A236206 -id:A236206 - cp's OEIS frontend

A270299 Numbers which are representable as a sum of eleven but no fewer consecutive nonnegative integers.

88, 176, 352, 704, 968, 1144, 1408, 1496, 1672, 1936, 2024, 2288, 2552, 2728, 2816, 2992, 3256, 3344, 3608, 3784, 3872, 4048, 4136, 4576, 4664, 5104, 5192, 5368, 5456, 5632, 5896, 5984, 6248, 6424, 6512, 6688, 6952, 7216, 7304, 7568, 7744, 7832, 8096, 8272

Offset: 1

Martin Renner, Mar 14 2016

nonn

			66 = 1 + 2 + 3 + ... + 9 + 10 + 11 = 21 + 22 + 23 (not in sequence);
88 = 3 + 4 + 5 + ... + 11 + 12 + 13;
176 = 11 + 12 + 13 + ... + 19 + 20 + 21;
352 = 27 + 28 + 29 + ... + 35 + 36 + 37.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A138591, A236206, A270296, A270297, A270298, A270300, A270301, A270302, A270303.

A163169(a(n)) = 11. - Ray Chandler, Mar 22 2016

a(n) = 88*A236206(n). - Hugo Pfoertner, Feb 04 2021

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Benoit Cloitre, Apr 18 2002

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd part.

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

New name from Peter Munn, Jul 06 2020

A236217 Numbers not divisible by 3, 5 or 11.

Original entry on oeis.org

1, 2, 4, 7, 8, 13, 14, 16, 17, 19, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119, 122, 124, 127, 128

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers coprime to 165. The asymptotic density of this sequence is 16/33. - Amiram Eldar, Oct 23 2020

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Intersection of: A160542 and A229829; A047201 and A229968; A001651, A047201 and A160542.

Cf. A236206, A236208.

Select[Range[200], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[200], Or @@ Divisible[#, {3, 5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)
Select[Range[130], CoprimeQ[165, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-80) - a(n-81) for n > 81. - Bruno Berselli, Mar 25 2014

A362012 Numbers k such that 1 < gcd(k, 105) < k and A007947(k) does not divide 105.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 28, 30, 33, 36, 39, 40, 42, 48, 50, 51, 54, 55, 56, 57, 60, 65, 66, 69, 70, 72, 77, 78, 80, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 108, 110, 111, 112, 114, 115, 117, 119, 120, 123, 126, 129, 130, 132, 133, 138, 140, 141, 144, 145, 150, 153, 154, 155, 156, 159, 160

Offset: 1

Michael De Vlieger, Apr 04 2023

nonn

The asymptotic density of this sequence is 19/35. - Amiram Eldar, Dec 02 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram showing numbers k in this sequence instead as k mod 105, in black, else white if k is coprime to 105, purple if k = 1, red if k | 105, and gold if rad(k) | 105, magnification 10X.

Cf. A007947, A108347, A236206.

With[{n = 105}, Select[Range[200], And[! CoprimeQ[#, n], ! Divisible[n, Times @@ FactorInteger[#][[All, 1]]]] & ] ]

This sequence is { N \ { A108347 U A236206 } }.

A381810 Array read by downward antidiagonals: A(n,k) is a generalization of odd columns of A125790 defined in Comments for n > 0, k >= 0.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 8, 36, 20, 10, 10, 64, 42, 84, 14, 12, 100, 72, 286, 100, 20, 14, 144, 110, 680, 322, 120, 26, 16, 196, 156, 1330, 744, 364, 140, 36, 18, 256, 210, 2300, 1430, 816, 406, 656, 46, 20, 324, 272, 3654, 2444, 1540, 888, 3396, 740, 60, 22, 400, 342, 5456, 3850, 2600, 1650, 10816, 3682, 840, 74

Offset: 1

Mikhail Kurkov, May 05 2025

This is generalization in the sense that first column of A125790 is A000123(2^(n-1)) while in this square array column zero is conjecturally A000123(n).
A(n,k) = v_{A001511(n)} where we start with vector v of fixed length L(n) = A070939(n) with elements v_i = A125790(i,2*k+1), pre-calculate A078121 up to L(n)-th row, reserve t as an empty vector of fixed length L(n) and for i=1..A119387(n+1), for j=1..L(n)-i+1 apply t := v (at the beginning of each cycle for i) and also apply v_j := Sum_{k=1..j+1} A078121(j,k-1)*t_k if R(n,L(n)-i) = 1, otherwise v_j := Sum_{k=1..j+1} A078121(j,k-1)*t_k*(-1)^(j+k+1). Here R(n,k) = floor(n/(2^k)) mod 2 is the (k+1)-th bit in the binary expansion of n.
Conjecture: sequence A(n,k) for fixed n is a polynomial of degree A070939(n).

