A118534 -id:A118534 - cp's OEIS frontend

A184830 a(n) = largest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.

0, 0, 2, 3, 3, 6, 7, 7, 9, 10, 15, 15, 15, 21, 23, 25, 27, 30, 27, 33, 39, 39, 45, 45, 47, 57, 58, 61, 63, 69, 67, 77, 79, 77, 81, 93, 99, 99, 105, 105, 105, 117, 123, 126, 125, 125, 135, 129, 147, 145, 151, 159, 165, 165, 167, 177, 171, 189, 189, 195

Offset: 1

Rémi Eismann, Jan 23 2011

From the definition, a(n) = A000961(n) - A057820(n) if A000961(n) - A057820(n) > A057820(n), 0 otherwise where A000961 are the prime powers and A057820 are the gaps between prime powers.

			For n = 1 we have A000961(1) = 1, A000961(2) = 2; for all k >= 2, 2 = 1 + (1 mod k), hence the largest k does not exist and a(1) = 0.
For n = 3 we have A000961(3) = 3, A000961(4) = 4; 2 is the largest k such that 4 = 3 + (3 mod k), hence a(3) = 2.
For n = 24 we have A000961(24) = 49, A000961(25) = 53; 45 is the largest k such that 53 = 49 + (49 mod k), hence a(24) = 45.

Rémi Eismann, Table of n, a(n) for n = 1..9790

Cf. A000961, A057820, A184829, A184831, A117078, A117563, A001223, A118534.

A184830 := proc(n)
    if A000961(n) > 2*A057820(n) then
        A000961(n)-A057820(n) ;
    else
        0;
    end if;
end proc:
seq(A184830(n),n=1..40) ; # R. J. Mathar, Sep 23 2016

nmax = 10000;
ppmax = 12*nmax; (* increase prime power max coef 12 in case of overflow *)
A000961 = Join[{1}, Select[Range[2, ppmax], PrimePowerQ]];
A057820 = Differences[A000961];
a[n_] := If[A000961[[n]] > 2*A057820[[n]], A000961[[n]] - A057820[[n]], 0];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 06 2023 *)

A184831 a(n) = A184830(n)/A184829(n) unless A184829(n) = 0 in which case a(n) = 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 1, 3, 2, 5, 5, 3, 7, 1, 5, 9, 15, 3, 3, 13, 3, 15, 9, 1, 19, 2, 1, 9, 23, 1, 11, 1, 11, 9, 3, 33, 11, 35, 21, 7, 13, 41, 63, 25, 5, 45, 3, 49, 5, 1, 3, 55, 33, 1, 59, 9, 63, 27, 65, 11, 1, 3, 75, 45, 1, 79, 1, 5, 41, 85, 1, 1, 89, 5, 39, 93, 1, 57, 9

Offset: 1

Rémi Eismann, Jan 23 2011

nonn

a(n) is the "level" of prime powers.
The decomposition of prime powers into weight * level + gap is A000961(n) = A184829(n) * a(n) + A057820(n) if a(n) > 0.
A184830(n) = A000961(n) - A057820(n) if A000961(n) - A057820(n) > A057820(n), 0 otherwise.

			For n = 1 we have A184829(1) = 0, hence a(1) = 0.
For n = 3 we have A184830(3)/A184829(3)= 2 / 2 = 1; hence a(3) = 1.
For n = 24 we have A184830(24)/A184829(24)= 45 / 5 = 9; hence a(24) = 9.

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

Cf. A000961, A057820, A184829, A184830, A117078, A117563, A001223, A118534.

