A090369 -id:A090369 - cp's OEIS frontend

A184727 a(n) = A005843(n-1)/A090369(n-1) for n > 2 and a(n) = 0 for n <= 2.

0, 0, 1, 2, 2, 2, 4, 2, 4, 6, 5, 2, 8, 2, 7, 10, 8, 2, 12, 2, 10, 14, 11, 2, 16, 10, 13, 18, 14, 2, 20, 2, 16, 22, 17, 14, 24, 2, 19, 26, 20, 2, 28, 2, 22, 30, 23, 2, 32, 14, 25, 34, 26, 2, 36, 22, 28, 38, 29, 2, 40, 2, 31, 42, 32, 26

Offset: 1

Rémi Eismann, Jan 20 2011

nonn

a(n) is the "level" of even numbers.

The decomposition of even numbers into weight * level + gap is A005843(n) = A090369(n-1) * a(n) + 2 if a(n) > 0.

			For n = 3 we have A005843(2)/A090369(2)= 4 / 4 = 1; hence a(3) = 1.
For n = 24 we have A005843(23)/A090369(23)= 46 / 23 = 2; hence a(24) = 2.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A005843, A090369, A117078, A117563, A001223.

A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 9, 14, 10, 27, 35, 22, 18, 65, 77, 18, 26, 119, 27, 38, 34, 27, 209, 46, 28, 55, 299, 36, 35, 377, 45, 62, 58, 45, 527, 40, 54, 629, 95, 54, 74, 779, 63, 86, 82, 63, 989, 94, 54, 161, 235, 68, 91, 265, 81, 65, 106, 81, 145, 118, 90, 1769, 1829

Offset: 1

Rémi Eismann, Aug 16 2007 - Jan 10 2011

nonn

a(n) is the weight of triangular numbers.

The decomposition of triangular numbers into weight * level + gap is A000217(n) = a(n) * A184219(n) + (n + 1) if a(n) > 0.

			For n = 1 we have A000217(n) = 1, A000217(n+1) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 5 we have A000217(n) = 15, A000217(n+1) = 21; 9 is the smallest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9.
For n = 22 we have A000217(n) = 253, A000217(n+1) = 276; 46 is the smallest k such that 276 - 253 = 23 = (253 mod k), hence a(22) = 46.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368.

A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 4, 7, 2, 4, 5, 13, 2, 7, 4, 19, 2, 4, 23, 2, 5, 2, 13, 4, 31, 2, 3, 2, 17, 37, 2, 19, 4, 43, 2, 4, 47, 2, 7, 2, 5, 53, 2, 5, 2, 4, 29, 61, 2, 3, 2, 4, 67, 2, 4, 5, 73, 2, 3, 2, 4, 79, 2, 4, 83, 2, 5, 2, 43, 89, 2, 7, 2, 3, 2, 47, 97

Offset: 1

Rémi Eismann, Aug 21 2007 - Jan 09 2011

nonn

a(n) is the "weight" of composite numbers.

The decomposition of composite numbers into weight * level + gap is A002808(n) = a(n) * A179621(n) + A073783(n) if a(n) > 0.

			For n = 1 we have A002808(n) = 4, A002808(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A002808(n) = 8, A002808(n+1) = 9; 7 is the smallest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 17 is the smallest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 17.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703.

A133150 a(n) = smallest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 14, 23, 17, 47, 31, 79, 49, 119, 71, 167, 97, 223, 127, 41, 46, 359, 199, 439, 241, 527, 82, 89, 337, 727, 391, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 217, 94, 1679, 881, 1847, 967, 119, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 1457

Offset: 1

Rémi Eismann, Sep 22 2007 - Jan 10 2011

nonn

a(n) is the "weight" of squares (A000290).

The decomposition of squares into weight * level + gap is A000217(n) = a(n) * A184221(n) + A005408(n) if a(n) > 0.

			For n = 1 we have A000290(n) = 1, A000290(n+1) = 4; there is no k such that 4 - 1 = 3 = (1 mod k), hence a(1) = 0.
For n = 5 we have A000290(n) = 25, A000290(n+1) = 36; 14 is the smallest k such that 36 - 25 = 11 = (25 mod k), hence a(5) = 14.
For n = 18 we have A000290(n) = 324, A000290(n+1) = 361; 41 is the smallest k such that 361 - 324 = 37 = (324 mod k), hence a(18) = 41.

Remi Eismann, Table of n, a(n) for n = 1..10000

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703, A130889, A130882.

A090370 Least m > 3 such that gcd(n-1, m*n - 1) = m-1.

