A071593 -id:A071593 - cp's OEIS frontend

A140811 a(n) = 6*n^2 - 1.

-1, 5, 23, 53, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615

Offset: 0

Paul Curtz, Jul 16 2008

Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).
a(n) is the set of all k such that 6*k + 6 is a perfect square. - Gary Detlefs, Mar 04 2010
The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as a(n+1)^2 - A157872(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012
Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
From Paul Curtz, Sep 17 2018: (Start)
Terms from center to right in the following spiral:
.
                    65--63--61--59
                    /             \
                  67  31--29--27  57
                  /   /         \   \
                69  33   9---7  25  55
                /   /   /     \   \   \
              71  35  11  -1===5==23==53==>
                  /   /   /   /   /   /
                37  13   1---3  21  51
                  \   \         /   /
                  39  15--17--19  49
                    \             /
                    41--43--45--47 (End)

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Barred Stars.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A157872, A060747, A103115(n+1), A141417 (array).
Cf. A000012, A001318, A005408, A091137, A140825.
Cf. A003154, A032528, A033581, A016777, A017593.

[6*n^2 - 1: n in [0..50]]; // Vincenzo Librandi, Jun 02 2011

LinearRecurrence[{3,-3,1},{-1,5,23},40] (* Vincenzo Librandi, Feb 05 2012 *)
CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

a(n)=6*n^2-1 \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 2*a(n-1) - a(n-2) + 12.
First differences: a(n+1) - a(n) = A017593(n).
Second differences: A071593(n+1) - A071593(n) = 12.
G.f.: (1-8*x-5*x^2)/(x-1)^3. - Jaume Oliver Lafont, Aug 30 2009
From Vincenzo Librandi, Feb 05 2012: (Start)
a(n) = a(n-1) + 12*n - 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A033581(n) - 1. - Omar E. Pol, Jul 18 2012
a(n) = A032528(2*n) - 1. - Adriano Caroli, Jul 21 2013
For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - Clark Kimberling, Oct 19 2014, corrected by M. F. Hasler, Oct 21 2014
a(-n) = a(n). - Paul Curtz, Sep 17 2018
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)
a(n) = A003154(n+1) - 2*A016777(n). - Leo Tavares, May 13 2022
E.g.f.: exp(x)*(6*x^2 + 6*x - 1). - Elmo R. Oliveira, Jan 16 2025

Edited and extended by R. J. Mathar, Aug 06 2008

Better description Ray Chandler, Feb 03 2009

A255401 Numbers n with the property that its k-th smallest divisor, for all 1 <= k <= tau(n), contains exactly k "1" digits in its binary representation.

Original entry on oeis.org

1, 3, 5, 17, 25, 39, 57, 201, 257, 289, 291, 323, 393, 579, 1083, 2307, 7955, 8815, 9399, 12297, 12909, 13737, 36867, 40521, 43797, 50349, 65537, 66049, 66291, 66531, 68457, 80457, 98313, 160329, 196617, 197633, 230691, 299559, 599079, 786441, 922179, 1278537

Offset: 1

Jaroslav Krizek, Feb 22 2015

For n>1; a(n) is a multiple of a Fermat prime (A019434). Subsequence of A071593.
For all divisors d_k of a(n) we have A000120(d_k) = k.
Subsequence of known numbers with k divisors:
for k = 2: 3, 5, 17, 257, 65537, ... - Fermat primes (A019434);
for k = 3: 25, 289, 66049, 4295098369, ... - some square of Fermat prime;
for k = 4: 39, 57, 201, 291, 323, 393, 579, 2307, 12297, 36867, 98313, 196617, 197633, 786441, 2359299, 805306377, 3221225481, 4295229443, 9663676419, 618475290627, 19791209299971, ... - some products of two distinct primes p*q, where p is a Fermat prime (A019434) and q is a term of sequence A081091, see (Magma) - Set(Sort([n*m: n in [A019434(n)], m in [A081091(m)] | n lt m and &+Intseq(n, 2) eq 2 and &+Intseq(m, 2) eq 3 and &+Intseq(n*m, 2) eq 4]));
for k = 6: 1083 - the only number with this property < 10^7;
for k = 8: 7955, 8815, 9399, 12909, 13737, 40521, 43797, 50349, 66291, 66531, 68457, 80457, 160329, 230691, 299559, 599079, 922179, 1278537, 2396199, 2556489, ...; see (Magma) - Set(Sort([n: n in [1..1000000] | [&+Intseq(d, 2): d in Divisors(n)] eq [1,2,3,4,5,6,7,8]])).
Conjectures: 1) Sequence is infinite. 2) 8 is the maximal value of k for numbers with this property.
Numbers 805306377, 3221225481, 4295098369, 4295229443, 9663676419, 618475290627 and 19791209299971 are also terms of this sequence.
Sequence of the smallest numbers n with k divisors having these properties for k >= 1 or 0 if no solution exists or has been found: 1, 3, 25, 39, 0, 1083, 0, 7955, ...; a(5) = a(7) = 0 if there are only 5 Fermat primes. Conjecture: a(k) = 0 for k > 8.

