A177713 -id:A177713 - cp's OEIS frontend

A124865 Numbers of the form p^2-q^2 with p > q prime.

5, 16, 21, 24, 40, 45, 48, 72, 96, 112, 117, 120, 144, 160, 165, 168, 192, 240, 264, 280, 285, 288, 312, 336, 352, 357, 360, 408, 432, 480, 504, 520, 525, 528, 552, 600, 648, 672, 720, 768, 792, 816, 832, 837, 840, 888, 912, 936, 952, 957, 960, 1008, 1032, 1080

Offset: 1

Alexander Adamchuk, Nov 10 2006

nonn

The only prime term is a(1) = 5.
All odd terms are of the form p^2-4.
All even terms are divisible by 8.
Numbers of the form (p^2-q^2)/8 (p, q odd primes, p>q) are listed in A124866.
Elliott & Richner call these "ans numbers". - Charles R Greathouse IV, Feb 17 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
N. E. Elliott and D. Richner, An investigation of the set of ans numbers, Missouri J. of Math. Sci. 15 (2003), pp. 189-199.
Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.

Apart from a(1), a subsequence of A177713.
Cf. A045636 (numbers of the form p^2+q^2, p, q primes).
Cf. A124866 (numbers of the form (p^2-q^2)/8, p, q odd primes, p>q).

With[{nn=60},Take[Union[#[[2]]^2-#[[1]]^2&/@Subsets[Prime[Range[nn]],{2}]],nn]] (* Harvey P. Dale, Aug 21 2015 *)

is(n)=if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4,d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0) \\ Charles R Greathouse IV, Feb 17 2014

a(n) >> n log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

Original entry on oeis.org

5, 6, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 53, 54, 55, 56, 57, 60, 61, 62, 63, 65, 66, 69, 70, 72, 73, 75, 76, 77, 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form sum_{i=j..2k+1} i where j>=1 and 2k+1>j and k>=1. Numbers of the form (2k+1+j)*(2k+2-j)/2, j>=1, k>=1, 2k+1>j. - R. J. Mathar, Dec 04 2011
Subsequences include the A000384 where >=6, the A014106 where >=5, A071355 where >=12, A130861 where >=9, A139577 where >=13, A139579 where >=17 etc. The sequence is the union of all odd-indexed rows of A141419, except its first column and numbers <=3: {5,6}, {9,12,14,15}, {13,18,22,25,27,28}, ... - R. J. Mathar, Dec 04 2011
Does this sequence have asymptotic density 1? - Robert Israel, Nov 27 2018

			5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...

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

Cf. A177712, A136724, A177713.
Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.
Contains A004766, A017137 and nonzero terms of A008588.
Disjoint from A002145.
Subsequence of A138591.

f:= proc(n) local r,k;
  for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
    k:= (r + 2*n/r - 3)/4;
    if k::posint and r >= 2*k+2 then return true fi
  od:
  false
end proc:
select(f, [$1..1000]); # Robert Israel, Nov 27 2018

z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1

A177732 The sums of two or more consecutive positive numbers, the largest being even.

Original entry on oeis.org

3, 7, 9, 10, 11, 15, 18, 19, 20, 21, 23, 26, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 43, 45, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 60, 63, 66, 67, 68, 69, 70, 71, 72, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 87, 90, 91, 93, 95, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form (j+2l)*(2l-j+1)/2 with j>=1 and 2l>j. Subsequences are A014105 where >=3, (j=1), A014107 where >=9 (j=2). - R. J. Mathar, Jul 14 2012

			3=1+2, 7=3+4, 9=2+3+4, 10=1+2+3+4, 11=5+6,..

Cf. A177712, A177713, A177731.

z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}];Union@lst2
With[{upto=108},Select[Union[Flatten[Table[Accumulate[Range[2n-1,1,-1]]+ 2n,{n,upto/4}]]],#<=upto&]] (* Harvey P. Dale, May 19 2019 *)

A177733 Integers that can be expressed as the sum of two or more positive consecutive numbers (the largest being even) AND also as the sum of two or more positive consecutive numbers (the largest being odd).

Original entry on oeis.org

9, 15, 18, 21, 27, 30, 33, 35, 36, 39, 42, 45, 49, 51, 54, 55, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 91, 93, 95, 98, 99, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 126, 129, 132, 133, 135, 138, 140, 141, 143, 144, 147, 150, 153, 154

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Intersection of A177732 and A177731.
From Robert Israel, May 02 2023: (Start)
Numbers k with odd divisors d_1, d_2 >= 2 such that k + (d_1+1)/2 is odd and
k + (d_2+1)/2 is even.
Contains no primes, powers of 2 or products of a prime and a power of 2.
Contains odd semiprime p*q iff at least one of p and q == 3 (mod 4).
(End)

			9 is in the sequence because 2+3+4=9=4+5.
15 is in the sequence because 7+8=15=1+2+3+4+5.

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

Cf. A000217, A177713.

filter:= proc(n) local a,b,x,y,todd,teven;
   todd:= false; teven:= false;
   for a in select(type,numtheory:-divisors(n),odd) minus {1} do
     b:= 2*n/a;
     x:= (a+b+1)/2;
       if x::odd then todd:= true; if teven then return true fi
       else teven:= true; if todd then return true fi
     fi od:
  false
end proc:
select(filter, [$1..200]); # Robert Israel, May 01 2023

z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1; z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}]; Intersection[lst1,lst2]

A357928 a(n) is the smallest c for which (s+c)^2-n is a square, where s = floor(sqrt(n)), or -1 if no such c exists.

