A051179 -id:A051179 - cp's OEIS frontend

A155877 Sums of three Fermat numbers.

9, 11, 13, 15, 23, 25, 27, 37, 39, 51, 263, 265, 267, 277, 279, 291, 517, 519, 531, 771, 65543, 65545, 65547, 65557, 65559, 65571, 65797, 65799, 65811, 66051, 131077, 131079, 131091, 131331, 196611, 4294967303, 4294967305, 4294967307

Offset: 1

Jonathan Vos Post, Jan 29 2009

nonn

			a(1) = 3 + 3 + 3 = 9.
a(2) = 3 + 3 + 5 = 11.
a(3) = 3 + 5 + 5 = 13.
a(4) = 5 + 5 + 5 = 15.
a(5) = 3 + 3 + 17 = 23.
a(6) = 3 + 5 + 17 = 25.
a(7) = 5 + 5 + 17 = 27.
a(8) = 3 + 17 + 17 = 37.
a(9) = 5 + 17 + 17 = 39.
a(10) = 17 + 17 + 17 = 51.
a(11) = 3 + 3 + 257 = 263.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A000215, A019434, A050922, A051179, A059919, A063486, A073617, A078303, A078304, A085866.

{(2^(2^a) + 1) + (2^(2^b) + 1) + (2^(2^c) + 1)} = {A000215(a) + A000215(b) + A000215(c)}.

More terms from R. J. Mathar, Feb 06 2009

A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

Original entry on oeis.org

1, 3, 1, 15, 5, 3, 9, 255, 85, 51, 153, 15, 165, 195, 105, 65535, 21845, 13107, 39321, 3855, 42405, 50115, 26985, 255, 43605, 52275, 26265, 61455, 23205, 15555, 38505, 4294967295, 1431655765, 858993459, 2576980377, 252645135, 2779096485, 3284386755

Offset: 0

Tilman Piesk, Aug 24 2013

T(n,k) is row k of the negated binary Walsh matrix of size 2^n read as reverse binary number. The left digit is always 1, so all entries are odd.
Most of these numbers are divisible by Fermat numbers (A000215): All entries in all rows beginning with row n are divisible by F_(n-1), except the entries 2^(n-1)...2^n-1. (This is the same in A228539.)
Divisibility by Fermat numbers:
All entries in rows n >= 1 are divisible by F_0 =  3, except those with k = 1.
All entries in rows n >= 3 are divisible by F_2 = 17, except those with k = 4..7.

			Negated binary Walsh matrix of size 4 and row 2 of the triangle:
1 1 1 1        15
1 0 1 0         5
1 1 0 0         3
1 0 0 1         9
Triangle starts:
      k  =  0     1     2     3    4     5     6     7   8     9    10    11 ...
n
0           1
1           3     1
2          15     5     3     9
3         255    85    51   153   15   165   195   105
4       65535 21845 13107 39321 3855 42405 50115 26985 255 43605 52275 26265 ...

Tilman Piesk, Rows 0..8 of the triangle, flattened
Tilman Piesk, Prime factorizations
Tilman Piesk, Negated binary Walsh matrix of size 256

A228539 (the same for the binary Walsh matrix, not negated)
A197818 (antidiagonals of the negated binary Walsh matrix converted to decimal).
A000215 (Fermat numbers), A023394 (Prime factors of Fermat numbers).

T(n,k) + A228539(n,k) = 2^2^n - 1
T(n,0) = A051179(n)
T(n,2^n-1) = A122569(n+1)
A211344(n,k) = T(n,2^(n-k))

A057755 Number of digits in n-th Fermat number (A000215).

Original entry on oeis.org

1, 1, 2, 3, 5, 10, 20, 39, 78, 155, 309, 617, 1234, 2467, 4933, 9865, 19729, 39457, 78914, 157827, 315653, 631306, 1262612, 2525223, 5050446, 10100891, 20201782, 40403563, 80807125, 161614249, 323228497, 646456994, 1292913987, 2585827973

Offset: 0

Robert G. Wilson v, Oct 30 2000

Also number of digits of A001146(n) and A051179(n). - Michel Marcus, Dec 21 2018

			a(6) = 20 because 2^(2^6) + 1 = 18446744073709551617 which is a twenty-digit number.

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus, an imprint of Springer-Verlag, NY, 1995, page 139.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 100 terms from Jinyuan Wang)
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A001146, A051179.

