A370972 -id:A370972 - cp's OEIS frontend

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

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

A370970 Numbers k which have a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.

Original entry on oeis.org

8596, 8790, 9360, 9380, 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

N. J. A. Sloane, Apr 13 2024, following emails from Ed Pegg Jr and Hans Havermann. The terms were computed by Hans Havermann

The total number of digits in k, f1, ..., fr is ten, and they are all distinct.

			The complete list of terms:
 8596 = 2*14*307
 8790 = 2*3*1465
 9360 = 2*4*15*78
 9380 = 2*5*14*67
 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 = 27*594 = 54*297
16704 = 9*32*58
17082 = 3*5694
17820 = 36*495 = 45*396
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

Hans Havermann, Pandigital Products, Apr 13 2024

Cf. A370972, A372106.

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.

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.

cp's OEIS Frontend

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

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A370970 Numbers k which have a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.

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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.

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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.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

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Python

A370970 Numbers k which have a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.

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Author

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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

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

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Comments

Examples

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A370970 Numbers k which have a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.

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.