A062846 -id:A062846 - cp's OEIS frontend

A062934 Numbers k that, when expressed in base 6 and then interpreted in base 7, give a multiple of k.

0, 1, 2, 3, 4, 5, 14027, 14028, 14103, 14112, 14867, 14876, 14951, 14952, 10099698, 10099846, 20210795, 30310661, 30311795, 50522741, 723825175, 142569349356, 482364801576, 486288289536, 972577541899, 1945641402768, 2474129673299

Offset: 1

Erich Friedman, Jul 21 2001

1.6 * 10^13 < a(28) <= 24335728984305. - Delbert L. Johnson, May 16 2024

			14027 in base 6 is 144535, which interpreted in base 7 is 28054 = 2*14027.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062929, A062930, A062931, A062937, A062939, A062942, A062943, A062944.

More terms from Naohiro Nomoto, Aug 06 2001
Offset changed to 1 and a(17)-a(20) from Georg Fischer, Mar 13 2023
a(21)-a(27) from Delbert L. Johnson, May 16 2024

A062937 Numbers k that, when expressed in base 6 and then interpreted in base 8, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1459, 1564, 3023, 3129, 3139, 3244, 4703, 4704, 4809, 6383, 6384, 8754, 9384, 18138, 18774, 18834, 19464, 28218, 28224, 28854, 38298, 38304, 79802, 326236, 2463293, 2628864, 14779758, 15773184, 22011172, 88678548, 94639104, 209918592

Offset: 1

Erich Friedman, Jul 21 2001

			1459 in base 6 is 10431, which interpreted in base 8 is 4377=3*1459.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062929, A062930, A062931, A062934, A062939, A062942, A062943, A062944.

Join[{0},Select[Range[21*10^7],Mod[FromDigits[IntegerDigits[#,6],8],#]==0&]] (* Harvey P. Dale, Apr 21 2024 *)

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 and a(33)-a(38) from Georg Fischer, Mar 13 2023

A062939 Numbers k that, when expressed in base 6 and then interpreted in base 9, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 51, 54, 105, 306, 324, 630, 2646, 6711, 8998, 9003, 19847, 29513, 30127, 30132, 67662, 71267, 314751, 314928, 405972, 427602, 1009394, 1347704, 1888506, 1889568, 2321838, 2840097, 5383299, 6056364, 7143622, 8086224, 11331036, 11337408, 14382561

Offset: 1

Erich Friedman, Jul 21 2001

			51 in base 6 is 123, which interpreted in base 9 is 102 = 2*51.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062929, A062930, A062931, A062934, A062937, A062942, A062943, A062944.

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 and a(33)-a(39) from Georg Fischer, Mar 13 2023

A062942 Numbers k that, when expressed in base 6 and then interpreted in base 10, give a multiple of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 308, 4920, 11284, 11914, 144393, 195453, 518659, 866358, 925148, 1010765, 1172718, 1369865, 2141968, 2557924, 4287428, 4296908, 6064590, 8219190, 15347544, 16891738, 18409156, 18532263, 21880744, 23693054, 25724568, 25781448, 88115915, 93066844

Offset: 1

Erich Friedman, Jul 21 2001

Zero followed by A032546. [From R. J. Mathar, Oct 02 2008]

			308 in base 6 is 1232, which interpreted in base 10 is 1232 = 4*308.

Cf. A062845, A062846, A062847, A062848, A062849, A062850, A062853, A062864, A062865, A062884, A062889, A062891, A062920, A062921, A062922, A062923, A062925, A062928, A062929, A062930, A062931, A062934, A062937, A062939, A062943, A062944.

More terms from Naohiro Nomoto, Aug 06 2001

Offset changed to 1 and a(29)-a(34) from Georg Fischer, Mar 13 2023

A155078 The representation of n=1,2,3,... in binary is a divisor of the same representation in another base. The sequence is the first such base.

