A017077 -id:A017077 - cp's OEIS frontend

A354938 Row 8 of A354940: Numbers k for which A345992(k) = 8, divided by 8.

3, 9, 11, 17, 19, 25, 27, 33, 41, 43, 49, 57, 59, 67, 73, 81, 83, 89, 97, 105, 107, 113, 121, 129, 131, 137, 139, 145, 161, 163, 169, 177, 179, 185, 193, 201, 209, 211, 217, 225, 227, 233, 241, 243, 249, 251, 257, 281, 283, 289, 297, 305, 307, 313, 321, 329, 331, 337, 345, 347, 353, 361, 377, 379, 393, 401, 409, 417

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Apparently, all terms are either of the form 8k+1 (in A017077) or 8k+3 (in A017101).

Row 8 of A354940.

Cf. A017077, A017101, A344005, A345992.

q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 8*n], m++]; GCD[8*n, m] == 8]; Select[Range[420], q] (* Amiram Eldar, Jun 17 2022 *)

PARI
```
A354938(n) = A354940sq(8,n);
```

A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

Original entry on oeis.org

1, 4, 6, 7, 9, 10, 15, 16, 17, 22, 23, 24, 25, 26, 28, 31, 33, 36, 38, 39, 40, 41, 42, 47, 49, 54, 55, 57, 58, 60, 63, 64, 65, 68, 70, 71, 73, 74, 79, 81, 86, 87, 88, 89, 90, 92, 95, 96, 97, 100, 102, 103, 104, 105, 106, 111, 112, 113, 118, 119, 121, 122, 124, 127, 129

Offset: 1

Peter Munn, Feb 13 2024

Construction by subgroup generation: (Start)
The set of numbers congruent to 1 modulo 8 (A017077) contains all the odd squares and generates a subgroup of the positive rational numbers (under multiplication) that contains no additional integers. The subgroup has an infinite number of cosets. The rest of the construction process extends the subgroup, reducing the number of cosets to 2, by choosing additional generators that are semiprime.
First we extend the subgroup to include all nonzero integer squares. As we already have the odd squares, we need only add 4, the square of the smallest prime, as a generator. The extended subgroup has only 8 cosets and its integer members are listed in A234000. To achieve a subgroup with 2 cosets we now add squarefree semiprime generators. The 2 smallest, 6 and 10, suffice.
The resulting subgroup has this sequence's terms as its integer members.
(End)
The equivalent process starting with numbers congruent to 1 modulo 3 (or 1 modulo 6) produces A189715. If we take its intersection with this sequence we get A370268, which starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). Similarly, if we start with numbers congruent to 1 modulo 5 (or 1 modulo 10) and take the resulting set's intersection with this sequence we get a set starting with the first 32 nonzero numbers of the form x^2 - 10y^2 (see A242664).
The construction process leads to a number of properties:
- The sequence is closed under multiplication and all integer ratios between terms are in the sequence.
- The sequence and its complement have the property that the terms of one can be generated by halving the even terms of the other. Each has asymptotic density 1/2.
Numbers whose squarefree part is congruent to {1,7} mod 8 or {6,10} mod 16.

			7 is prime, so 7 is its only prime factor, which has the form 8m-1. So 7 has an even number (zero) of prime factors not of the form 8m+-1, and therefore is in the sequence. In terms of the subgroup generators described at the start of the comments, (13*8+1) * 4 / (6*10) = 105 * 4/60 = 7.
110 = 2 * 5 * 11, so it has 3 prime factors and all 3 do not have the form 8m+-1. 3 is odd, so 110 is not in the sequence.

Disjoint union of A004215, A055042, A055043 and A234000.
See the comments for the relationships with A002481, A017077, A189715, A242664, A370268.
Cf. A042999 (primes), A059897.

isok(k) = {c = core(k); c%8 == 1 || c%8 == 7 || c%16 == 6 || c%16 == 10}

def A370267(n):
    def f(x): return n+x-sum(((y:=x>>(i<<1))-7>>3)+(y-1>>3)+2 for i in range((x.bit_length()>>1)+1))-sum(((z:=x>>(i<<1)+1)-5>>3)+(z-3>>3)+2 for i in range(x.bit_length()-1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Mar 19 2025

{a(n) : n >= 1} = {A059897(i,j) : i in A234000, j in {1, 6, 10, 15}}.