A184832 a(n) = smallest k such that A005117(n+1) = A005117(n) + (A005117(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 2, 5, 4, 3, 3, 2, 13, 13, 3, 17, 2, 3, 4, 23, 2, 29, 29, 2, 3, 3, 2, 37, 37, 2, 41, 4, 3, 43, 7, 3, 53, 2, 3, 3, 2, 59, 2, 5, 5, 2, 3, 3, 2, 71, 2, 7, 4, 3, 3, 2, 5, 5, 3, 89, 2, 3, 3, 31, 2, 101, 101, 2, 3, 3, 2, 109, 109, 2, 113, 4, 3, 4, 11, 7, 5, 2, 3, 3, 2

Offset: 1

Rémi Eismann, Jan 23 2011

nonn

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

The decomposition of squarefree numbers into weight * level + gap is A005117(n) = a(n) * A184834(n) + A076259(n) if a(n) > 0.

			For n = 1 we have A005117(1) = 1, A005117(2) = 2; there is no k such that 2 - 1 = 1 = (1 mod k), hence a(1) = 0.
For n = 4 we have A005117(4) = 5, A005117(5) = 6; 2 is the smallest k such that 6 - 5 = 1 = (5 mod k), hence a(4) = 2.
For n = 23 we have A005117(23) = 35, A005117(24) = 37; 3 is the smallest k such that 37 - 35 = 2 = (35 mod k), hence a(23) = 3.

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

Cf. A005117, A076259, A184834, A184833, A117078, A117563, A001223, A118534.

A184833 a(n) = largest k such that A005117(n+1) = A005117(n) + (A005117(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 4, 5, 4, 9, 9, 12, 13, 13, 15, 17, 20, 21, 20, 23, 28, 29, 29, 32, 33, 33, 36, 37, 37, 40, 41, 40, 45, 43, 49, 51, 53, 56, 57, 57, 60, 59, 64, 65, 65, 68, 69, 69, 72, 71, 76, 77, 76, 81, 81, 84, 85, 85, 87, 89, 92, 93, 93, 93, 100, 101, 101, 104, 105

Offset: 1

Rémi Eismann, Jan 23 2011

From the definition, a(n) = A005117(n) - A076259(n) if A005117(n) - A076259(n) > A076259(n), 0 otherwise where A005117 are the squarefree numbers and A076259 are the gaps between squarefree numbers.

			For n = 1 we have A005117(1) = 1, A005117(2) = 2; there is no k such that 2 - 1 = 1 = (1 mod k), hence a(1) = 0.
For n = 4 we have A005117(4) = 5, A005117(5) = 6; 4 is the largest k such that 6 - 5 = 1 = (5 mod k), hence a(4) = 2; a(3) = 5 - 1 = 4.
For n = 23 we have A005117(23) = 35, A005117(24) = 37; 33 is the largest k such that 37 - 35 = 2 = (35 mod k), hence a(23) = 33; a(24) = 35 - 2 = 33.

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

Cf. A005117, A076259, A184832, A184834, A117078, A117563, A001223, A118534.

A184834 a(n) = A184833(n)/A184832(n) unless A184832(n) = 0 in which case a(n) = 0.

Original entry on oeis.org

0, 0, 0, 2, 1, 1, 3, 3, 6, 1, 1, 5, 1, 10, 7, 5, 1, 14, 1, 1, 16, 11, 11, 18, 1, 1, 20, 1, 10, 15, 1, 7, 17, 1, 28, 19, 19, 30, 1, 32, 13, 13, 34, 23, 23, 36, 1, 38, 11, 19, 27, 27, 42, 17, 17, 29, 1, 46, 31, 31, 3, 50, 1, 1, 52, 35, 35, 54, 1, 1, 56, 1, 28, 39

Offset: 1

Rémi Eismann, Jan 23 2011

nonn

a(n) is the "level" of squarefree numbers.
The decomposition of squarefree numbers into weight * level + gap is A005117(n) = A184832(n) * a(n) + A076259(n) if a(n) > 0.
A184833(n) = A005117(n) - A076259(n) if A005117(n) - A076259(n) > A076259(n), 0 otherwise.

