A196032 -id:A196032 - cp's OEIS frontend

A337250 Numbers having at least one 3 in their representation in base 4.

3, 7, 11, 12, 13, 14, 15, 19, 23, 27, 28, 29, 30, 31, 35, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 75, 76, 77, 78, 79, 83, 87, 91, 92, 93, 94, 95, 99, 103, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 1

François Marques, Sep 19 2020

Complementary sequence of A023717.

			18 is not in the sequence since it is 102_4 in base 4, but 19 is in the sequence since it is 103_4 in base 4.

François Marques, Table of n, a(n) for n = 1..10000

Cf. A196032 (at least one 0 in base 4).
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), this sequence, A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

seq(`if`(numboccur(3, convert(n, base, 4))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 4 ], 3 ]>0)& ]

isok(m) = #select(x->(x==3), digits(m, 4)) >= 1; \\ Michel Marcus, Sep 20 2020

from gmpy2 import digits
def A337250(n):
    def f(x):
        l = (s:=digits(x,4)).find('3')
        if l >= 0: s = s[:l]+'2'*(len(s)-l)
        return n+int(s,3)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A382413 Numbers with at least one zero in their base-7 representation.

Original entry on oeis.org

0, 7, 14, 21, 28, 35, 42, 49, 50, 51, 52, 53, 54, 55, 56, 63, 70, 77, 84, 91, 98, 99, 100, 101, 102, 103, 104, 105, 112, 119, 126, 133, 140, 147, 148, 149, 150, 151, 152, 153, 154, 161, 168, 175, 182, 189, 196, 197, 198, 199, 200, 201, 202, 203, 210, 217, 224, 231, 238

Offset: 1

Paolo Xausa, Mar 24 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007093, A043393, A382412 (complement).

Select[Range[0, 250], DigitCount[#, 7, 0] > 0 &]

A382415 Numbers with at least one zero in their base-5 representation.

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 26, 27, 28, 29, 30, 35, 40, 45, 50, 51, 52, 53, 54, 55, 60, 65, 70, 75, 76, 77, 78, 79, 80, 85, 90, 95, 100, 101, 102, 103, 104, 105, 110, 115, 120, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382416 (base 6), A382413 (base 7), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007091, A023721 (complement), A023722.

Select[Range[0, 150], DigitCount[#, 5, 0] > 0 &]

A382416 Numbers with at least one zero in their base-6 representation.

Original entry on oeis.org

0, 6, 12, 18, 24, 30, 36, 37, 38, 39, 40, 41, 42, 48, 54, 60, 66, 72, 73, 74, 75, 76, 77, 78, 84, 90, 96, 102, 108, 109, 110, 111, 112, 113, 114, 120, 126, 132, 138, 144, 145, 146, 147, 148, 149, 150, 156, 162, 168, 174, 180, 181, 182, 183, 184, 185, 186, 192, 198

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382413 (base 7), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007092, A043369, A248910 (complement).

Select[Range[0, 200], DigitCount[#, 6, 0] > 0 &]

A382417 Numbers with at least one zero in their base-8 representation.

Original entry on oeis.org

0, 8, 16, 24, 32, 40, 48, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 80, 88, 96, 104, 112, 120, 128, 129, 130, 131, 132, 133, 134, 135, 136, 144, 152, 160, 168, 176, 184, 192, 193, 194, 195, 196, 197, 198, 199, 200, 208, 216, 224, 232, 240, 248, 256, 257, 258, 259, 260

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382413 (base 7), A382418 (base 9), A011540 (base 10).

Cf. A007094, A043421, A255805 (complement).

Select[Range[0, 300], DigitCount[#, 8, 0] > 0 &]

A382418 Numbers with at least one zero in their base-9 representation.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 99, 108, 117, 126, 135, 144, 153, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 180, 189, 198, 207, 216, 225, 234, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 261, 270, 279, 288, 297

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382413 (base 7), A382417 (base 8), A011540 (base 10).

Cf. A007095, A043453, A255808 (complement).

Select[Range[0, 300], DigitCount[#, 9, 0] > 0 &]

A259568 Numbers following gaps in the sequence of base-4 numbers that don't contain 0.

Original entry on oeis.org

1, 5, 9, 13, 21, 25, 29, 37, 41, 45, 53, 57, 61, 85, 89, 93, 101, 105, 109, 117, 121, 125, 149, 153, 157, 165, 169, 173, 181, 185, 189, 213, 217, 221, 229, 233, 237, 245, 249, 253, 341, 345, 349, 357, 361, 365, 373, 377, 381, 405, 409, 413, 421, 425, 429, 437, 441, 445, 469, 473, 477, 485, 489, 493, 501, 505, 509, 597, 601, 605

Offset: 1

Sean Oneil, Jun 30 2015

Partial sums for the convergent modified harmonic series in base 4 excluding 0 = Sum of 1/a(n) + 1/(a(n) + 1) + 1/(a(n) + 2) = Sum of (3*a(n)^2 + 6*a(n) + 2)/(a(n)*(a(n) + 1)*(a(n) + 2)).

			Pattern of numbers of skipped terms (numbers in base 4 with at least one zero) is 1 (4 = 10_4), 1 (8 = 20_4), 1 (12 = 30_4), 4+1 (16 = 100_4, 17 = 101_4, 18 = 102_4, 19 = 103_4, 20 = 110_4), 1, 1, 4+1, 1, 1, 4+1, 1, 1, 16+4+1, ...

Robert Baillie, Sums of Reciprocals of Integers Missing a Given Digit, American Mathematical Monthly (Washington, DC: Mathematical Association of America) 86 (5): 372-374, May 1979. doi:10.2307/2321096. ISSN 0002-9890. JSTOR 2321096.

Subset of A016813 (congruent to 1 mod 4). a(n) = A023705(3n - 2). Each term is one more than the numbers that follow gaps in A196032.

lista(nn)=prec0 = 1; for(n=1, nn, if (vecmin(digits(n, 4)), if (prec0, print1(n,, ", ")); prec0 = 0, prec0 = 1);); \\ Michel Marcus, Aug 03 2015

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A337250 Numbers having at least one 3 in their representation in base 4.

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A382413 Numbers with at least one zero in their base-7 representation.

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A382415 Numbers with at least one zero in their base-5 representation.

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A382416 Numbers with at least one zero in their base-6 representation.

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A382417 Numbers with at least one zero in their base-8 representation.

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A382418 Numbers with at least one zero in their base-9 representation.

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A259568 Numbers following gaps in the sequence of base-4 numbers that don't contain 0.

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