A017087 a(n) = (8*n + 1)^11.

Original entry on oeis.org

1, 31381059609, 34271896307633, 2384185791015625, 50542106513726817, 550329031716248441, 3909821048582988049, 20635899893042801193, 87507831740087890625, 313726685568359708377, 984770902183611232881

Offset: 0

N. J. A. Sloane

Composition of A008455(n) and A017077(n). - Wesley Ivan Hurt, Jul 17 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).

Cf. A008455, A017077.

[(8*n+1)^11: n in [0..20]]; // Vincenzo Librandi, Jul 11 2011

(8*Range[0,10]+1)^11 (* or *) LinearRecurrence[{12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{1,31381059609,34271896307633,2384185791015625,50542106513726817,550329031716248441,3909821048582988049,20635899893042801193,87507831740087890625,313726685568359708377,984770902183611232881,2775173073766990340489},20] (* Harvey P. Dale, Sep 08 2017 *)
CoefficientList[Series[(1 + 31381059597*x + 33895323592391*x^2 + 1974994185258003*x^3 + 24186918344729610*x^4 + 93655732195384290*x^5 + 134070558743608110*x^6 + 73557591075608934*x^7 + 14545208676272997*x^8 + 849143191166465*x^9 + 8626027938459*x^10 + 1977326743*x^11)/(-1 + x)^12, {x, 0, 15}], x] (* Wesley Ivan Hurt, Jul 17 2025 *)

a(n)=(8*n+1)^11 \\ Charles R Greathouse IV, Aug 11 2014

From Wesley Ivan Hurt, Jul 17 2025: (Start)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12).
G.f.: (1 + 31381059597*x + 33895323592391*x^2 + 1974994185258003*x^3 +24186918344729610*x^4 + 93655732195384290*x^5 + 134070558743608110*x^6 + 73557591075608934*x^7 + 14545208676272997*x^8 + 849143191166465*x^9 + 8626027938459*x^10 + 1977326743*x^11)/(-1 + x)^12.
a(n) = A008455(A017077(n)). (End)

A047400 Numbers that are congruent to {1, 3, 6} mod 8.

Original entry on oeis.org

1, 3, 6, 9, 11, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 41, 43, 46, 49, 51, 54, 57, 59, 62, 65, 67, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 97, 99, 102, 105, 107, 110, 113, 115, 118, 121, 123, 126, 129, 131, 134, 137, 139, 142, 145, 147, 150, 153, 155, 158

Offset: 1

N. J. A. Sloane

Union of A017077, A017101 and A017137. - R. J. Mathar, Apr 14 2008

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A004773.

Cf. A017077, A017101, A017137.

[n: n in [1..300] | n mod 8 in [1, 3, 6]]; // Vincenzo Librandi, Mar 27 2011

A047400:=n->2*(12*n-9+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047400(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Select[Range[0, 150], MemberQ[{1, 3, 6}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 10 2016 *)

a(n) = {x=8*floor((n-1)/3);if(n%3==1,x=x+1);if(n%3==2,x=x+3);if(n%3==0,x=x+6);x} \\ Michael B. Porter, Oct 02 2009

a(n) = A004773(n-1) + A004773(n). - Gary W. Adamson, Sep 13 2007
G.f.: x*(1+x)*(2x^2+x+1)/((-1+x)^2*(x^2+x+1)). a(n) = a(n-3)+8 for n>3. - R. J. Mathar, Apr 14 2008
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-9+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-5, a(3k-2) = 8k-7. (End)

A047594 Numbers that are congruent to {0, 2, 3, 4, 5, 6, 7} mod 8.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76

Offset: 1

N. J. A. Sloane

A004774 without the 1. - R. J. Mathar, Oct 18 2008

Complement of A017077. - Michel Marcus, Sep 11 2015

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A004774, A017077.