			For n = 1 we have A184832(1) = 0, hence a(1) = 0.
For n = 4 we have A184833(4)/A184832(4)= 4 / 2 = 1; hence a(4) = 2.
For n = 23 we have A184833(23)/A184832(23)= 33 / 3 = 11; hence a(23) = 11.

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

Cf. A005117, A076259, A184832, A184833, A117078, A117563, A001223, A118534.

A162175 Primes classified by weight.

Original entry on oeis.org

11, 17, 29, 41, 59, 67, 71, 79, 83, 89, 101, 103, 107, 109, 137, 149, 167, 179, 191, 193, 197, 227, 229, 239, 241, 251, 269, 277, 281, 283, 311, 331, 347, 349, 359, 367, 379, 383, 409, 419, 431, 433, 439, 443, 449, 461, 463, 467, 487, 491, 499, 503, 521, 557

Offset: 1

Rémi Eismann, Jun 27 2009

nonn

Conjecture: primes classified by level are rarefying among prime numbers.
A000040(n) = 2, 3, 7, A162174(n), a(n). - Rémi Eismann, Jun 27 2009
By definition, primes classified by weight have a prime gap g(n) < sqrt(p(n)) (or more precisely, for primes classified by weight, we have A001223(n) <= sqrt(A118534(n)) - 1 ). So by definition, prime numbers classified by weight follow Legendre's conjecture and Andrica's conjecture - Rémi Eismann, Aug 26 2013

			For prime(5)=11, A117078(5)=3 <= A117563(5)=3 ; prime(5)=11 is classified by weight. For prime(170)=1013, A117078(170)=19 <= A117563(170)=53 ; prime(170)=1013 is classified by weight.

R. Eismann, Table of n, a(n) for n = 1..10000
Remi Eismann, Decomposition of natural numbers into weight * level + jump and application to a new classification of prime numbers
OEIS Wiki, Decomposition into weight * level + jump

Cf. A117078, A117563, A000040, A162174.

If for prime(n), A117078(n) (the weight) <= A117563(n) (the level) and A117078(n) <> 0 then prime(n) is classified by weight. If for prime(n), A117078(n) (the weight) > A117563(n) (the level) then prime(n) is classified by level.

A179620 a(n) = largest 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, 8, 8, 10, 13, 14, 14, 16, 19, 20, 20, 23, 24, 25, 26, 26, 28, 31, 32, 33, 34, 34, 37, 38, 38, 40, 43, 44, 44, 47, 48, 49, 50, 50, 53, 54, 55, 56, 56, 58, 61, 62, 63, 64, 64, 67, 68, 68, 70, 73, 74, 75, 76, 76, 79, 80, 80, 83, 84, 85, 86, 86, 89

Offset: 1

Rémi Eismann, Jan 09 2011

a(n) = A002808(n) - A073783(n) if A002808(n) - A073783(n) > A073783(n), 0 otherwise.

A002808(n): composite numbers; A073783(n): first difference of composite numbers.

			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 largest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7; a(3) = A002808(3) - A073783(3) = 8 - 1 = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 34 is the largest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 34; a(24) = A002808(24) - A073783(24) = 34.

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

Cf. A002808, A073783, A130882, A179621, A118534, A117078, A117563, A001223.

A184218 a(n) = largest 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, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377

Offset: 1

Rémi Eismann, Jan 10 2011

From the definition, a(n) = A000217(n) - (n + 1) if A000217(n) - (n + 1) > (n + 1), or 0 otherwise, where A000217 are the triangular numbers.

			For n = 3 we have A000217(3) = 6, A000217(4) = 10; there is no k such that 10 - 6 = 4 = (6 mod k), hence a(3) = 0.
For n = 5 we have A000217(5) = 15, A000217(6) = 21; 9 is the largest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9; a(5) = A000217(5) - (5 + 1) = 15 - 6 = 9.
For n = 24 we have A000217(24) = 300, A000217(25) = 325; 275 is the largest k such that 325 - 300 = 25 = (300 mod k), hence a(24) = 275; a(24) = A000217(24) - (24 + 1) = 275.

