A352172 -id:A352172 - cp's OEIS frontend

A352260 Integers that need 2 iterations of the map x->A352172(x) to reach 1.

25, 52, 125, 152, 205, 215, 250, 251, 455, 502, 512, 520, 521, 545, 554, 1025, 1052, 1125, 1152, 1205, 1215, 1250, 1251, 1455, 1502, 1512, 1520, 1521, 1545, 1554, 2005, 2015, 2050, 2051, 2105, 2115, 2150, 2151, 2255, 2500, 2501, 2510, 2511, 2525, 2552, 4055, 4155, 4505

Offset: 1

Michel Marcus, Mar 10 2022

			25 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352261, A352262, A352263, A352264, A352265, A352266, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[4505], q[#, 2] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok2(n) = {for (k=1, 2, n = f(n); if ((n==1), return(k==2)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x):
    x = A352172(x)
    return x != 1 and A352172(x) == 1
print([k for k in range(4506) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352261 Integers that need 3 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

5, 8, 15, 18, 24, 42, 50, 51, 80, 81, 105, 108, 115, 118, 124, 142, 150, 151, 180, 181, 204, 214, 222, 240, 241, 255, 258, 285, 402, 412, 420, 421, 445, 454, 500, 501, 510, 511, 525, 528, 544, 552, 582, 800, 801, 810, 811, 825, 852, 1005, 1008, 1015, 1018, 1024, 1042, 1050

Offset: 1

Michel Marcus, Mar 10 2022

			5 -> 125 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352262, A352263, A352264, A352265, A352266, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[1050], q[#, 3] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok3(n) = {for (k=1, 3, n = f(n); if ((n==1), return(k==3)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=3):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(1051) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352262 Integers that need 4 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

2, 12, 20, 21, 45, 54, 102, 112, 120, 121, 145, 154, 200, 201, 210, 211, 225, 252, 405, 415, 450, 451, 504, 514, 522, 540, 541, 558, 585, 855, 1002, 1012, 1020, 1021, 1045, 1054, 1102, 1112, 1120, 1121, 1145, 1154, 1200, 1201, 1210, 1211, 1225, 1252, 1405, 1415, 1450, 1451

Offset: 1

Michel Marcus, Mar 10 2022

			2 -> 8 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352263, A352264, A352265, A352266, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[1451], q[#, 4] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok4(n) = {for (k=1, 4, n = f(n); if ((n==1), return(k==4)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=4):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(1452) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352263 Integers that need 5 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

679, 697, 769, 796, 967, 976, 1679, 1697, 1769, 1796, 1967, 1976, 2379, 2397, 2739, 2793, 2937, 2973, 3279, 3297, 3367, 3376, 3637, 3673, 3729, 3736, 3763, 3792, 3927, 3972, 6079, 6097, 6179, 6197, 6337, 6373, 6709, 6719, 6733, 6790, 6791, 6907, 6917, 6970, 6971, 7069, 7096

Offset: 1

Michel Marcus, Mar 10 2022

			679 -> 54010152 -> 8000000 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352262, A352264, A352265, A352266, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[7096], q[#, 5] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok5(n) = {for (k=1, 5, n = f(n); if ((n==1), return(k==5)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=5):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(7100) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352264 Integers that need 6 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

377, 737, 773, 1377, 1737, 1773, 3077, 3177, 3707, 3717, 3770, 3771, 3889, 3898, 3988, 4689, 4698, 4869, 4896, 4968, 4986, 5677, 5767, 5776, 6489, 6498, 6577, 6668, 6686, 6757, 6775, 6849, 6866, 6894, 6948, 6984, 7037, 7073, 7137, 7173, 7307, 7317, 7370, 7371, 7567, 7576

Offset: 1

Michel Marcus, Mar 10 2022

			377 -> 3176523 -> 54010152000 -> 8000000 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352262, A352263, A352265, A352266, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[7576], q[#, 6] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok6(n) = {for (k=1, 6, n = f(n); if ((n==1), return(k==6)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=6):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(7577) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352265 Integers that need 7 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

478, 487, 748, 784, 847, 874, 1478, 1487, 1748, 1784, 1847, 1874, 2278, 2287, 2447, 2474, 2728, 2744, 2782, 2827, 2872, 4078, 4087, 4178, 4187, 4247, 4274, 4427, 4472, 4708, 4718, 4724, 4742, 4780, 4781, 4807, 4817, 4870, 4871, 5788, 5878, 5887, 7048, 7084, 7148, 7184, 7228

Offset: 1

Michel Marcus, Mar 10 2022

			478 -> 11239424 -> 5159780352 -> 54010152000000000 -> 8000000 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352262, A352263, A352264, A352266, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[7228], q[#, 7] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok7(n) = {for (k=1, 7, n = f(n); if ((n==1), return(k==7)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=7):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(7229) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352266 Integers that need 8 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

27, 57, 72, 75, 127, 157, 172, 175, 207, 217, 270, 271, 355, 457, 475, 507, 517, 535, 547, 553, 570, 571, 574, 702, 705, 712, 715, 720, 721, 745, 750, 751, 754, 1027, 1057, 1072, 1075, 1127, 1157, 1172, 1175, 1207, 1217, 1270, 1271, 1355, 1457, 1475, 1507, 1517, 1535, 1547

Offset: 1

Michel Marcus, Mar 10 2022

			27 -> 2744 -> 11239424 -> 5159780352 -> 54010152000000000 -> 8000000 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352262, A352263, A352264, A352265, A352267, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[1547], q[#, 8] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok8(n) = {for (k=1, 8, n = f(n); if ((n==1), return(k==8)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=8):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(1548) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352267 Integers that need 9 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

