A091072 -id:A091072 - cp's OEIS frontend

A173256 Partial sums of A001481.

0, 1, 3, 7, 12, 20, 29, 39, 52, 68, 85, 103, 123, 148, 174, 203, 235, 269, 305, 342, 382, 423, 468, 517, 567, 619, 672, 730, 791, 855, 920, 988, 1060, 1133, 1207, 1287, 1368, 1450, 1535, 1624, 1714, 1811, 1909, 2009, 2110, 2214, 2320, 2429, 2542, 2658, 2775

Offset: 1

Jonathan Vos Post, Feb 14 2010

nonn

The subsequence of primes in this sequence begins 3, 7, 29, 103, 269, 619, 1811, 3271.

			a(66) = 0 + 1 + 2 + 4 + 5 + 8 + 9 + 10 + 13 + 16 + 17 + 18 + 20 + 25 + 26 + 29 + 32 + 34 + 36 + 37 + 40 + 41 + 45 + 49 + 50 + 52 + 53 + 58 + 61 + 64 + 65 + 68 + 72 + 73 + 74 + 80 + 81 + 82 + 85 + 89 + 90 + 97 + 98 + 100 + 101 + 104 + 106 + 109 + 113 + 116 + 117 + 121 + 122 + 125 + 128 + 130 + 136 + 137 + 144 + 145 + 146 + 148 + 149 + 153 + 157 + 160 = 4876.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001481, A022544, A004018, A000161, A002654, A064533, A000404, A002828, A000378, A025284-A025320, A125110, A091072.

N:= 1000:
A001481:= sort(convert({seq(seq(x^2+y^2, y=0..floor(sqrt(N-x^2))),x=0..floor(sqrt(N)))},list)):
ListTools:-PartialSums(A001481); # Robert Israel, Mar 15 2016

from itertools import count, accumulate, islice
from sympy import factorint
def A173256_gen(): # generator of terms
    return accumulate(filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0)))
A173256_list = list(islice(A173256_gen(),30)) # Chai Wah Wu, Jun 27 2022

a(n) = Sum_{i=1..n} A001481(i) = Sum_{i=1..n} (numbers that are the sum of 2 nonnegative squares) = Sum_{i=1..n} (numbers n such that i = x^2 + y^2 has a solution in nonnegative integers x, y).

a(21) corrected by Robert Israel, Mar 15 2016

A225836 Numbers of form 2^j*(4k+1), j >= 0, k >= 1.

Original entry on oeis.org

5, 9, 10, 13, 17, 18, 20, 21, 25, 26, 29, 33, 34, 36, 37, 40, 41, 42, 45, 49, 50, 52, 53, 57, 58, 61, 65, 66, 68, 69, 72, 73, 74, 77, 80, 81, 82, 84, 85, 89, 90, 93, 97, 98, 100, 101, 104, 105, 106, 109, 113, 114, 116, 117, 121, 122, 125, 129, 130, 132, 133

Offset: 1

Ralf Stephan, May 16 2013

A091072 without the powers of 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A091067.

[n: n in [1..200] | d gt 1 and IsOne(d mod 4) where d is n div 2^Valuation(n,2)]; // Bruno Berselli, May 16 2013

mx = 149; t = {}; Do[n = 2^j (4 k + 1); If[n <= mx, AppendTo[t, n]], {j, 0, Log[2, mx]}, {k, mx/4}]; Union[t] (* T. D. Noe, May 16 2013 *)

for(n=1,200,t=n/2^valuation(n,2);if(t>1&&(t%4==1),print1(n,",")))

is(n)=n>>=valuation(n,2); n%4==1 && n>1 \\ Charles R Greathouse IV, Sep 27 2015

list(lim)=my(v=List(),t); forstep(n=5,lim,4, listput(v,t=n); while((t<<=1)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Sep 27 2015

a(n) ~ 2n. - Charles R Greathouse IV, Sep 27 2015

A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

Original entry on oeis.org

1, 3, 2, 6, 4, 7, 8, 12, 5, 11, 9, 14, 16, 15, 13, 19, 21, 22, 10, 23, 17, 24, 18, 27, 29, 30, 26, 35, 52, 38, 25, 31, 32, 28, 33, 47, 34, 46, 20, 39, 40, 44, 36, 43, 37, 48, 41, 75, 53, 59, 61, 54, 57, 55, 58, 60, 64, 51, 65, 56, 66, 62, 50, 71, 45, 78, 42