List([0..18],n->Size(ListOfDigits(2^(2^n)+1))); # Muniru A Asiru, Dec 20 2018

[Floor(2^n*Log(10,2)/Log(10,10))+1: n in [0..40]]; // Vincenzo Librandi, Nov 08 2018

seq(length(2^(2^n)),n=0..20); # Zerinvary Lajos, Apr 20 2008

Table[ Floor[ 2^n * N[ Log[ 10, 2 ], 24 ] + 1 ], {n, 0, 43} ]

for(n=0, 50, print(n, " ", floor(2^n*log(2)/log(10))+1); ) \\ Jinyuan Wang, Nov 07 2018

a(n) = floor(log_10(F_n)+1) (F_n is the n-th Fermat number). - Ivan Panchenko, Sep 06 2009

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

A236693 Numbers k such that 2^sigma(k) == 1 (mod k).

Original entry on oeis.org

1, 3, 15, 35, 51, 65, 105, 119, 195, 255, 315, 323, 357, 377, 455, 459, 585, 595, 663, 969, 1045, 1071, 1105, 1131, 1189, 1365, 1455, 1469, 1485, 1547, 1615, 1785, 1799, 1885, 1887, 1911, 2261, 2295, 2385, 2639, 2795, 2907, 3135, 3145, 3185, 3213, 3315, 3339

Offset: 1

Joseph L. Pe, Jan 30 2014

This sequence is infinite since A051179(n) is a term. - Jinyuan Wang, Mar 13 2020

			2^sigma(15) = 2^24 = 16777216 is congruent to 1 (mod 15), so 15 is a term of the sequence.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Florian Luca, Positive integers n such that n| a^sigma(n) - 1, Novi Sad Journal of Mathematics, Vol 33, No. 2 (2003), pp. 49-66.

Supersequence of A015715.

Cf. A000203, A051179, A020492, A276238.

l = {1};
For[i = 1, i <= 10^4, i++,
    If[Mod[2^DivisorSigma[1, i], i] == 1, l = Append[l, i]]];
l

s=[1]; for(n=1, 10000, if(2^sigma(n)%n==1, s=concat(s, n))); s \\ Colin Barker, Jan 30 2014

isok(n) = Mod(2, n)^sigma(n)==1; \\ Altug Alkan, Sep 19 2017

a(1) = 1 added by Amiram Eldar, Sep 19 2017

A345135 Number of ordered rooted binary trees with n leaves and with minimal Sackin tree balance index.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 6, 4, 1, 8, 28, 56, 70, 56, 28, 8, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 32, 496, 4960, 35960, 201376, 906192, 3365856, 10518300, 28048800, 64512240, 129024480, 225792840, 347373600, 471435600

Offset: 0

Mareike Fischer, Jun 09 2021

Ordered rooted binary trees are trees with two descendants per inner node where left and right are distinguished.
The Sackin tree balance index is also known as total external path length.
a(0) = 1 by convention.

			a(1) = a(2) = 1 because there are only the trees (o) and (o,o) which get counted. a(3) = 2 because the trees ((o,o),o) and (o,(o,o)) get counted. a(4) = 1 because only the tree ((o,o),(o,o)) is counted. Note that the other possible rooted binary ordered trees with four leaves, namely the different orderings of (((o,o),o),o), are not Sackin minimal. a(5) = 4 because the following trees get counted: (((o,o),o),(o,o)), ((o,(o,o)),(o,o)), ((o,o),((o,o),o)), ((o,o),(o,(o,o))).

Alois P. Heinz, Table of n, a(n) for n = 0..4096
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index. Ann. Comb. 25, 515-541 (2021), Theorem 7.

Cf. A000079, A000108, A011782, A037293, A051179, A053644, A299037.

a:= n-> (b-> binomial(b, n-b))(2^ilog2(n)):
seq(a(n), n=0..46);  # Alois P. Heinz, Jun 09 2021

a[0] := 1; a[n_] := Module[{k = 2^(BitLength[n] - 1)}, Binomial[k, n - k]];
Table[a[n], {n, 0, 46}]

a(n) = binomial(2^(ceiling(log_2(n))-1),n-2^(ceiling(log_2(n))-1)).
a(n) = binomial(A053644(n),n-A053644(n)).
a(2^n) = 1.
a(2^n-1) = A011782(n).
a(2^n+1) = A000079(n).
From Alois P. Heinz, Jun 09 2021: (Start)
max({ a(k) | k = 2^n..2^(n+1) }) = A037293(n).
Sum_{i=2^n..2^(n+1)-1} a(i) = 2^(2^n) - 1 = A051179(n). (End)
Conjecture: Sum_{n>=0} a(n)*x^n = 1 + x + Sum_{n>=0} x^(2^n) * ((1 + x)^(2^n) - 1). - Paul D. Hanna, Jul 18 2024

A211344 Atomic Boolean functions interpreted as binary numbers.

