A370970 -id:A370970 - cp's OEIS frontend

A372249 The smallest number k which, when written in base n, has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

297, 440, 1440, 8596, 15552, 121638, 282240, 1039104, 5757696, 24108000

Offset: 7

Scott R. Shannon, Apr 24 2024

The offset begins at 7 as in bases 2..6 no term exists.

Assumes all factors f_i > 1. If f_i = 1 is allowed, then a(7) = 104 = 1*4*26, a(10) = 4830 = 1*2*5*7*69, a(12) = 72240 = 1*4*6*7*430, a(15) = 4244940 = 1*2*3*7*9*10*1123, ... - Chai Wah Wu, Apr 24 2024

			The terms and their factorizations are:
a(7) = 297 = [9, 33] = 603_7 = [12_7, 45_7] = "6031245" which contains all digits 0..6 once.
a(8) = 440 = [2, 4, 5, 11] = 670_8 = [2_8, 4_8, 5_8, 13_8] = "67024513" which contains all digits 0..7 once.
a(9) = 1440 = [3, 4, 5, 24] = 1870_9 = [3_9, 4_9, 5_9, 26_9] = "187034526" which contains all digits 0..8 once.
a(10) = 8596 = [2, 14, 307] = "8596214307" which contains all digits 0..9 once. See also A370970.
a(11) = 15552 = [2, 3, 6, 8, 54] = 10759_11 = [2_11, 3_11, 6_11, 8_11, 4a_11] = "1075923684a" which contains all digits 0..a once.
a(12) = 121638 = [2, 3, 11, 1843] = 5a486_12 = [2_12, 3_12, b_12, 1097_12] = "5a48623b1097" which contains all digits 0..b once.
a(13) = 282240 = [2, 3, 5, 7, 21, 64] = 9b60a_13 = [2_13, 3_13, 5_13, 7_13, 18_13, 4c_13] = "9b60a2357184c" which contains all digits 0..c once.
a(14) = 1039104 = [2, 3, 4, 6, 8, 11, 82] = 1d097a_14 = [2_14, 3_14, 4_14, 6_14, 8_14, b_14, 5c_14] = "1d097a23468b5c" which contains all digits 0..d once.
From _Chai Wah Wu_, Apr 24 2024: (Start)
a(15) = 5757696 = [2, 3, 4, 12, 84, 238] = 78aeb6_15 = [2_15, 3_15, 4_15, c_15, 59_15, 10d_15] = "78aeb6234c5910d" which contains all digits 0..e once.
a(16) = 24108000 = [3, 4, 5, 7, 10, 41, 140] = 16fdbe0_16 = [3_16, 4_16, 5_16, 7_16, a_16, 29_16, 8c_16] = "16fdbe03457a298c" which contains all digits 0..f once. (End)

Cf. A372053, A370970, A371958.

a(15)-a(16) from Chai Wah Wu, Apr 24 2024

Added escape clause to definition at the suggestion of Chai Wah Wu. - N. J. A. Sloane, Apr 25 2024

A371958 The smallest number k that has a factorization k = f1f2...*fr where the numbers k, f1, f2, ..., fr together contain every number from 0 to n, without overlap, as substrings.

Original entry on oeis.org

10, 10, 30, 102, 120, 240, 1260, 1680, 8596, 34580, 113760, 576840, 3579840, 14938560, 109133640

Offset: 1

Scott R. Shannon, Apr 14 2024

In the first thirteen terms the 'perfect' solutions (ones without any excess digits) are for n = 6, 9, 10, 11, 12. It is likely such solutions become very rare as n increases.

			a(1) = 10 as 10 = 2 * 5, and {10, 2, 5} contains the numbers 0 and 1 as non-overlapping substrings, and no smaller number has a similar factorization.
a(2) = 10 as {10, 2, 5} also contains the number 0, 1, and 2.
a(3) = 30 as 30 = 2 * 15, and {30, 2, 15} contains 0,..,3.
a(4) = 102 as 102 = 3 * 34, which contains 0,..,4.
a(5) = 120 as 120 = 2 * 3 * 4 * 5, which contains 0,..,5.
a(6) = 240 as 240 = 3 * 5 * 16, which contains 0,..,6. The first perfect solution.
a(7) = 1260 as 1260 = 3 * 3 * 4 * 5 * 7, which contains 0,..,7.
a(8) = 1680 as 1680 = 2 * 2 * 3 * 4 * 5 * 7, which contains 0,..,8.
a(9) = 8596 as 8596 = 2 * 14 * 307, which contains 0,..,9. A perfect solution.
a(10) = 34580 = 7 * 10 * 19 * 26, which contains 0,..,10. A perfect solution. Note that all three of 0, 1, and 10 must appear as separate nonoverlapping substrings.
a(11) = 113760 as 113760 = 2 * 4 * 9 * 10 * 158, which contains 0,..,11. A perfect solution.
a(12) = 576840 as 576840 = 10 * 11 * 12 * 19 * 23, which contains 0,..,12. A perfect solution.
a(13) = 3579840 as 3579840 = 2 * 2 * 6 * 10 * 11 * 12 * 113, which contains 0,..,13.
a(14) = 14938560 as 14938560 = 7 * 10 * 12 * 12 * 13 * 114, which contains 0,...,14. A perfect solution.
a(15) = 109133640 as 109133640 = 2 * 11 * 14 * 18 * 127 * 155, which contains 0,...,15.