3, 13, 30, 31, 56, 65, 103, 113, 130, 131, 156, 165, 235, 253, 300, 301, 310, 311, 325, 352, 506, 516, 523, 532, 560, 561, 605, 615, 650, 651, 1003, 1013, 1030, 1031, 1056, 1065, 1103, 1113, 1130, 1131, 1156, 1165, 1235, 1253, 1300, 1301, 1310, 1311, 1325, 1352, 1506, 1516

Offset: 1

Michel Marcus, Mar 10 2022

			3 -> 27 -> 2744 -> 11239424 -> 5159780352 -> 54010152000000000 -> 8000000 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352262, A352263, A352264, A352265, A352266, A352268.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[1516], q[#, 9] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok9(n) = {for (k=1, 9, n = f(n); if ((n==1), return(k==9)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=9):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(1517) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A352268 Integers that need 10 iterations of the map x->A352172(x) to reach 1.

Original entry on oeis.org

55, 155, 505, 515, 550, 551, 1055, 1155, 1505, 1515, 1550, 1551, 2555, 5005, 5015, 5050, 5051, 5105, 5115, 5150, 5151, 5255, 5500, 5501, 5510, 5511, 5525, 5552, 10055, 10155, 10505, 10515, 10550, 10551, 11055, 11155, 11505, 11515, 11550, 11551, 12555, 15005, 15015, 15050

Offset: 1

Michel Marcus, Mar 10 2022

			55 -> 15625 -> 27000000 -> 2744 -> 11239424 -> 5159780352 -> 54010152000000000 -> 8000000 -> 512 -> 1000 -> 1.

Cf. A352172. Subsequence of A351876.

Cf. A352260, A352261, A352262, A352263, A352264, A352265, A352266, A352267.

f[n_] := (Times @@ Select[IntegerDigits[n], # > 1 &])^3; q[n_, len_] := (v = Nest[f, n, len - 1]) != 1 && f[v] == 1; Select[Range[15050], q[#, 10] &] (* Amiram Eldar, Mar 10 2022 *)

f(n) = vecprod(apply(x->x^3, select(x->(x>1), digits(n)))); \\ A352172
isok10(n) = {for (k=1, 10, n = f(n); if ((n==1), return(k==10)););}

from math import prod
def A352172(n): return prod(int(d)**3 for d in str(n) if d != '0')
def ok(x, iters=10):
    i = 0
    while i < iters and x != 1: i, x = i+1, A352172(x)
    return i == iters and x == 1
print([k for k in range(15051) if ok(k)]) # Michael S. Branicky, Mar 10 2022

A351876 Numbers whose trajectory under iteration of the product of cubes of nonzero digits map includes 1 (conjectured).

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 11, 12, 13, 15, 18, 20, 21, 24, 25, 27, 30, 31, 42, 45, 50, 51, 52, 54, 55, 56, 57, 65, 72, 75, 80, 81, 100, 101, 102, 103, 105, 108, 110, 111, 112, 113, 115, 118, 120, 121, 124, 125, 127, 130, 131, 142, 145, 150, 151, 152, 154, 155, 156, 157, 165, 172, 175, 180, 181

Offset: 1

Luca Onnis, Feb 23 2022

To determine whether a given number k is a term of this sequence, start with k, take the cube of the product of its nonzero digits, apply the same process to the result, and continue until 30 iterations are reached. If 1 is reached during the process, k is a term of this sequence. If not, k is not a term of this sequence.
Every power 10^k is a term of this sequence.
If k is a term, the numbers obtained by inserting zeros anywhere in k are terms.
If k is a term, the numbers obtained by inserting ones anywhere in k are terms.
If k is a term, each distinct permutation of the digits of k gives another term.
If k is a term, the number of iterations required to converge to 1 is less than or equal to 10 (conjectured).

			217 is a term of the sequence; its trajectory is 217 -> 2744 -> 11239424 -> 5159780352 -> 54010152000000000 -> 8000000 -> 512 -> 1000 -> 1.
4 is not a term of the sequence; its trajectory begins with 4 -> 64 -> 13824 -> 7077888 -> 5416169448144896 -> 188436971148778297205194752000 -> 1545896640285238037724131582088286996267008000000 -> ... Subsequent terms in the trajectory get larger and larger, rather than reaching 1. However, it is not yet known whether it eventually reaches 1 after some number of iterations > 30.

Cf. A351327, A051801.

Cf. A352172 (product of cubes of nonzero digits).

Select[Range[1000], FixedPoint[ Product[ReplaceAll[0 -> 1][IntegerDigits[#]][[i]]^3, {i, 1, Length[ReplaceAll[0 -> 1][IntegerDigits[#]]]}] &, #, 12] == 1 &]

cp's OEIS Frontend

A352260 Integers that need 2 iterations of the map x->A352172(x) to reach 1.

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A352261 Integers that need 3 iterations of the map x->A352172(x) to reach 1.

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A352262 Integers that need 4 iterations of the map x->A352172(x) to reach 1.

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A352263 Integers that need 5 iterations of the map x->A352172(x) to reach 1.

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A352264 Integers that need 6 iterations of the map x->A352172(x) to reach 1.

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A352265 Integers that need 7 iterations of the map x->A352172(x) to reach 1.

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A352266 Integers that need 8 iterations of the map x->A352172(x) to reach 1.

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A352267 Integers that need 9 iterations of the map x->A352172(x) to reach 1.

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