Original entry on oeis.org

3, 4, 5, 4, 3, 3, 4, 4, 5, 3, 13, 3, 15, 4, 8, 4, 8, 5, 21, 8, 4, 22, 25, 6, 27, 26, 5, 4, 31, 3, 4, 4, 8, 8, 32, 3, 39, 38, 38, 8, 43, 4, 45, 22, 8, 23, 19, 6, 9, 50, 8, 26, 24, 5, 46, 4, 5, 29, 18, 3, 63, 4, 5, 4, 7, 6, 69, 8, 25, 30, 47, 6, 4, 74, 17, 38, 79, 12, 60, 8, 79, 82, 85, 4, 8, 43

Offset: 1

J. M. Bergot, Jan 19 2009

The pattern of solutions for each binary representation is notable. For 1001= decimal 9, the bases as solutions are 5,8,11,14,... whereas the pattern for 111=decimal 7 is 4,9,11,16,18,....
The binary representation of n corresponds to the unique polynomial p_n(x) with coefficients in {0,1} such that p(2) = n. a(n) is the least x >= 3 such that p_n(x) == 0 mod n. Thus 3 <= a(n) <= n + 2. - Robert Israel, Dec 15 2014
From Rémy Sigrist, Mar 15 2017: (Start)
If n is even then a(n) <= max(4, n).
If n is odd then a(n) <= n + 2.
If n is odd then n and a(n) are coprime.
If a(n)=4 then n belongs to A062846.
(End)

			The n-th term is solved by converting the decimal n to binary then asking to what other base is this representation a multiple of n. For the 5th term, the binary representation is 101; if this is converted to base 3, 101 = 9+0+1 = 10, a multiple of 5. The base 3 is the first base producing a multiple of n: the 5th term is therefore 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062846.

A155078 := proc(n) local bdgs,b ; bdgs := convert(n,base,2) ; for b from 3 do add(op(i,bdgs)*b^(i-1),i=1..nops(bdgs)) ; if mod n = 0 then RETURN(b); fi; od: end: seq(A155078(n),n=1..100) ; # R. J. Mathar, Mar 14 2009
# second Maple program:
f:= proc(n) local b, L,r, sols;
     L:= convert(n,base,2);
     r:= add(L[i]*b^(i-1),i=1..nops(L));
     sols:= subs(0=n,1=n+1,2=n+2,map(t -> rhs(op(t)),{msolve(r,n)})) ;
     min(sols);
end proc:
3, seq(f(n),n=2..100); # Robert Israel, Dec 15 2014

It is simply a matter of converting a binary number to another base to see if the resulting number is a multiple of n. The lowest other base is listed.

Corrected and extended by R. J. Mathar, Mar 14 2009

A283751 Least b>1 such that n, when expressed in base b and then interpreted in base b^2, yields a multiple of n.

Original entry on oeis.org

2, 2, 2, 3, 2, 5, 3, 2, 2, 3, 5, 11, 4, 3, 2, 5, 2, 17, 3, 19, 5, 2, 11, 23, 4, 5, 3, 3, 2, 29, 6, 2, 2, 11, 3, 7, 6, 37, 19, 3, 8, 41, 2, 6, 5, 9, 23, 3, 4, 7, 5, 17, 13, 53, 3, 11, 2, 7, 29, 59, 6, 61, 2, 4, 2, 13, 11, 67, 17, 23, 10, 71, 6, 2, 37, 5, 19, 11

Offset: 0

Rémy Sigrist, Mar 15 2017

a(n) <= max(2, n) for any n >= 0.
a(n*a(n)) <= a(n) for any n >= 0 (see also A283927).
a(n) = 2 iff n belongs to A062846.
Conjecture: if n is composite, then a(n) < n (see also A283937).
Theorem: If n is composite, then a(n) < n. Proof: If n=ab with 1Michael R Peake, Mar 25 2017
First occurrence of b > 1: 1, 4, 13, 6, 31, 36, 41, 46, 71, 12, 133, 53, 155, 106, 161, 18, 199, 20, 261, ..., . - Robert G. Wilson v, Mar 19 2017

			The number 5, when expressed in base 2 and then interpreted in base 4, yields 17, not a multiple of 5.
The number 5, when expressed in base 3 and then interpreted in base 9, yields 11, not a multiple of 5.
The number 5, when expressed in base 4 and then interpreted in base 16, yields 17, not a multiple of 5.
The number 5, when expressed in base 5 and then interpreted in base 25, yields 25, a multiple of 5.
Hence, a(5)=5.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A062846, A155078, A283927, A283937.

f[n_] := Block[{b = 2}, While[ Mod[ FromDigits[ IntegerDigits[n, b], b^2], n] > 0, b++]; b]; Array[f, 80, 0] (* Robert G. Wilson v, Mar 19 2017 *)

a(n) = my (b=2); if (n>0, while (fromdigits(digits(n,b), b^2)%n, b++)); return (b)

A343550 Numbers k > 9 such that the number m formed by inserting a digit 0 between each pair of digits in k is divisible by k.