Original entry on oeis.org

1, 3, 5, 15, 51, 85, 255, 3855, 13107, 21845, 65535, 16711935, 252645135, 858993459, 1431655765, 4294967295, 281470681808895, 71777214294589695, 1085102592571150095, 3689348814741910323, 6148914691236517205

Offset: 0

Tilman Piesk, Jul 24 2012

Row n of the triangle shows the atoms among n-ary Boolean functions:
            1                                  01
         3     5                       0011          0101
     15    51     85          00001111      00110011      01010101
Often n-ary x_k = T(n,k), e.g. for 2-ary functions x_1=0011, x_2=0101 and for 3-ary functions x_1=00001111, x_2=00110011, x_3=01010101.
An easier generalized way is the enumeration from right to left, here shown with k starting from 0, so that n-ary x_k = T(n, n-k-1). As numbers in the diagonals on the right have the same bit pattern, this corresponds to the infinitary definition of x_k as a binary fraction 1/A000215(k) = 1/(2^2^k + 1):
2-ary x_0=0101=5, 3-ary x_0=01010101=85, infinitary x_0 = 1/3 = .010101...
2-ary x_1=0011=3, 3-ary x_1=00110011=51, infinitary x_1 = 1/5 = .001100110011...

Tilman Piesk, Table of n, a(n) for n = 0..65
Tilman Piesk, Atomic Boolean functions in Sierpinski triangle (Wikimedia Commons)

A001317, A089633, A051179 (left diagonal)

Seq = sym(zeros(55,1)) ;
Filledlines = 0 ;
for m=1:10
    for n=1:m
        Sum = sym(0) ;
        for k=0:2^m-1
            if mod(  floor( k/2^(m-n) )  ,2) == 0
               Sum = Sum + 2^sym(k) ;
            end
        end
        Seq( Filledlines + n ) = Sum ;
    end
    Filledlines = Filledlines + m ;
end

from itertools import count, islice
def A211344_gen(): # generator of terms
    return (sum((bool(~(m:=(1<A211344_list = list(islice(A211344_gen(),20)) # Chai Wah Wu, May 03 2023

def arity_and_atom_to_integer(arity, atom):
    result = 0
    max_place = (1 << arity) - (1 << atom) - 1
    for exponent in range(max_place + 1):
        if not bool(~max_place & max_place - exponent):
            place_value = 1 << exponent
            result += place_value
    return result
def A211344(n, k):
    return arity_and_atom_to_integer(n, n-k-1) # Tilman Piesk, Jan 25 2025

a = A001317( A089633 )

A218384 Number of nonempty subsets S of the powerset of a set of size n that have the even intersection property.

Original entry on oeis.org

1, 7, 71, 3071, 1966207, 270499994623, 2342736474457787596799, 86772003564839307784895323681111305093119, 59169757600268575861444773339439520883460632949720404019392912099891777942585343

Offset: 1

Michel Marcus, Oct 27 2012

nonn

If A is a set, an element S of P(P(A)) \ {{}} has the even intersection property (eip) if there exists a set B (necessarily nonempty) included in A with |B∩S| even for each s in S.

For instance for S={{},{1}} of A={1,2}, we can take B={2}, and then |{}∩{2}|=0 (even) and |{1}∩{2}|=0 (even), so S has eip.

			For |A|=2, A = {1,2} and P(A) = {{}, {1}, {2}, {1,2}}
S can be
  {{}, {1}, {2}, {1,2}}
  {{}, {1}, {2}}
  {{}, {1}, {1,2}}
  {{}, {2}, {1,2}}
  {{1}, {2}, {1,2}}
  {{}, {1}} has eip, with B={2}
  {{}, {2}} has eip, with B={1}
  {{}, {1,2}} has eip, with B={1,2}
  {{1}, {1,2}}
  {{2}, {1,2}}
  {{1}, {2}}
  {{}} has eip, with B={1,2}
  {{1}} has eip, with B={2}
  {{2}} has eip, with B={1}
  {{1,2}} has eip, with B={1,2}
So we have 7 S with eip.

Tilman Piesk, relationship to Boolean functions (Wikiversity)
V. Scharaschkin, The Odd and Even Intersection Properties, The Electronic Journal of Combinatorics, Volume 18, Issue 1 (2011), #P185.
Steve Wright, Some enumerative combinatorics arising from a problem on quadratic nonresidues, Australas. J. Combin. 44 (2009), 301-315.

Cf. A218383, A051179, A305737.

A218384:=n->1+2*add((-1)^(n-i-1)*(2^(2^i-1)-1)* product((2^(n-j+1)-1)/(2^j-1), j=1..i)*2^binomial(n-i, 2), i=0..n-1): seq(A218384(n), n=1..10); # Wesley Ivan Hurt, Dec 11 2015

Table[1 + 2 Sum[((-1)^(n - i - 1)) (2^(2^i - 1) - 1) Product[(2^(n - j + 1) - 1)/(2^j - 1), {j, 1, i}] 2^Binomial[n - i, 2], {i, 0, n - 1}], {n, 9}] (* Michael De Vlieger, Dec 11 2015 *)

e(m) = {for (n=1, m, v = 1+2*sum(i=0, n-1, ((-1)^(n-i-1))*(2^(2^i-1)-1)* prod(j=1,i,(2^(n-j+1)-1)/(2^j-1))*2^binomial(n-i,2));print1(v, ", "););}

a(n) = 1 + 2*Sum_{i=0..n-1} (-1)^(n-i-1)*(2^(2^i-1)-1)*(Product_{j=1..i} (2^(n-j+1)-1)/(2^j-1)) * 2^binomial(n-i,2).
a(n) ~ 2^(n + 2^(n-1)). - Vaclav Kotesovec, Apr 03 2021
Apparently, a(n) = A051179(n) - A305737(n). - Tilman Piesk, Jan 26 2025

A250405 Numbers k such that all values of Euler phi (A000010) of all divisors of k are pairwise distinct and represent all proper divisors of k+1.

Original entry on oeis.org

1, 3, 15, 255, 65535, 4294967295

Offset: 1

Jaroslav Krizek, Nov 22 2014

Numbers k such that {phi(d) : d|k} = {d : d|(k+1), d<(k+1)} as multisets.
Conjecture: last term is 4294967295.
First six terms coincide with A051179. - Omar E. Pol, Apr 12 2025

			15 is in the sequence because {phi(d) : d|15} = {1, 2, 4, 8} = {d : d|16, d<16}.
2 is not in the sequence because {phi(d) : d|2} = {1, 1}, but {d : d|2, d<2} = {1}.

Subsequence of A250404 and A203966.
Sequence differs from A051179, A050474 and A116518.
Cf. A000010.

[n: n in [1..100000] | ([EulerPhi(d): d in Divisors(n)]) eq ([d: d in Divisors(n+1) | d lt n+1 ])];

Edited and a(6) added by Max Alekseyev, May 04 2024

A305737 Number of subsets S of vectors in GF(2)^n such that span(S) = GF(2)^n.

Original entry on oeis.org

1, 2, 8, 184, 62464, 4293001088, 18446743803209556992, 340282366920938461120638132973980614656, 115792089237316195423570985008687907766497981100801256254562260326801824546816

Offset: 0

Geoffrey Critzer, Jun 22 2018

nonn

Asymptotic to A001146(n) = 2^(2^n).

R. P. Stanley, Enumerative Combinatorics Vol 1, Cambridge, 1997, page 127.

Andrew Howroyd, Table of n, a(n) for n = 0..10
Tilman Piesk, relationship to Boolean functions (Wikiversity)

Cf. A001146, A022166, A158474, A051179.

Table[Sum[QBinomial[n, k, q] (-1)^(n - k) q^Binomial[n - k, 2] (2^(q^k) - 1) /. q -> 2, {k, 0, n}], {n, 0, 8}]

\\ here U(n,k) is A022166(n,k).
U(n,k)={polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
a(n)={sum(k=0, n, U(n,k)*(-1)^(n-k)*2^binomial(n-k,2)*(2^(2^k)-1))} \\ Andrew Howroyd, Mar 01 2020

a(n) = Sum_{k=0..n} A022166(n,k)*(-1)^(n-k)*2^binomial(n-k,2)*(2^(2^k)-1).
Sum_{k=0..n} a(k)* A022166(n,k) = 2^(2^n) - 1. - Geoffrey Critzer, Apr 25 2024
a(n) = Sum_{k=0..n} A158474(n,k) * A051179(n-k). - Tilman Piesk, Mar 12 2025

a(8) corrected by Andrew Howroyd, Mar 01 2020

cp's OEIS Frontend

A155877 Sums of three Fermat numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A057755 Number of digits in n-th Fermat number (A000215).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A236693 Numbers k such that 2^sigma(k) == 1 (mod k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A345135 Number of ordered rooted binary trees with n leaves and with minimal Sackin tree balance index.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A211344 Atomic Boolean functions interpreted as binary numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

Python

Python

Formula