[n + Floor((n-2)/7) : n in [1..100]]; // Wesley Ivan Hurt, Sep 11 2015

[n: n in [0..100] | n mod 8 in [0,2,3,4,5,6,7]]; // Vincenzo Librandi, Sep 12 2015

A047594:=n->n+floor((n-2)/7): seq(A047594(n), n=1..100); # Wesley Ivan Hurt, Sep 11 2015

Table[n+Floor[(n-2)/7], {n,100}] (* Wesley Ivan Hurt, Sep 11 2015 *)
Select[Range[0, 100], MemberQ[{0, 2, 3, 4, 5, 6, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, Sep 12 2015 *)
DeleteCases[Range[0,70],?(Mod[#,8]==1&)] (* _Harvey P. Dale, Dec 19 2015 *)

a(n)=(8*n-2)\7 \\ Charles R Greathouse IV, Jul 21 2016

From R. J. Mathar, Mar 03 2009: (Start)
G.f.: x^2*(2+x+x^2+x^3+x^4+x^5+x^6)/((1-x)^2*(x^6+x^5+x^4+x^3+x^2+x+1)).
a(n) = a(n-7) + 8 for n>7. (End)
a(n) = n + floor((n-2)/7). - Wesley Ivan Hurt, Sep 11 2015
From Wesley Ivan Hurt, Jul 21 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.
a(n) = (56*n - 35 + (n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) - 6*((n+5) mod 7) + ((n+6) mod 7))/49.
a(7k) = 8k-1, a(7k-1) = 8k-2, a(7k-2) = 8k-3, a(7k-3) = 8k-4, a(7k-4) = 8k-5, a(7k-5) = 8k-6, a(7k-6) = 8k-8. (End)

A137190 Lucky numbers (A000959) which are congruent to 1 mod 8.

Original entry on oeis.org

1, 9, 25, 33, 49, 73, 105, 129, 169, 193, 201, 241, 273, 289, 297, 321, 361, 385, 393, 409, 433, 489, 529, 537, 553, 577, 601, 673, 729, 745, 769, 777, 801, 841, 873, 897, 937, 961, 993, 1009, 1041, 1057, 1105, 1201, 1209, 1233, 1249, 1281, 1329, 1369, 1401, 1417, 1441

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

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

Intersection of A000959 and A017077.

A138393 Numbers of form 8k+1 which are not squares.

Original entry on oeis.org

17, 33, 41, 57, 65, 73, 89, 97, 105, 113, 129, 137, 145, 153, 161, 177, 185, 193, 201, 209, 217, 233, 241, 249, 257, 265, 273, 281, 297, 305, 313, 321, 329, 337, 345, 353, 369, 377, 385, 393, 401, 409, 417, 425, 433, 449, 457, 465, 473, 481, 489, 497, 505

Offset: 1

Artur Jasinski, Mar 18 2008

nonn

Cf. A004768, A017077.

a = {}; Do[If[Sqrt[8k + 1] == Floor[Sqrt[8k + 1]],[null],AppendTo[a, 8k + 1]], {k, 0, 100}]; a
Select[8*Range[80]+1,!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, May 03 2018 *)

A165220 Numbers n such that 8*n+1 is a cube.

Original entry on oeis.org

0, 91, 614, 1953, 4492, 8615, 14706, 23149, 34328, 48627, 66430, 88121, 114084, 144703, 180362, 221445, 268336, 321419, 381078, 447697, 521660, 603351, 693154, 791453, 898632, 1015075, 1141166, 1277289, 1423828, 1581167, 1749690, 1929781

Offset: 0

Vincenzo Librandi, Sep 08 2009

For every even n, n^4+(n/2)^3 is a cube.

In effect, a(n) = n*(24*n+3+64*n^2) and 8*a(n)+1 = (8*n+1)^3. [R. J. Mathar, Oct 18 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A017077.

I:=[0, 91, 614, 1953]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Apr 06 2013

LinearRecurrence[{4, -6, 4, -1}, {0, 91, 614, 1953}, 100] (* Vincenzo Librandi, Apr 06 2013 *)

G.f.: x*(91+250*x+43*x^2)/(1-x)^4. [Colin Barker, Jun 15 2012]

Typo corrected by Zak Seidov, Sep 14 2009

A168390 a(n) = 1 + 8*floor(n/2).