Cf. A027746, A001055, A370970,

a(14)-a(15) from David Consiglio, Jr., Apr 25 2024

A370972 Composite numbers with properties that its digits (which may appear with multiplicity) may not appear in any of its factors (wherein the digits may also appear with multiplicity) and the combined digits of the product and the factors must have at least one of each of the ten digits.

Original entry on oeis.org

8596, 8790, 9360, 9380, 9870, 10752, 10764, 10854, 10968, 11760, 12780, 13608, 13860, 14760, 14780, 14820, 15628, 15678, 16038, 16704, 16920, 17082, 17280, 17340, 17640, 17820, 17920, 18090, 18096, 18690, 18720, 18960, 19068, 19084, 19240, 19440, 19460, 19608, 19740, 19780, 19800, 19980, 20457, 20574, 20748, 20754

Offset: 1

N. J. A. Sloane, Apr 15 2024. Terms were computed by Hans Havermann

See A370970 for another version.

Ed Pegg Jr noted that 1476395008 is the smallest term composed of nine distinct digits. See A372106 for subsequent terms. - Hans Havermann, Apr 19 2024

			996880 = 2*2*4*5*17*733: 8 and 9 appear twice each in the product. 2, 3, and 7 appear twice each in the factors. The digits in the product are distinct from the digits in the factors and, ignoring the duplicates, we have a combined 9680245173, one of each of the ten digits. -  _Hans Havermann_, Apr 15 2024

Ed Pegg Jr, Posting to Math-Fun Mailing List, April 2024.

Hans Havermann, Table of a(n) and corresponding factorization(s) for all terms <= 100000.

Cf. A370970, A372106.

A372106 A370972 terms composed of nine distinct digits which may repeat.

Original entry on oeis.org

1476395008, 116508327936, 505627938816, 640532803911, 1207460451879, 1429150367744, 1458956660623, 3292564845031, 3820372951296, 5056734498816, 6784304541696, 8090702381056, 9095331446784, 10757095489536, 10973607685048, 13505488366293, 14913065975808, 38203732951296

Offset: 1

Hans Havermann, Apr 18 2024

Each factorization is necessarily composed of multipliers that use only the single missing digit.
The single missing digit cannot be 0, 1, 5, or 6. Terms missing 2, 3, 4, 7, and 8 appear within a(1)-a(6). 52612606387341 = 9^6 * 99 * 999999 is an example of a term missing 9. - Michael S. Branicky, Apr 18 2024
Some terms are equal to the sum of two distinct smaller terms:
a(741) = a(635) + a(673)
a(1202) = a(1081) + a(1144)
a(1273) = a(1110) + a(1169)
a(1493) = a(1335) + a(1374)
a(2753) = a(2478) + a(2528)
a(2793) = a(2512) + a(2583)
a(3581) = a(3234) + a(3317)
a(4199) = a(3808) + a(3921)
a(4803) = a(4510) + a(4607) = a(4557) + a(4568)
a(5756) = a(5256) + a(5362)
a(6083) = a(5718) + a(5847)
a(7262) = a(6761) + a(6779)
a(7331) = a(6786) + a(6904)
a(9204) = a(8723) + a(8886)
a(9364) = a(8858) + a(8982)
a(9453) = a(8972) + a(8983) - Hans Havermann, Apr 21 2024

			10973607685048 = 22222*22222*22222 is in the sequence because it has nine distinct digits and may be factored using only its missing digit.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1016 from Hans Havermann)
Michael S. Branicky and Hans Havermann, Table of n, a(n) for n = 1..10000, fully factored

Cf. A370970, A370972.