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000 [Originally computed by Remi Eismann]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. essentially the same as A000096, A000217, A000027, A130703, A184219, A118534, A117078, A117563, A001223.

[0,0,0,0] cat [(n+1)*(n-2)/2: n in [5..60]]; // Vincenzo Librandi, Jun 22 2016

Join[{0, 0, 0, 0}, LinearRecurrence[{3, -3, 1}, {9, 14, 20}, 100]] (* G. C. Greubel, Jun 22 2016 *)
lim = 10^4; Table[SelectFirst[Reverse@ Range@ lim, Function[k, PolygonalNumber[n + 1] == # + Mod[#, k] &@ PolygonalNumber@ n]], {n, 53}] /. {k_ /; MissingQ@ k -> 0, k_ /; k == lim -> 0} (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)

a(n) = (n+1)*(n-2)/2 = A000096(n-2) for n >= 5 and a(n) = 0 for n <= 4. - M. F. Hasler, Jan 10 2011
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x^5*(5*x^2 - 13*x + 9)/(1 - x)^3. (End)

A184220 a(n) = largest 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, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322, 359, 398, 439, 482, 527, 574, 623, 674, 727, 782, 839, 898, 959, 1022, 1087, 1154, 1223, 1294, 1367, 1442, 1519, 1598, 1679, 1762

Offset: 1

Rémi Eismann, Jan 10 2011

From the definition, a(n) = A000290(n) - A005408(n) if A000217(n) - A005408(n) > A005408(n), 0 otherwise, where A000290 are the squares and A005408 are the gaps between squares: 2n + 1.

			For n = 3 we have A000290(3) = 9, A000290(4) = 16; there is no k such that 16 - 9 = 7 = (9 mod k), hence a(3) = 0.
For n = 5 we have A000290(5) = 25, A000290(6) = 36; 14 is the largest k such that 36 - 25 = 11 = (25 mod k), hence a(5) = 14; a(5) = A000290(5) - A005408(5) = 25 - 11 = 14.
For n = 25 we have A000217(25) = 625, A000217(26) = 676; 574 is the largest k such that 676 - 625 = 51 = (625 mod k), hence a(25) = 574; a(25) = A000290(25) - A005408(25) = 574.

Rémi Eismann, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. essentially the same as A008865, A000290, A005408, A133150, A184221, A118534, A117078, A117563, A001223.

a(n) = (n-1)^2-2 = A008865(n-1) for n >= 5 and a(n) = 0 for n <= 4.

A106752 Numbers of prime factors of k, k defined in A117078.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Rémi Eismann, Jun 22 2007, Feb 14 2008

nonn

a(n) = 0 only for n = 1, 2 and 4.

			For a(1), k=0 thus a(1)=0,
For a(3), k=3 thus a(3)=1,
For a(11), k=25=5*5 thus a(11)=2.

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

Cf. A117078, A117563, A118534, A118144, A118123.

a(n) = numbers of factors of A117078(n). A117078(n) : smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

cp's OEIS Frontend

A184830 a(n) = largest k such that A000961(n+1) = A000961(n) + (A000961(n) mod k), or 0 if no such k exists.

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Maple

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A184831 a(n) = A184830(n)/A184829(n) unless A184829(n) = 0 in which case a(n) = 0.

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A184832 a(n) = smallest k such that A005117(n+1) = A005117(n) + (A005117(n) mod k), or 0 if no such k exists.

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A184833 a(n) = largest k such that A005117(n+1) = A005117(n) + (A005117(n) mod k), or 0 if no such k exists.

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A184834 a(n) = A184833(n)/A184832(n) unless A184832(n) = 0 in which case a(n) = 0.

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A162175 Primes classified by weight.

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A179620 a(n) = largest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

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A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

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Magma

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A184220 a(n) = largest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists.

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