Original entry on oeis.org

0, 0, -1, 1, 0, 1, -1, 2, 1, 0, -1, 3, 1, 4, -1, 1, 0, 5, -1, 6, 2, 1, -1, 8, 1, 0, -1, 1, 3, 10, -1, 11, 1, 2, -1, 1, 0, 13, -1, 2, 1, 15, -1, 16, 6, 1, -1, 18, 1, 0, -1, 3, 7, 20, -1, 1, 2, 4, -1, 23, 1, 24, -1, 1, 0, 1, -1, 26, 10, 5, -1, 28, 1, 29, -1, 2, 12, 1, -1, 32

Offset: 0

Darío Clavijo, Oct 20 2022

sign

c exists iff n != 2 (mod 4), and it allows n to be written as the difference of two perfect squares.
This gives a factorization n = x*y where x and y may or may not be primes: let s = floor(sqrt(n)), u = a(n) + s and v = u^2 - n; then w = sqrt(v), x = u - w, y = u + w and x*y == n.
The Fermat factorization algorithm seeks such a form, starting from s, so that a(n) is the number of steps it must take for n != 2 (mod 4).
a(n) >= 1 if n is not square and is writable as a difference of squares.
a(n) = 0 if n is square.
a(n) = -1 if n is not writable as a difference of squares.

			n   prime  square  n == 2 (mod 4)   c   s  v=(s+c)^2-n   u   w   x   y  x*y
--  -----  ------  --------------  --  --  -----------  --  --  --  --  ---
76      F       F               F  12   8          324  20  68   2  38   76
13      T       F               F   4   3           36   7   6   1  13   13
25      F       T               F   0   0            0   5   0   5   5   25
7       T       F               T  -1   -            -   -   -   -   -    -

Wikipedia, Fermat's factorization method

Cf. A177713, A037074.

a(n) = if ((n%4)==2, -1, my(s=sqrtint(n), c=0); while (!issquare((s+c)^2-n), c++); c); \\ Michel Marcus, Oct 24 2022

from gmpy2 import *
def fermat(n):
    a, rem = isqrt_rem(n)
    b2 = -rem
    c0 = (a << 1) + 1
    c = c0
    while not is_square(b2):
        b2 += c
        c += 2
    return (c-c0) >> 1
def A357928(n):
  if is_square(n):
      return 0
  elif ((n-2) % 4) != 0:
      return fermat(n)
  else:
      return -1

from math import isqrt
from itertools import takewhile
from sympy import divisors
def A357928(n): return -1 if n&3==2 else min((m>>1 for d in takewhile(lambda d:d**2<=n,divisors(n)) if not((m:=n//d+d) & 1)),default=0) - isqrt(n) # Chai Wah Wu, Oct 26 2022

A357945 Numbers k which are not square but D = (b+c)^2 - k is square, where b = floor(sqrt(k)) and c = k - b^2.

Original entry on oeis.org

5, 13, 28, 65, 69, 76, 125, 128, 189, 205, 300, 305, 325, 352, 413, 425, 532, 533, 544, 565, 693, 725, 793, 828, 860, 1025, 1036, 1045, 1105, 1141, 1248, 1449, 1469, 1504, 1525, 1708, 1885, 1917, 1965, 2125, 2240, 2353, 2380, 2501, 2533, 2548, 2812, 2816, 2825, 2829, 2844, 2873, 2893

Offset: 1

Darío Clavijo, Oct 21 2022

nonn

All composite terms are included in A177713.

Terms are the difference of two perfect squares k = (b+c)^2 - d^2, where d = sqrt(D), and so if composite are factorizable by Fermat's method k = ((b+c) + d) * ((b+c) - d).

			8525 is a term since it's not square and b = floor(sqrt(k)) = 92 and c = k - b^2 = 61 gives D = (b+c)^2 - k = 14884 which is square (122^2).

Subsequence of A042965 and of A000037.
A211412 is a subsequence.
Cf. A053186, A177713.

isok(k) = if (!issquare(k), my(b=sqrtint(k), c=k-b^2); issquare((b+c)^2 - k)); \\ Michel Marcus, Oct 23 2022

from gmpy2 import *
def is_A357945(n):
  if not is_square(n):
    b,c = isqrt_rem(n)
    return is_square(c*(2*b+c-1))
  else:
    return False

1.6*n < a(n) <= 4n^4 + 1. (Improvements welcome!) - Charles R Greathouse IV, Oct 23 2022

cp's OEIS Frontend

A124865 Numbers of the form p^2-q^2 with p > q prime.

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A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

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A177732 The sums of two or more consecutive positive numbers, the largest being even.

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A177733 Integers that can be expressed as the sum of two or more positive consecutive numbers (the largest being even) AND also as the sum of two or more positive consecutive numbers (the largest being odd).

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A357928 a(n) is the smallest c for which (s+c)^2-n is a square, where s = floor(sqrt(n)), or -1 if no such c exists.

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A357945 Numbers k which are not square but D = (b+c)^2 - k is square, where b = floor(sqrt(k)) and c = k - b^2.

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