Offset: 1

Rémy Sigrist, Jun 28 2017

A144795 gives the numbers without isolated 1's in base-2 representation.
This sequence is conjectured to be a permutation of the natural numbers.
This sequence has similarities with A269361: here we require that the product of two consecutive terms has no isolated 1, there the product of two consecutive terms has only isolated 1's, in base-2 representation.
For any k > 0:
- a(2*k-1) belongs to A091072,
- a(2*k) belongs to A091067.

			The first terms, alongside a(n)*a(n+1) in binary, are:
n       a(n)    a(n)*a(n+1) in binary
--      ----    ---------------------
1       1       11
2       3       110
3       2       1100
4       6       11000
5       4       11100
6       7       111000
7       8       1100000
8       12      111100
9       5       110111
10      11      1100011
11      9       1111110
12      14      11100000
13      16      11110000
14      15      11000011
15      13      11110111
16      19      110001111
17      21      111001110
18      22      11011100
19      10      11100110
20      23      110000111

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A289194

Cf. A091067, A091072, A144795, A269361.

a = {1}; Do[k = 1; While[Nand[! MemberQ[a, k], ! MemberQ[Length /@ DeleteCases[Split[IntegerDigits[k Last[a], 2]], s_ /; First@ s == 0], 1]], k++]; AppendTo[a, k], {n, 2, 67}]; a (* Michael De Vlieger, Jun 29 2017 *)

A379925 Numbers k for which nonnegative integers x and y exist such that x^2 + y^2 = k and x + y is a square.

Original entry on oeis.org

0, 1, 8, 10, 16, 41, 45, 53, 65, 81, 128, 130, 136, 146, 160, 178, 200, 226, 256, 313, 317, 325, 337, 353, 373, 397, 425, 457, 493, 533, 577, 625, 648, 650, 656, 666, 680, 698, 720, 746, 776, 810, 848, 890, 936, 986, 1040, 1098, 1160, 1201, 1205, 1213, 1225, 1226

Offset: 1

Felix Huber, Jan 25 2025

Numbers k for which exists at least one solution to k = x^2 + (z^2 - x)^2 in integers x and z with x >= 0 and z >= sqrt(2*x).

Subsequence of A001481.

			10 is in the sequence because 10 = 1^2 + 3^2 and 1 + 3 = 2^2.
81 is in the sequence because 81 = 0^2 + 9^2 and 0 + 9 = 3^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Sum of two squares theorem

Cf. A000161, A000290, A001481, A004018, A025284-A025320, A091072, A125110, A118882, A125022, A380072, A380073, A380074.

# Calculates the first 10005 terms.
A379925:=proc(K)
    local i,j,L;
    L:={};
    for i from 0 to floor(sqrt((K+1)^2)/2) do
        for j from 0 to floor(sqrt((K+1)^2/2-i^2)) do
            if issqr(i+j) then
                L:=L union {i^2+j^2}
            fi
        od
    od;
    return op(L)
end proc;
A379925(1737);

isok(n)=my(x=0, r=0); while(x<=sqrt(n) && r==0, if(issquare(n-x^2) && issquare(x+sqrtint(n-x^2)), r=1); x++); r; \\ Michel Marcus, Feb 10 2025

k = m^(4*j) is in the sequence for nonnegative integers m and j (not both 0) because x = 0 and z = m^j is a solution to m^(4*j) = x^2 + (z^2 - x)^2.

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A173256 Partial sums of A001481.

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A225836 Numbers of form 2^j*(4k+1), j >= 0, k >= 1.

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A289194 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has no isolated 1 in its base-2 representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A379925 Numbers k for which nonnegative integers x and y exist such that x^2 + y^2 = k and x + y is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula