A047241 -id:A047241 - cp's OEIS frontend

A087444 Numbers that are congruent to {1, 4} mod 9.

1, 4, 10, 13, 19, 22, 28, 31, 37, 40, 46, 49, 55, 58, 64, 67, 73, 76, 82, 85, 91, 94, 100, 103, 109, 112, 118, 121, 127, 130, 136, 139, 145, 148, 154, 157, 163, 166, 172, 175, 181, 184, 190, 193, 199, 202, 208, 211, 217, 220, 226, 229, 235, 238, 244, 247, 253

Offset: 1

Paul Barry, Sep 04 2003

3*a(n) is conjectured to be the total number of sides (straight double bonds (long side) and single bond (short side)) of a certain equilateral triangle expansion shown in one of the links. The pattern is supposed to become the planar Archimedean net 3.3.3.3.6 when n -> infinity. 3*a(n) is also conjectured to be the total number of sided (bonds) in another version of an equilateral triangle expansion that is supposed to become the planar Archimedean net 3.6.3.6. See the illustrations in the links. - Kival Ngaokrajang, Nov 30 2014

David Lovler, Table of n, a(n) for n = 1..10000
Kival Ngaokrajang, Illustration of initial terms (3.3.3.3.6), (3.6.3.6)
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A047241, A087445, A087446.

Select[Range[300],MemberQ[{1,4},Mod[#,9]]&] (* or *) LinearRecurrence[ {1,1,-1},{1,4,10},60] (* Harvey P. Dale, Jan 22 2019 *)

a(n) = (18*n - 17 - 3*(-1)^n)/4 \\ David Lovler, Aug 20 2022

G.f.: x*(1+3*x+5*x^2)/((1+x)*(1-x)^2).
E.g.f.: 5 + ((9*x - 17/2)*exp(x) - (3/2)*exp(-x))/2.
a(n) = (18*n - 17 - 3*(-1)^n)/4.
a(n) = 9*n - a(n-1) - 13 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010

Kival Ngaokrajang's comment reworded by Wolfdieter Lang, Dec 05 2014

E.g.f. and formula adapted to offset by David Lovler, Aug 20 2022

A258011 Numbers remaining after the third stage of Lucky sieve.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37, 43, 45, 49, 51, 55, 57, 63, 67, 69, 73, 75, 79, 85, 87, 91, 93, 97, 99, 105, 109, 111, 115, 117, 121, 127, 129, 133, 135, 139, 141, 147, 151, 153, 157, 159, 163, 169, 171, 175, 177, 181, 183, 189, 193, 195, 199, 201, 205, 211, 213, 217, 219, 223, 225, 231, 235, 237, 241, 243, 247, 253, 255

Offset: 1

Antti Karttunen, Jul 27 2015

Equal to A047241 with its every seventh term (A258016) removed.

Numbers congruent to {1, 3, 7, 9, 13, 15, 21, 25, 27, 31, 33, 37} modulo 42. - Jianing Song, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1)

Row 3 of A258207.
Setwise difference of A047241 \ A258016.
Cf. also A260440 (Every ninth term).

gf := (x*(1 + x*(2 + x*(4 + x*(2 + x*(4 + x*(2 + x*(6 + x*(4 + x*(2 + x*(4 + x*(2 + x*(4 + 5*x)))))))))))))/(1 - x*(1 + (1 - x)*x^11)): ser:= series(gf, x, 112):
seq(coeff(ser, x, k), k = 1..74); # Peter Luschny, Apr 29 2022

(define (A258011 n) (A258207bi 3 n)) ;; A258207bi given in A258207.

From Jianing Song, Apr 27 2022: (Start)
a(n) = a(n-12) + 42.
a(n) = a(n-1) + a(n-12) - a(n-13).
G.f.:(x+2*x^2+4*x^3+2*x^4+4*x^5+2*x^6+6*x^7+4*x^8+2*x^9+4*x^10+2*x^11+4*x^12+5*x^13)/(1-x-x^12+x^13). (End)

A047262 Numbers that are congruent to {0, 2, 4, 5} mod 6.

Original entry on oeis.org

0, 2, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 65, 66, 68, 70, 71, 72, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047233 with A016789(n-1). - Guenther Schrack, Feb 14 2019

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for two-way infinite sequences.

Cf. A016789, A047233, A047237, A047273.

Complement: A047241.

[n : n in [0..100] | n mod 6 in [0, 2, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

A047262:=n->(6*n-4-I^(1-n)+I^(1+n))/4: seq(A047262(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Select[Range[0,100], MemberQ[{0,2,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{2,-2,2,-1}, {0,2,4,5}, 70] (* Harvey P. Dale, Dec 09 2015 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(2+x^2)/((1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(2+x^2)/((1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

From R. J. Mathar, Oct 08 2011: (Start)
G.f.: x^2*(2+x^2) / ( (1+x^2)*(1-x)^2 ).
a(n) = 3*n/2 - 1 - sin(Pi*n/2)/2. (End)
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n > 4.
a(n) = (6*n - 4 - i^(1-n) + i^(1+n))/4, where i = sqrt(-1).
a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047233(n).
a(2-n) = - A047237(n), a(n-1) = A047273(n) - 1 for n > 1. (End)
From Guenther Schrack, Feb 14 2019: (Start)
a(n) = (6*n - 4 - (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=2, a(3)=4, a(4)=5, for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = log(3)/4 + log(2)/3 - sqrt(3)*Pi/36. - Amiram Eldar, Dec 17 2021

More terms from Wesley Ivan Hurt, May 21 2016

A087446 Numbers that are congruent to {1, 6} mod 15.

Original entry on oeis.org

1, 6, 16, 21, 31, 36, 46, 51, 61, 66, 76, 81, 91, 96, 106, 111, 121, 126, 136, 141, 151, 156, 166, 171, 181, 186, 196, 201, 211, 216, 226, 231, 241, 246, 256, 261, 271, 276, 286, 291, 301, 306, 316, 321, 331, 336, 346, 351, 361, 366, 376, 381, 391, 396, 406

Offset: 1

Paul Barry, Sep 04 2003

3*a(n) is conjectured to be the number of edges (bonds) visited when walking around the boundary of a certain equilateral triangle construction at the n-th iteration. See the illustration in the link. Note that isthmus edges (bridges) are counted twice. The pattern is supposed to become the planar Archimedean net 3.12.12 when n -> infinity. - Kival Ngaokrajang, Nov 30 2014

Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (1,1,-1)

Cf. A001651, A047241, A087444, A087445.

#+{1,6}&/@(15*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,6,16},60] (* Harvey P. Dale, Dec 05 2018 *)

G.f.: x*(1 + 5*x + 9*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (30*x-1)*exp(x)/4 + 5*exp(-x)/4.
a(n) = (18*n-1)/4 + 5*(-1)^n/4.
a(n) = 15*n - a(n-1) - 23, with a(1)=1. - Vincenzo Librandi, Aug 08 2010

Editing: rewording of Kival Ngaokrajang's comment. - Wolfdieter Lang, Dec 06 2014

A203016 Numbers congruent to {1, 2, 3, 4} mod 6, multiplied by 3.

Original entry on oeis.org

3, 6, 9, 12, 21, 24, 27, 30, 39, 42, 45, 48, 57, 60, 63, 66, 75, 78, 81, 84, 93, 96, 99, 102, 111, 114, 117, 120, 129, 132, 135, 138, 147, 150, 153, 156, 165, 168, 171, 174, 183, 186, 189, 192, 201, 204, 207, 210, 219, 222, 225, 228, 237, 240, 243, 246, 255, 258, 261, 264, 273, 276, 279, 282, 291, 294, 297

Offset: 1

N. J. A. Sloane, Dec 27 2011

Appears to coincide with the list of numbers n such that A006600(n) is not a multiple of n. Equals A047227 multiplied by 3.

Colin Foster, Peripheral mathematical knowledge, For the Learning of Mathematics, vol. 31, #3 (November, 2011), pp. 24-28.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A006600, A047227, A047235, A047241.

[3*n : n in [0..100] | n mod 6 in [1..4]]; // Wesley Ivan Hurt, Jun 07 2016

A203016:=n->3*(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A203016(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

3 Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]] &] (* Wesley Ivan Hurt, Jun 07 2016 *)

From Wesley Ivan Hurt, Jun 07 2016: (Start)
G.f.: 3*x*(1+x+x^2+x^3+2*x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = 3*(6*n-5-i^(2*n)+(1+i)*i^(1-n)+(1-i)*i^(n-1))/4 where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = 3*A047235(k), a(2k-1) = 3*A047241(k). (End)
E.g.f.: 3*(4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 07 2016

A258016 Unlucky numbers removed at the stage three of Lucky sieve.

Original entry on oeis.org

19, 39, 61, 81, 103, 123, 145, 165, 187, 207, 229, 249, 271, 291, 313, 333, 355, 375, 397, 417, 439, 459, 481, 501, 523, 543, 565, 585, 607, 627, 649, 669, 691, 711, 733, 753, 775, 795, 817, 837, 859, 879, 901, 921, 943, 963, 985, 1005, 1027, 1047, 1069, 1089, 1111, 1131, 1153, 1173, 1195, 1215, 1237, 1257, 1279, 1299, 1321, 1341, 1363, 1383, 1405

Offset: 1

Antti Karttunen, Jul 27 2015

Numbers congruent to 19 or 39 modulo 42. - Jianing Song, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1)

Row 3 of A255543. Every seventh term of A047241.

Cf. also A258011.

a(n) = A047241(7*n).
a(n) = A260436(A255413(1+n)).
From Jianing Song, Apr 27 2022: (Start)
a(n) = a(n-2) + 42.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: (19*x+20*x^2+3*x^3)/(1-x-x^2+x^3).
E.g.f.: 3 + (21*x-3)*cosh(x) + (21*x-2)*sinh(x). (End)

A271508 Numbers that are congruent to {1,4} mod 10.

Original entry on oeis.org

1, 4, 11, 14, 21, 24, 31, 34, 41, 44, 51, 54, 61, 64, 71, 74, 81, 84, 91, 94, 101, 104, 111, 114, 121, 124, 131, 134, 141, 144, 151, 154, 161, 164, 171, 174, 181, 184, 191, 194, 201, 204, 211, 214, 221, 224, 231, 234, 241, 244, 251, 254, 261, 264, 271, 274

Offset: 1

Wesley Ivan Hurt, Apr 08 2016

Numbers ending in 1 or 4, Union of A017281 and A017317.

a(n+3) gives the sum of 5 consecutive terms of A004442 starting at A004442(n) for n>0. (i.e., a(4) = 14 = 0+3+2+5+4 = Sum_{i=0..4} A004442(n+i)).

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

Cf. A001622, A004442, A017281, A017317, A042948, A047241.

Magma
```
[5*n-5-(-1)^n : n in [1..100]];
```

A271508:=n->5*n-5-(-1)^n: seq(A271508(n), n=1..100);

Table[5 n - 5 - (-1)^n, {n, 60}] (* or *)
Select[Range[0, 300], MemberQ[{1, 4}, Mod[#, 10]] &]

my(x='x+O('x^99)); Vec(x*(1+3*x+6*x^2)/((-1+x)^2*(1+x))) \\ Altug Alkan, Apr 09 2016

G.f.: x*(1+3*x+6*x^2)/((-1+x)^2*(1+x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 5*n - 5 - (-1)^n.
a(n) = -n + 2*A047241(n).
a(n+1) = n + 1 + 2*A042948(n).
Shifted bisections: a(2n+2) = A017317(n), a(2n+1) = A017281(n).
E.g.f.: 5*(x-1)*exp(x) - exp(-x). - G. C. Greubel, Apr 08 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A308378 Numbers k such that phi(2k+1) = phi(2k+2).