Original entry on oeis.org

4, 5, 6, 4, 8, 5, 4, 6, 12, 4, 14, 8, 4, 5, 18, 4, 20, 5, 4, 12, 24, 4, 6, 14, 4, 5, 30, 4, 32, 5, 4, 18, 6, 4, 38, 20, 4, 5, 42, 4, 44, 5, 4, 24, 48, 4, 8, 6, 4, 5, 54, 4, 6, 5, 4, 30, 60, 4, 62, 32, 4, 5, 6, 4

Offset: 4

Lekraj Beedassy, Nov 27 2003

nonn

Choosing a pair (m, n) so as to redefine 1 hour = m*n minutes and 1 minute = m*n seconds, then the three hands of a fictitious n-hour clock coincide in exactly m-1 equally spaced positions, including that of the n o'clock position. For instance, in the cases where we select (m, n) as (6, 11), (8, 15), (4, 25), with m*n respectively equal to 66, 120, 100 (implying 1 hour = 66 minutes, 1 minute = 66 seconds; 1 hour = 120 minutes, 1 minute = 120 seconds; 1 hour = 100 minutes, 1 minute = 100 seconds), the hands coincide in exactly 6-1=5, 8-1=7, 4-1=3 equally spaced positions on a 11-hour, 15-hour, 25-hour clock respectively.

			We have a(50)=8 because 50*8 = 400 is the least multiple of 50 such that gcd(50-1, 400-1) = 8 - 1 = 7.

Giovanni Resta, Table of n, a(n) for n = 4..10000

Cf. A090368, A090369.

A090370:=proc(n) local m; m:=4; while  (gcd(n-1, m*n - 1) <> m-1) do m:=m+1; end;  return m; end; # Søren Eilers, Aug 09 2018

a[n_] := Block[{m=4}, While[GCD[n-1, n*m-1] != m-1, m++]; m]; Table[a[k], {k, 4, 67}] (* Giovanni Resta, Aug 09 2018 *)

a(n) = {m = 4; while (gcd(n-1,m*n - 1) != m-1, m++); return (m);} \\ Michel Marcus, Jul 27 2013

a(n) = 1 + A090368(k) for n=2k. [corrected by Søren Eilers, Aug 09 2018]

a(n) = 1 + A090369(k) for n=2k+1.

a(46) and a(49) corrected by Søren Eilers, Aug 09 2018

A133346 a(n) = smallest k such that prime(n+2) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 11, 0, 15, 21, 21, 31, 7, 11, 35, 9, 17, 17, 61, 9, 21, 23, 23, 77, 7, 19, 97, 101, 91, 19, 13, 41, 25, 127, 47, 139, 21, 17, 31, 11, 167, 13, 37, 11, 61, 25, 39, 7, 13, 73, 9, 227, 25, 239, 35, 15, 9, 29, 271, 269, 37, 25, 7, 61, 59, 27, 21, 13, 11, 113, 113

Offset: 1

Rémi Eismann, Oct 20 2007

nonn

			For n = 1 we have prime(n) = 2, prime(n+2) = 5; there is no k such that 5 - 2 = 3 = (2 mod k), hence a(1) = 0.
For n = 6 we have prime(n) = 13, prime(n+2) = 19; 7 is the smallest k such that 19 - 13 = 6 = (13 mod k), hence a(6) = 7.
For n = 30 we have prime(n) = 113, prime(n+2) = 131; 19 is the smallest k such that 131 - 113 = 18 = (113 mod k), hence a(30) = 19.

Remi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000040, A117078, A117563, A001223, A118534, A020639, A090369, A090368, A130533, A130650, A130703, A130889, A130882.

A133347 a(n) = smallest k such that prime(n+3) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 19, 27, 29, 27, 33, 39, 47, 49, 55, 59, 19, 61, 65, 15, 29, 31, 31, 29, 29, 89, 23, 113, 41, 121, 15, 27, 47, 21, 17, 31, 15, 33, 61, 25, 57, 57, 193, 71, 43, 31, 43, 221, 73, 233, 27, 83, 257, 37, 29, 51, 51, 21, 11, 97, 289, 41, 313, 107, 67

Offset: 1

Rémi Eismann, Oct 20 2007

nonn

			For n = 1 we have prime(n) = 2, prime(n+3) = 7; there is no k such that 7 - 2 = 5 = (2 mod k), hence a(1) = 0.
For n = 10 we have prime(n) = 29, prime(n+3) = 41; 17 is the smallest k such that 41 - 29 = 12 = (29 mod k), hence a(10) = 17.
For n = 53 we have prime(n) = 241, prime(n+3) = 263; 73 is the smallest k such that 263 - 241 = 22 = (241 mod k), hence a(53) = 73.

Remi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000040, A117078, A117563, A001223, A118534, A020639, A090369, A090368, A130533, A130650, A130703, A130889, A130882.

cp's OEIS Frontend

A184727 a(n) = A005843(n-1)/A090369(n-1) for n > 2 and a(n) = 0 for n <= 2.

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A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

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A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

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A133150 a(n) = smallest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists.

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A090370 Least m > 3 such that gcd(n-1, m*n - 1) = m-1.

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A133346 a(n) = smallest k such that prime(n+2) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

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A133347 a(n) = smallest k such that prime(n+3) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

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