import heapq
from itertools import islice
def agen(): # generator of terms
    allowed = [2, 3, 4, 7, 8, 9]
    v, oldt, h, repunits, bigr = 1, 0, list((d, d) for d in allowed), [1], 1
    while True:
        v, d = heapq.heappop(h)
        if (v, d) != oldt:
            s = set(str(v))
            if len(s) == 9 and str(d) not in s:
                yield v
            oldt = (v, d)
            while v > bigr:
                bigr = 10*bigr + 1
                repunits.append(bigr)
                for c in allowed:
                    heapq.heappush(h, (bigr*c, c))
            for r in repunits:
                heapq.heappush(h, (v*d*r, d))
print(list(islice(agen(), 100))) # Michael S. Branicky, Apr 19 2024

A371993 The smallest number k that has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain all, and only, the digits 0..n, where n<=9.

Original entry on oeis.org

100, 20, 120, 102, 120, 240, 1260, 1680, 8598

Offset: 1

Scott R. Shannon, Apr 15 2024

			The factorizations are:
a(1) = 100 = [10, 10]
a(2) = 20 = [2, 10]
a(3) = 120 = [2, 2, 3, 10]
a(4) = 102 = [3, 34]
a(5) = 120 = [2, 3, 4, 5]
a(6) = 240 = [3, 5, 16]
a(7) = 1260 = [3, 3, 4, 5, 7]
a(8) = 1680 = [2, 2, 3, 4, 5, 7]
a(9) = 8596 = [2, 14, 307]

Cf. A371958, A370970, A370972.

A371982 Numbers k which are the sum k = b+c where the digits of {k, b ,c} together give 0,1,...,9 exactly once.

Original entry on oeis.org

1026, 1035, 1053, 1062, 1089, 1098, 1206, 1305, 1503, 1602, 2034, 2043, 3015, 3051, 5013, 6012, 6021

Offset: 1

Jean-Marc Rebert, Apr 15 2024

			The complete list of terms:
1026 = 437+589 = 439+587 = 487+539 = 489+537
1035 = 246+789 = 249+786 = 286+749 = 289+746
1053 = 264+789 = 269+784 = 284+769 = 289+764
1062 = 473+589 = 479+583 = 483+579 = 489+573
1089 = 324+765 = 325+764 = 364+725 = 365+724 = 432+657 = 437+652 = 452+637 = 457+632
1098 = 342+756 = 346+752 = 352+746 = 356+742 = 423+675 = 425+673 = 473+625 = 475+623
1206 = 347+859 = 349+857 = 357+849 = 359+847
1305 = 426+879 = 429+876 = 476+829 = 479+826
1503 = 624+879 = 629+874 = 674+829 = 679+824
1602 = 743+859 = 749+853 = 753+849 = 759+843
2034 = 56+1978 = 58+1976 = 76+1958 = 78+1956
2043 = 56+1987 = 57+1986 = 65+1978 = 68+1975 = 75+1968 = 78+1965 = 86+1957 = 87+1956
3015 = 47+2968 = 48+2967 = 67+2948 = 68+2947
3051 = 64+2987 = 67+2984 = 84+2967 = 87+2964
5013 = 26+4987 = 27+4986 = 86+4927 = 87+4926
6012 = 34+5978 = 38+5974 = 74+5938 = 78+5934
6021 = 34+5987 = 37+5984 = 43+5978 = 48+5973 = 73+5948 = 78+5943 = 84+5937 = 87+5934

Cf. A370970.

A372259 Numbers k which have a factorization k = f_1f_2...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together give 0,1,...,9 exactly once.

Original entry on oeis.org

4830, 6970, 7056, 7096, 7290, 7690, 7830, 8370, 8596, 8652, 8790, 8970, 9076, 9360, 9370, 9380, 9670, 9706, 9720, 9730, 9870, 10752, 12780, 14760, 14820, 15628, 15678, 16038, 16704, 17082, 17820, 17920, 18720, 19084, 19240, 20457, 20574, 20754, 21658, 24056, 24507, 25803, 26180, 26910, 27504, 28156, 28651, 30296, 30576, 30752, 31920, 32760, 32890, 34902, 36508, 47320, 58401, 65128, 65821

Offset: 1

Chai Wah Wu, Apr 24 2024

A370970 is a subsequence. In contrast to A370970, here the factors f_i are allowed to be equal to 1.

			The complete list of terms:
  4830 = 1*2*5*7*69
  6970 = 1*2*3485
  7056 = 1*3*24*98 = 1*3*8*294
  7096 = 1*2*3548
  7290 = 1*3*5*486
  7690 = 1*2*3845
  7830 = 1*6*29*45
  8370 = 1*2*9*465
  8596 = 2*14*307
  8652 = 1*4*7*309
  8790 = 2*3*1465
  8970 = 1*26*345
  9076 = 1*2*4538
  9360 = 1*5*24*78 = 2*4*15*78
  9370 = 1*2*4685
  9380 = 2*5*14*67
  9670 = 1*2*4835
  9706 = 1*2*4853
  9720 = 1*3*5*648
  9730 = 1*2*4865
  9870 = 2*3*1645
 10752 = 3*4*896
 12780 = 4*5*639
 14760 = 5*9*328
 14820 = 5*39*76
 15628 = 4*3907
 15678 = 39*402
 16038 = 54*297 = 27*594
 16704 = 9*32*58
 17082 = 3*5694
 17820 = 45*396 = 36*495
 17920 = 8*35*64
 18720 = 4*5*936
 19084 = 52*367
 19240 = 8*37*65
 20457 = 3*6819
 20574 = 6*9*381
 20754 = 3*6918
 21658 = 7*3094
 24056 = 8*31*97
 24507 = 3*8169
 25803 = 9*47*61
 26180 = 4*7*935
 26910 = 78*345
 27504 = 3*9168
 28156 = 4*7039
 28651 = 7*4093
 30296 = 7*8*541
 30576 = 8*42*91
 30752 = 4*8*961
 31920 = 5*76*84
 32760 = 8*45*91
 32890 = 46*715
 34902 = 6*5817
 36508 = 4*9127
 47320 = 8*65*91
 58401 = 63*927
 65128 = 7*9304
 65821 = 7*9403

Cf. A370970, A370972, A372106.

A372294 The smallest number k which, when written in base n, has a factorization k = f_1f_2...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

Original entry on oeis.org

-1, -1, -1, -1, -1, 104, 440, 1440, 4830, 15552, 72240, 282240, 1039104, 4244940, 24108000

Offset: 2

Chai Wah Wu, Apr 25 2024

Similar to A372249, except that here the factors are allowed to be equal to 1. Differs from A372249 at n = 7, 10, 12, 15, ...

			a(7)  =      104 = 1*4*26
a(8)  =      440 = 2*4*5*11
a(9)  =     1440 = 3*4*5*24
a(10) =     4830 = 1*2*5*7*69
a(11) =    15552 = 2*3*6*8*54
a(12) =    72240 = 1*4*6*7*430
a(13) =   282240 = 2*3*5*7*21*64
a(14) =  1039104 = 2*3*4*6*8*11*82
a(15) =  4244940 = 1*2*3*7*9*10*1123
a(16) = 24108000 = 3*4*5*7*10*41*140

Cf. A372249, A372053, A370970, A371958.

a(n) <= A372249(n).

cp's OEIS Frontend

A372249 The smallest number k which, when written in base n, has a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A371958 The smallest number k that has a factorization k = f1*f2*...*fr where the numbers k, f1, f2, ..., fr together contain every number from 0 to n, without overlap, as substrings.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A370972 Composite numbers with properties that its digits (which may appear with multiplicity) may not appear in any of its factors (wherein the digits may also appear with multiplicity) and the combined digits of the product and the factors must have at least one of each of the ten digits.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A372106 A370972 terms composed of nine distinct digits which may repeat.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A371993 The smallest number k that has a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together contain all, and only, the digits 0..n, where n<=9.

Views

Author

Keywords

Examples

Crossrefs

A371982 Numbers k which are the sum k = b+c where the digits of {k, b ,c} together give 0,1,...,9 exactly once.

Views

Author

Keywords

Examples

Crossrefs

A372259 Numbers k which have a factorization k = f_1*f_2*...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together give 0,1,...,9 exactly once.

Views

Author

Keywords

Comments

Examples

Crossrefs

A372294 The smallest number k which, when written in base n, has a factorization k = f_1*f_2*...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A372249 The smallest number k which, when written in base n, has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

A371958 The smallest number k that has a factorization k = f1f2...*fr where the numbers k, f1, f2, ..., fr together contain every number from 0 to n, without overlap, as substrings.

A371993 The smallest number k that has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain all, and only, the digits 0..n, where n<=9.

A372259 Numbers k which have a factorization k = f_1f_2...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together give 0,1,...,9 exactly once.

A372294 The smallest number k which, when written in base n, has a factorization k = f_1f_2...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.