A000079 -id:A000079 - cp's OEIS frontend

A153279 Eigentriangle by rows, T(n,k) = A000079(n-k) * (diagonalized matrix of (1,1,3,9,27,81,...)).

1, 2, 1, 4, 2, 3, 8, 4, 6, 9, 16, 8, 12, 18, 27, 32, 16, 24, 36, 54, 81, 64, 32, 48, 72, 108, 162, 243, 128, 64, 96, 144, 216, 324, 486, 729, 256, 128, 192, 288, 432, 648, 972, 1458, 2187, 512, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561

Offset: 0

Gary W. Adamson, Dec 23 2008

Row sums = 3^n
Sum of n-th row terms = rightmost term of next row.
Eigensequence of the triangle = A153280: (1, 3, 15, 165, 4785, 397155,...)

			First few rows of the triangle =
1;
2, 1;
4, 2, 3;
8, 4, 6, 9;
16, 8, 12, 18, 27;
32, 16, 24, 36, 54, 81;
64, 32, 48, 72, 108, 162, 243;
128, 64, 96, 144, 216, 324, 486, 729;
256, 128, 192, 288, 432, 648, 972, 1458, 2187;
512, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561;
...
Row 3 = (8, 4, 6, 9) = termwise products of (8, 4, 2, 1) and (1, 1, 3, 9).

Cf. A000079, A000244, A153280

Triangle read by rows, M*Q. M = triangle T(n,k) = A000079(n-k); powers of 2 in every column. Q = an infinite lower triangular matrix with powers of 3 prefaced with a 1: (1,1,3,9,27,...) as the main diagonal and the rest zeros.

A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).

Original entry on oeis.org

1, 2, 3, 4, 18, 16, 8, 81, 192, 125, 16, 324, 1536, 2500, 1296, 32, 1215, 10240, 31250, 38880, 16807, 64, 4374, 61440, 312500, 699840, 705894, 262144, 128, 15309, 344064, 2734375, 9797760, 17294403, 14680064, 4782969

Offset: 0

Peter Luschny, Jan 14 2009

Formatted as a square array:
1st row is A000079(n). Subsets of an n-set.
2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.
2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.
1st column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).
Alternating sum of rows in the triangle, Sum_{k=0..n} (-1)^(n-k) * T(n,k) = n! = A000142(n).
This triangle gives the coefficient of Sidi's polynomials D_{n,2,n}(-z)/(-z), for n >= 0. See [Sidi 1980]. - Wolfdieter Lang, Oct 27 2022

			Triangle begins as:
   1;
   2,    3;
   4,   18,    16;
   8,   81,   192,    125;
  16,  324,  1536,   2500,   1296;
  32, 1215, 10240,  31250,  38880,  16807;
  64, 4374, 61440, 312500, 699840, 705894, 262144;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874. Eq. (4.10), p. 862.

Cf. A000079, A000272, A036290, A066274, A075513.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*(k+2)^n ))); # G. C. Greubel, May 09 2019

[[Binomial(n,k)*(k+2)^n: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019

T := proc(n,k) binomial(n,k)*(k+2)^n end;

Table[Binomial[n, k]*(k+2)^n, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 09 2019 *)

{T(n, k) = binomial(n,k)*(k+2)^n}; \\ G. C. Greubel, May 09 2019

[[binomial(n,k)*(k+2)^n for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019

T(n,k) = binomial(n,k)*(k+2)^n, where n >= 0, and k >= 0.
From Wolfdieter Lang, Oct 20 2022: (Start)
O.g.f. of column k: (-x)^k*(k + 2)^k/(1 - (k + 2)*x)^(k+1), for k >= 0. See |A075513| with offset 0.
E.g.f. of column k: exp((k+2)*x)*((k+2)*x)^k/k!, for k >= 0. (End)
E.g.f. of triangle (of row polynomials in y): exp(2*x)*substitute(z = x*y*exp(x), LambertW(-z)^2/(-z)*2*(1 + LambertW(-z)))). - Wolfdieter Lang, Oct 24 2022

A181174 The "Row2" sums of the powers-of-2 triangle A000079.