			Array begins:
===========================================================
n\k|  0    1     2      3      4      5       6       7 ...
---+-------------------------------------------------------
1  |  2,   4,    6,     8,    10,    12,     14,     16 ...
2  |  4,  16,   36,    64,   100,   144,    196,    256 ...
3  |  6,  20,   42,    72,   110,   156,    210,    272 ...
4  | 10,  84,  286,   680,  1330,  2300,   3654,   5456 ...
5  | 14, 100,  322,   744,  1430,  2444,   3850,   5712 ...
6  | 20, 120,  364,   816,  1540,  2600,   4060,   5984 ...
7  | 26, 140,  406,   888,  1650,  2756,   4270,   6256 ...
8  | 36, 656, 3396, 10816, 26500, 55056, 102116, 174336 ...
  ...

Cf. A000123, A001511, A007814, A053645, A062383, A070939, A078121, A106400, A119387, A125790, A236206.

upto1(n) = my(v1); v1 = vector(n+1, i, vector(i, j, j==1 || j==i)); for(i=2, n, for(j=1, i-1, v1[i+1][j+1] = sum(k=j-1, i-1, v1[i][k+1]*v1[k+1][j]))); v1
A(n,m) = my(L = logint(n,2), A = valuation(n,2), B = logint(n>>A,2), v1, v2, v3); v1 = upto1(L+2); v2 = vector(L+2, i, vecsum(v1[i])); for(i=1, 2*m, v2 = vector(L+2, i, sum(j=1, i, v1[i][j]*v2[j]))); for(i=1, B, v3 = v2; for(j=1, L-i+1, v2[j+1] = sum(k=1, j+1, v1[j+1][k]*v3[k+1]*if(!bittest(n,L-i+1), (-1)^(j+k+1), 1)))); v2[A+2]

upto1(n) = my(v1); v1 = vector(n+1, i, vector(i, j, j==1 || j==i)); for(i=2, n, for(j=1, i-1, v1[i+1][j+1] = sum(k=j-1, i-1, v1[i][k+1]*v1[k+1][j]))); v1
upto2(n,m) = my(L = logint(n,2), A = valuation(n,2), B = logint(n>>A,2), v1, v2, v3, v4, v5); v1 = upto1(L+2); v2 = vector(L+2, i, 1); v3 = vector(m+1, i, 0); for(s=0, m, for(i=1, min(s+1,2), v2 = vector(L+2, i, sum(j=1, i, v1[i][j]*v2[j]))); v4 = v2; for(i=1, B, v5 = v4; for(j=1, L-i+1, v4[j+1] = sum(k=1, j+1, v1[j+1][k]*v5[k+1]*if(!bittest(n,L-i+1), (-1)^(j+k+1), 1)))); v3[s+1] = v4[A+2]); v3 \\ slightly modified version of the first program, some kind of memoization; generates A(n,k) for k=0..m

A(2^(n-1),k) = A125790(n,2*k+1) for n > 0, k >= 0.
Conjectured formulas: (Start)
A(n,0) = A000123(n) for n > 0.
A(n,k) = Sum_{j=0..k} A000123(A062383(n)*j+n)*A106400(k-j) for n > 0, k >= 0.
If we change v_i = A125790(i,2*k+1) to v_i = A125790(i,2*k) to get similar generalization of even columns, then for resulting array B(n,k) we have B(n,k) = Sum_{j=0..k} A000123(A062383(n)*j+A053645(n))*A106400(k-j) for n > 0, k >= 0.
2*(k+1) divides A(n,k) for n > 0 if (k+1) is a term of A236206.
G.f. for n-th row is f(A070939(n)+1,n) for n > 0 where f(n,k) = (Sum_{(c_0 + c_1 + ... + c_{n-1}) == 2*k (mod 2^n), 0 <= c_i < 2^n, 2^i divides c_i} x^((c_0 + c_1 + ... + c_{n-1} - 2*k)/2^n))/(1-x)^n for n > 0, k >= 0. Similarly, g.f. for n-th row of B(n,k) is f(A070939(n)+1,A053645(n)).
G.f. for n-th row is (Sum_{i=0..L(n)-1} x^i * Sum_{j=0..i} binomial(L(n)+1,j)*A(n,i-j)*(-1)^j)/(1-x)^(L(n)+1) for n > 0 where L(n) = A070939(n).
s(4*n+1) = 1 for n >= 0, s(4*n) = s(4*n+2) = 1 if A010060(n) = 1 for n > 0 where s(n) = A007814(Sum_{k=0..n-1} A(k+1,n-k-1)). (End)

cp's OEIS Frontend

A270299 Numbers which are representable as a sum of eleven but no fewer consecutive nonnegative integers.

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A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

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Mathematica

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A236217 Numbers not divisible by 3, 5 or 11.

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A362012 Numbers k such that 1 < gcd(k, 105) < k and A007947(k) does not divide 105.

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A381810 Array read by downward antidiagonals: A(n,k) is a generalization of odd columns of A125790 defined in Comments for n > 0, k >= 0.

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PARI

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