Original entry on oeis.org

10, 15, 18, 20, 30, 40, 45, 50, 60, 70, 80, 90, 100, 111, 120, 126, 150, 180, 200, 222, 240, 250, 285, 300, 333, 360, 400, 444, 450, 480, 500, 555, 600, 666, 700, 750, 777, 800, 888, 900, 999, 1000, 1041, 1110, 1185, 1200, 1260, 1395, 1443, 1500, 1554, 1665

Offset: 1

Lars Blomberg, Apr 19 2021

One-digit terms are not considered since no 0 digits can be inserted.
If k is a term then so is k*10^i, i > 0.
If k is a term then so is k*i, 2 <= i <= 9 as long as no carry occurs in the multiplication.
The number of terms with n digits is (12, 29, 51, 107, 149, 240, 308, 438, 566, 789, 1007), 2 <= n <= 12.

			18 is a term because 108/18=6, and so is 1185 because 1010805/1185=853.
10101/111=91, 1010100/1110=910, 101010000/11100=9100, ... so 111, 1110, 11100, ... are all terms.
1000401/1041=961 and 2000802/2082=961 so 1041 and 2082 are terms but 3123 is not since it does not divide 3010203.

Cf. A051022, A285176, A343551, A343552.

Cf. A062846 (binary), A062891 (ternary).

A331841 When expressed in base 2 and then interpreted in base 5, is a multiple of the original number.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 18, 21, 27, 30, 54, 57, 60, 63, 89, 90, 108, 114, 126, 130, 178, 180, 189, 228, 300, 356, 378, 390, 630, 712, 780, 900, 1170, 1299, 1300, 1890, 1953, 2340, 2370, 2730, 3510, 3900, 3906, 4740, 7020, 7110, 7410, 7800, 8100, 8190, 9261, 11700

Offset: 1

Dimiter Skordev, Jan 29 2020

			30 = 11110_2; 11110_5 = 780 = 26*30.

Dimiter Skordev, Table of n, a(n) for n = 1..138 (terms less than 10^7).
Dimiter Skordev, Python script

Cf. (with base 2 and b): A062845 (b=3), A062846 (b=4), A062847 (b=6), A062848 (b=7), A062849 (b=8), A062850 (b=9), A032533 (b=10).

[0] cat [k:k in [1..12000]|Seqint(Intseq(Seqint(Intseq(k, 2))), 5) mod k eq 0]; // Marius A. Burtea, Jan 29 2020

Prepend[Select[Range[12000], Divisible[FromDigits[IntegerDigits[#, 2], 5], #] &], 0] (* Amiram Eldar, Jan 29 2020 *)

isok(n) = (n == 0) || (fromdigits(digits(n, 2), 5) % n) == 0; \\ Michel Marcus, Jan 29 2020

cp's OEIS Frontend

A062934 Numbers k that, when expressed in base 6 and then interpreted in base 7, give a multiple of k.

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A062937 Numbers k that, when expressed in base 6 and then interpreted in base 8, give a multiple of k.

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A062939 Numbers k that, when expressed in base 6 and then interpreted in base 9, give a multiple of k.

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A062942 Numbers k that, when expressed in base 6 and then interpreted in base 10, give a multiple of k.

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A155078 The representation of n=1,2,3,... in binary is a divisor of the same representation in another base. The sequence is the first such base.

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A283751 Least b>1 such that n, when expressed in base b and then interpreted in base b^2, yields a multiple of n.

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A343550 Numbers k > 9 such that the number m formed by inserting a digit 0 between each pair of digits in k is divisible by k.

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A331841 When expressed in base 2 and then interpreted in base 5, is a multiple of the original number.

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Magma

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