Original entry on oeis.org

1, 2, 24, 320, 11264, 688128, 90177536, 22817013760, 11751030521856, 11997870882291712, 24607668363952390144, 100719222642454151823360, 825394103341888069030641664, 13520781109074923362448234774528

Offset: 0

Johannes W. Meijer, Oct 10 2010

The a(n) represent the "Row2" sums, see A180662, of the powers-of-2 triangle A000079. This sequence is related to the Jacobsthal and triangular numbers.

f[k_] := (2^k) (-1)^(k + 1)
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]    (* A181174 signed *)
(* Clark Kimberling, Dec 30 2011 *)

a(n) = 2^(n*(n+1)/2)*(2^(n+1)+(-1)^n)/3.

a(n) = A001045(n+1)*2^A000217(n).

A269383 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260738(n+1)-1) * ((2 * a(A260739(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 12, 31, 14, 29, 128, 63, 256, 25, 18, 19, 512, 21, 24, 127, 30, 33, 20, 23, 1024, 61, 26, 27, 2048, 57, 4096, 255, 22, 125, 8192, 511, 28, 49, 34, 35, 16384, 37, 48, 1023, 62, 41, 40, 47, 32768, 253, 58, 59, 36, 65, 65536, 39, 126, 45, 131072, 2047, 96, 121, 50, 51, 262144, 53, 60

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269384.
Cf. A000035, A000079, A260738, A260739.
Cf. also A249813, A269373.
Differs from both A246683 and A249813 for the first time at n=18, which here a(18)=12, instead of 128.

a(1) = 1, a(n) = A000079(A260738(n+1)-1) * ((2 * a(A260739(n+1))) - 1).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A082902 a(n) = gcd(2^n, sigma(2,n)) = gcd(A000079(n), A001157(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 2, 4, 2, 2, 2, 1, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 1, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 2, 1, 1, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 2, 1, 4, 4, 2, 2, 4, 4, 2, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 1, 2, 1, 2, 4, 2, 2, 8

Offset: 1

Labos Elemer, Apr 22 2003

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079, A001157, A006519.

a[n_] := 2^IntegerExponent[DivisorSigma[2, n], 2]; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

A082902(n) = gcd(2^n, sigma(n, 2)); \\ Antti Karttunen, Sep 27 2018

a(n) = A006519(A001157(n)). - Antti Karttunen, Sep 27 2018

Multiplicative with a(2^e) = 1, and a(p^e) = A006519(e+1) for an odd prime p. - Amiram Eldar, Oct 01 2023

A095361 Ratio A095110(n)/A000079(n-2) rounded down.

Original entry on oeis.org

2, 6, 12, 20, 37, 52, 106, 200, 369, 657, 1226, 2191, 4268, 8076, 15291, 28931, 55124, 105982, 202482, 391505

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

This is the average length of maximum Motzkin path prefix (i.e. non-diving portion) found in the "Jacobi-vectors" of all 4k+3 integers in range ]2^n,2^(n+1)]. See A095269-A095270.

The ratios before rounding are: 2, 6, 12, 20, 37.875, 52.9375, 106.28125, 200.25, 369.179687, 657.445312, 1226.675781, 2191.126953, 4268.283691, 8076.054443, 15291.317139, 28931.598755, 55124.513184, 105982.564758, 202482.488968, 391505.689705.

A095362 gives the same ratios rounded to nearest integer. A095357 gives similar ratios computed only for 4k+3 primes.

a(1) = 2, a(n) = floor(A095110(n)/A000079(n-2)) for n > 1.

A268675 Self-inverse permutation of natural numbers: a(1) = 1; a(n) = A000079(A193231(A007814(n))) * A250469(a(A268674(n))).

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 4, 21, 10, 11, 24, 13, 14, 15, 32, 17, 42, 19, 40, 9, 22, 23, 12, 55, 26, 27, 56, 29, 30, 31, 16, 69, 34, 35, 168, 37, 38, 39, 20, 41, 18, 43, 88, 93, 46, 47, 96, 91, 110, 123, 104, 53, 54, 25, 28, 117, 58, 59, 120, 61, 62, 63, 64, 65, 138, 67, 136, 33, 70, 71, 84, 73, 74, 75, 152, 77, 78, 79, 160

Offset: 1

Antti Karttunen, Feb 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1914
Index entries for sequences that are permutations of the natural numbers

Cf. A000079, A000265, A007814, A193231, A250469, A268674.