Original entry on oeis.org

1, 9, 9, 17, 17, 25, 25, 33, 33, 41, 41, 49, 49, 57, 57, 65, 65, 73, 73, 81, 81, 89, 89, 97, 97, 105, 105, 113, 113, 121, 121, 129, 129, 137, 137, 145, 145, 153, 153, 161, 161, 169, 169, 177, 177, 185, 185, 193, 193, 201, 201, 209, 209, 217, 217, 225, 225, 233, 233

Offset: 1

Vincenzo Librandi, Nov 24 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017077, A168378, A168381.

[1+8*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

RecurrenceTable[{a[1]==1,a[n]==8n-a[n-1]-6},a,{n,60}] (* or *) LinearRecurrence[{1,1,-1},{1,9,9},60] (* or *) With[{c=Table[8n+1,{n,0,40}]},Rest[Riffle[c,c]]] (* Harvey P. Dale, Jul 28 2012 *)
Table[1 + 8 Floor[n/2], {n, 60}] (* Bruno Berselli, Sep 18 2013 *)
CoefficientList[Series[(1 + 8 x - x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 18 2013 *)

a(n) = 8*n - a(n-1) - 6, with n>1, a(1)=1.
a(1)=1, a(2)=9, a(3)=9; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Jul 28 2012
G.f.: x*(1 + 8*x - x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
a(n) = A168381(n) - 1 = A168378(n) - 2. - Bruno Berselli, Sep 18 2013
E.g.f.: (2 - exp(x) + (4*x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 19 2016

New definition by Vincenzo Librandi, Sep 18 2013

A185379 Product of exactly three distinct primes congruent to 1 mod 8 (A007519).

Original entry on oeis.org

50881, 62033, 67609, 78761, 95489, 110449, 120377, 134521, 140233, 146761, 162401, 167977, 170017, 170969, 179129, 186337, 195857, 207281, 218161, 225913, 234889, 239513, 246041, 263177, 266377, 279497, 285073, 289153, 290321, 292009, 299081, 301801, 312953

Offset: 1

Jonathan Vos Post, Feb 20 2011

Subset of numbers that are divisible by exactly 3 primes (counted with multiplicity), also known as triprimes or 3-almost primes, A014612. Subset of {d = p_1 * p_2 * ... * p_m where p_i == 1 (mod 8), 1 <= i <= m are distinct primes} as occurs in Wei, p.2.

			a(12) = 170017 = 17 * 73 * 137 = A007519(1) * A007519(3) * A007519(7).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Dasheng Wei, On the equation x^2-Dy^2=n, Feb 18, 2011.

Cf. A007519, A014612, A017077, A185377.

p = Select[Prime[Range[100]], Mod[#, 8] == 1 &]; Sort[Reap[Do[n = p[[i]] p[[j]] p[[k]]; If[n <= p[[1]] p[[2]] p[[-1]], Sow[n]], {i, 2, Length[p]}, {j, i - 1}, {k, j - 1}]][[2, 1]]]

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\697, if(p%8==1, listput(u,p))); for(i=1,#u-2, for(j=i+1, #u-1, if(u[i]*u[j]*u[j+1]>lim, break); for(k=j+1,#u, t=u[i]*u[j]*u[k]; if(t>lim, break); listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

{A007519(i) * A007519(j) * A007519(k) for i < j < k}. {A000040(i) * A000040(j) * A000040(k) for i < j < k, and A000040(i) in A017077 and A000040(j) in A017077 and A000040(k) in A017077}.

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A354938 Row 8 of A354940: Numbers k for which A345992(k) = 8, divided by 8.

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A370267 Numbers with an even number of prime factors not of the form 8m+-1 (counting repetitions).

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A017087 a(n) = (8*n + 1)^11.

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A047400 Numbers that are congruent to {1, 3, 6} mod 8.

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A047594 Numbers that are congruent to {0, 2, 3, 4, 5, 6, 7} mod 8.

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A137190 Lucky numbers (A000959) which are congruent to 1 mod 8.

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A138393 Numbers of form 8k+1 which are not squares.

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A165220 Numbers n such that 8*n+1 is a cube.

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