prod(divmod(k,s)) for k in c for i in range(1,len(str(k))) if k%(s:=(r:=10**(i-1))*10)>=r or i==1)
        d |= c
        if (r:=len(d)) == m:
            return sum(1 for q in d if q<10)
        m = r
def A361349_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:A361338(n)==10,count(max(startvalue,1)))
A361349_list = list(islice(A361349_gen(),10)) # Chai Wah Wu, Apr 05-06 2023

from functools import lru_cache
from itertools import count, islice
@lru_cache(maxsize=None)
def f(n):
    if n < 10: return {n}
    s = str(n)
    return {e for i in range(1, len(s)) if s[i]!="0" or i==len(s)-1 for e in f(int(s[:i])*int(s[i:]))}
def agen(): # generator of terms
    yield from (k for k in count() if len(f(k)) == 10)
print(list(islice(agen(), 38))) # Michael S. Branicky, Apr 05 2023

a(4)-a(38) from Chai Wah Wu, Apr 05 2023

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

A361341 Numbers k such that A361338(k) = 2.

Original entry on oeis.org

112, 113, 114, 115, 116, 117, 119, 122, 123, 124, 126, 127, 128, 129, 132, 133, 134, 135, 136, 137, 138, 142, 143, 144, 146, 147, 153, 155, 157, 159, 162, 163, 166, 168, 169, 172, 173, 175, 176, 177, 178, 182, 183, 184, 186, 193, 198, 199, 211, 213, 221, 224, 228, 229, 231, 233, 234, 235, 241, 243, 244, 248, 253, 259, 264, 268, 272, 273, 275, 281, 282

Offset: 1

N. J. A. Sloane, Apr 05 2023

			From _M. F. Hasler_, Apr 08 2023: (Start)
From 112 we can get 1*12 = 12 and 11*2 = 22, then 1*2 = 2 and 2*2 = 4.
All smaller numbers have only one possible outcome: for n = 111 the only possible outcome is 1, for 99 < n < 111, the outcome is always 0, and for 2-digit numbers there is only one possibility for the split-and-multiply operation and the result is always smaller than the initial value. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..7920

Cf. A361338-A361349.

-1 + Position[#, 2][[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_A361341(n)=A361338(n)==2}, [0..300]) \\ M. F. Hasler, Apr 08 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

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A361349 Numbers k such that A361338(k) = 10.

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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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A361341 Numbers k such that A361338(k) = 2.

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

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