Cf. also A268385.

a(1) = 1, and for n > 1, a(n) = A000079(A193231(A007814(n))) * A250469(a(A268674(n))).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A309530 Number of power-of-two-divisors of sum of divisors of sum of divisors of powers of two: a(n) = A001511(A051027(A000079(n))).

Original entry on oeis.org

1, 3, 4, 4, 6, 4, 8, 5, 5, 10, 5, 6, 14, 12, 12, 6, 18, 10, 20, 11, 9, 9, 6, 8, 8, 18, 6, 15, 7, 16, 32, 7, 11, 22, 17, 14, 7, 24, 22, 13, 5, 13, 11, 12, 20, 10, 7, 11, 9, 16, 33, 22, 6, 10, 15, 17, 28, 12, 6, 20, 62, 36, 12, 9, 24, 16, 5, 26, 12, 26, 10, 18, 6, 12, 16, 28, 19, 26

Offset: 0

Juri-Stepan Gerasimov, Aug 06 2019

nonn

			a(0) = A001511(A051027(A000079(0))) = A001511(A051027(A000079(2^0))) = A001511(A051027(1)) = A001511(1) = 1.

Juri-Stepan Gerasimov, x = A001511(A051027(A000079(x))), SeqFan list, Aug 19 2019.

Cf. A000043 (numbers m such that m - 1 divides a(m - 1) - 2), A000079, A001511, A051027, A090748.

[Valuation(2*SumOfDivisors(SumOfDivisors(2^n)),2): n in [0..89]];

a(n) = valuation(2*sigma(sigma(2^n)), 2); \\ Michel Marcus, Aug 06 2019

from sympy import divisor_sigma
def A309530(n): return ((m:=int(divisor_sigma((1<Chai Wah Wu, Jul 13 2022

a(n) = A001511(A051027(A000079(n))).

A378144 a(n) = P(n) * 2^floor(log_2(prime(n+1))) = A002110(n) * A000079(A098388(n+1)).

Original entry on oeis.org

1, 4, 24, 120, 1680, 18480, 480480, 8168160, 155195040, 3569485920, 103515091680, 6417935684160, 237463620313920, 9736008432870720, 418648362613440960, 19676473042831725120, 1042853071270081431360, 61528331204934804450240, 7506456407002046142929280, 502932579269137091576261760

Offset: 0

Michael De Vlieger, Nov 17 2024

Last term in row n of A378133.

a(n) is the largest product of a power of 2 and A002110(n) less than A002110(n+1).

Michael De Vlieger, Table of n, a(n) for n = 0..349

Cf. A000079, A002110, A060735, A098388, A378133.

{1}~Join~Table[Product[Prime[i], {i, n}]*2^Floor[Log2[Prime[n + 1]]], {n, 120}]

from sympy import primorial, prime
def A378144(n): return primorial(n)<Chai Wah Wu, Nov 19 2024

a(n) = A002110(n)*A000079(A098388(n+1)).

A092176 A067076 + A000079/2.

Original entry on oeis.org

0, 1, 3, 6, 12, 21, 39, 72, 138, 269, 526, 1041, 2067, 4116, 8214, 16409, 32796, 65565, 131104, 262178, 524323, 1048614, 2097192, 4194347, 8388655, 16777265, 33554482, 67108916, 134217781

Offset: 0

David G. Williams (davwill24(AT)aol.com), Apr 03 2004

Cf. A067076, A000079.

cp's OEIS Frontend

A153279 Eigentriangle by rows, T(n,k) = A000079(n-k) * (diagonalized matrix of (1,1,3,9,27,81,...)).

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A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).

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A181174 The "Row2" sums of the powers-of-2 triangle A000079.

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Mathematica

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A269383 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260738(n+1)-1) * ((2 * a(A260739(n+1))) - 1).

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A082902 a(n) = gcd(2^n, sigma(2,n)) = gcd(A000079(n), A001157(n)).

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Mathematica

PARI

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A095361 Ratio A095110(n)/A000079(n-2) rounded down.

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A268675 Self-inverse permutation of natural numbers: a(1) = 1; a(n) = A000079(A193231(A007814(n))) * A250469(a(A268674(n))).

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A309530 Number of power-of-two-divisors of sum of divisors of sum of divisors of powers of two: a(n) = A001511(A051027(A000079(n))).

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A378144 a(n) = P(n) * 2^floor(log_2(prime(n+1))) = A002110(n) * A000079(A098388(n+1)).

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