			The divisors of 1083, expressed in base 2 and listed in ascending order as 1, 11, 10011, 111001, 101101001, 10000111011, contain 1, 2, 3, 4, 5 and 6 "1" digits, respectively.

Jaroslav Krizek and Amiram Eldar, Table of n, a(n) for n = 1..50 (terms 1..45 from Jaroslav Krizek)

Cf. A019434, A071593, A081091.

Set(Sort([n: n in [1..1000000] | [&+Intseq(d, 2): d in Divisors(n)] eq [1..NumberOfDivisors(n)]]))

Select[Range[10^6], Total @ IntegerDigits[#, 2] & /@ (d = Divisors[#]) == Range @ Length[d] &] (* Amiram Eldar, Dec 29 2019 *)

isok(n) = {my(d = divisors(n)); for (i=1, #d, if (hammingweight(d[i]) != i, return (0));); return (1);} \\ Michel Marcus, Feb 22 2015

A152088 Positive integers k that when written in binary have exactly the same number of (non-leading) 0's as the number of divisors of k.

Original entry on oeis.org

19, 33, 34, 43, 49, 53, 69, 74, 79, 82, 103, 107, 109, 141, 142, 166, 177, 178, 201, 202, 209, 226, 261, 268, 292, 295, 299, 301, 302, 309, 314, 327, 334, 339, 341, 346, 355, 358, 362, 367, 379, 388, 391, 395, 398, 403, 422, 431, 439, 443, 451, 453, 454, 458

Offset: 1

Leroy Quet, Nov 23 2008

			34 written in binary is 100010, which has four 0's. Also, 34 has 4 divisors (1,2,17,34). Since the number of binary 0's equals the number of divisors, then 34 is included in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A023416, A071593, A080791.

Select[Range[500], DigitCount[#, 2, 0] == DivisorSigma[0, #] &] (* Amiram Eldar, Dec 28 2019 *)

Extended by Ray Chandler, Nov 26 2008

A253204 a(1) = 1; for n>1, a(n) is a prime power p^h (h>=1) with the property that its k-th smallest divisor, for all 1 <= k <= tau(p^h), contains exactly k "1" digits in its binary representation.

Original entry on oeis.org

1, 3, 5, 17, 25, 257, 289, 65537, 66049, 4295098369

Offset: 1

Jaroslav Krizek, Mar 25 2015

Supersequence of A019434 (Fermat primes). Subsequence of A071593 and A255401.

Sequence is finite if there is only 5 Fermat primes (A019434).

			The divisors of 4295098369, expressed in base 2 and listed in ascending order as 1, 10000000000000001, 100000000000000100000000000000001, contain 1, 2 and 3, "1" digits, respectively.

Cf. A019434, A071593, A255401.

Set(Sort([1] cat [n: n in [2..1000000] | [&+Intseq(d, 2): d in Divisors(n)] eq [1..NumberOfDivisors(n)] and #(PrimeDivisors(n)) eq 1]));

cp's OEIS Frontend

A140811 a(n) = 6*n^2 - 1.

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A255401 Numbers n with the property that its k-th smallest divisor, for all 1 <= k <= tau(n), contains exactly k "1" digits in its binary representation.

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Magma

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A152088 Positive integers k that when written in binary have exactly the same number of (non-leading) 0's as the number of divisors of k.

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A253204 a(1) = 1; for n>1, a(n) is a prime power p^h (h>=1) with the property that its k-th smallest divisor, for all 1 <= k <= tau(p^h), contains exactly k "1" digits in its binary representation.

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Magma