Original entry on oeis.org

0, 1, 7, 127, 247, 487, 1312, 1627, 1852, 2593, 5857, 6682, 9157, 11467, 12772, 23107, 24607, 24667, 28822, 32767, 82087, 92317, 99157, 107887, 143497, 153697, 159637, 194122, 198742, 207637, 245767, 284407, 294703, 343492, 420127

Offset: 1

Torlach Rush, May 24 2019

nonn

For n > 0, 2*a(n) + 1 is a term of A020884. This is because 2*a(n) + 1 is odd and every odd number is the difference of the squares of two consecutive numbers and hence are coprime.
For n > 0, (2*a(n) + 1) * (2*a(n) + 2) is a term of A024364. This is because (2*a(n) + 1) * (2*a(n) + 2) = 2*((a(n) + 1)^2 + (a(n) + 1) * a(n)) and gcd((a(n) + 1), a(n)) = 1.
For n > 0, a(n) is congruent to 1 or 4 mod 6.
2*a(n) + 1 is congruent to 1 or 3 mod 6 and is a term of A047241.
2*a(n) + 2 is congruent to 2 or 4 mod 6 and is a term of A047235.

			0 is a term because phi(1) = phi(2) = 1.
1 is a term because phi(3) = phi(4) = 2.
7 is a term because phi(15) = phi(16) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..5416 (calculated using the b-file at A001274)

Cf. A000010, A004767, A020884, A024364, A047235, A047241, A299535.
Subset of A047234.
Subset of A001274.

Select[Range[0, 9999], EulerPhi[2# + 1] == EulerPhi[2# + 2] &] (* Alonso del Arte, Jul 05 2019 *)
Select[(#-1)/2&/@SequencePosition[EulerPhi[Range[900000]],{x_,x_}][[All,1]],IntegerQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2019 *)

lista(nn) = for(n=0, nn, if(eulerphi(2*n+1) == eulerphi(2*n+2), print1(n, ", ")));
lista(430000)

a(n) = (A299535(n) - 2) / 2.

A379020 Numbers k such that 2^k - 25 is prime.

Original entry on oeis.org

5, 7, 9, 13, 33, 37, 57, 63, 93, 127, 129, 165, 189, 369, 717, 3079, 3087, 3925, 6709, 7633, 18001, 21961, 55557, 60415, 63589, 69463, 75949, 98265, 212295, 416773, 647545, 824325, 1538959, 2020893, 2421175

Offset: 1

Boyan Hu, Dec 13 2024

Except for a(1), all terms are congruent to 1 or 3 mod 6.

a(36) > 3400000. - Boyan Hu, Jun 16 2025

			7 is in the sequence because 2^7-25=103 is prime.
8 is not in the sequence because 2^8-25=231=3*7*11 is not prime.

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-25, PRP Top Records.
Boyan Hu, 2^n-25 PRP searching progress

Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Except for a(1), subsequence of A047241.

Do[ If[ PrimeQ[ 2^n - 25 ], Print[ n ] ], { n, 1, 15000} ]

PARI
```
is(n)=ispseudoprime(2^n-25)
```

a(1)=5 inserted by Max Alekseyev, May 28 2025

A051391 Number of nonisomorphic Steiner triple systems (STS's) S(2,3,v) on v = 6n+1 or 6n+3 points.

Original entry on oeis.org

1, 1, 1, 1, 2, 80, 11084874829

Offset: 1

N. J. A. Sloane

			There are 2 nonisomorphic STS's on 13 points.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 304.
CRC Handbook of Combinatorial Designs, 1996, p. 70.

Petteri Kaski and Patric R. J. Östergård, The Steiner triple systems of order 19
Petteri Kaski and Patric R. J. Östergård, The Steiner triple systems of order 19, Mathematics of Computation, Vol. 73, No. 248 (Oct., 2004), pp. 2075-2092.
Index entries for sequences related to Steiner systems

Indexed by A047241. Cf. A001201, A030128, A030129, A051390, A124118, A124119.

cp's OEIS Frontend

A087444 Numbers that are congruent to {1, 4} mod 9.

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A258011 Numbers remaining after the third stage of Lucky sieve.

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A047262 Numbers that are congruent to {0, 2, 4, 5} mod 6.

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A087446 Numbers that are congruent to {1, 6} mod 15.

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A203016 Numbers congruent to {1, 2, 3, 4} mod 6, multiplied by 3.

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A258016 Unlucky numbers removed at the stage three of Lucky sieve.

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A271508 Numbers that are congruent to {1,4} mod 10.

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