cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A354934 Row 4 of A354940: Numbers k for which A345992(k) = 4, divided by 4.

Original entry on oeis.org

5, 9, 13, 17, 21, 25, 29, 37, 41, 49, 53, 57, 61, 73, 81, 85, 89, 93, 97, 101, 109, 113, 117, 121, 125, 129, 137, 149, 157, 169, 173, 181, 185, 193, 197, 201, 217, 229, 233, 237, 241, 253, 257, 269, 277, 281, 289, 293, 297, 301, 309, 313, 317, 337, 341, 349, 353, 361, 373, 381, 389, 397, 401, 409, 413, 417, 421, 425
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Comments

Apparently a subsequence of A016813.

Crossrefs

Row 4 of A354940.
Cf. A016813, A344005, A345992.

Programs

  • Mathematica
    q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 4*n], m++]; GCD[4*n, m] == 4]; Select[Range[425], q] (* Amiram Eldar, Jun 17 2022 *)
  • PARI
    A354934(n) = A354940sq(4,n);

A354935 Row 5 of A354940: Numbers k for which A345992(k) = 5, divided by 5.

Original entry on oeis.org

3, 6, 8, 11, 13, 16, 23, 26, 31, 36, 41, 43, 46, 51, 53, 56, 61, 71, 73, 81, 83, 86, 91, 96, 101, 103, 106, 113, 116, 121, 128, 131, 141, 146, 151, 163, 166, 171, 173, 176, 181, 191, 193, 196, 206, 211, 223, 226, 233, 241, 243, 251, 256, 263, 271, 276, 281, 283, 293, 301, 311, 313, 321, 326, 331, 343, 346, 353, 356, 361
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Comments

Apparently, all terms are either of the form 5k+1 (in A016861), or of the form 5k+3 (in A016885).

Crossrefs

Row 5 of A354940.
Cf. A016861, A016885, A344005, A345992.

Programs

  • Mathematica
    q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 5*n], m++]; GCD[5*n, m] == 5]; Select[Range[360], q] (* Amiram Eldar, Jun 17 2022 *)
  • PARI
    A354935(n) = A354940sq(5,n);

A354936 Row 6 of A354940: Numbers k for which A345992(k) = 6, divided by 6.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 43, 49, 61, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151, 157, 163, 169, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 289, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 529, 541, 547, 571, 577, 601, 607, 613, 619, 625, 631, 643, 661
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Crossrefs

Row 6 of A354940.
Apparently a subsequence of A016921.
Cf. A344005, A345992.

Programs

  • Mathematica
    q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 6*n], m++]; GCD[6*n, m] == 6]; Select[Range[660], q] (* Amiram Eldar, Jun 17 2022 *)
  • PARI
    A354936(n) = A354940sq(6,n);

A354937 Row 7 of A354940: Numbers k for which A345992(k) = 7, divided by 7.

Original entry on oeis.org

4, 5, 8, 11, 15, 19, 22, 25, 29, 32, 39, 43, 47, 50, 53, 57, 61, 64, 67, 71, 78, 81, 89, 92, 95, 99, 103, 106, 109, 113, 127, 131, 134, 137, 141, 151, 155, 162, 169, 173, 176, 179, 183, 190, 193, 197, 211, 218, 229, 232, 239, 243, 256, 257, 263, 267, 271, 274, 277, 281, 291, 295, 302, 309, 313, 316, 323, 337, 344
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Comments

Apparently, all terms are either of the form 7k+1 (in A016993), 7k+4 (A017029) or 7k+5 (A017041).

Crossrefs

Row 7 of A354940.
Cf. A016993, A017029, A017041, A344005, A345992.

Programs

  • Mathematica
    q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 7*n], m++]; GCD[7*n, m] == 7]; Select[Range[345], q] (* Amiram Eldar, Jun 17 2022 *)
  • PARI
    A354937(n) = A354940sq(7,n);

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Comments

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

Crossrefs

Row 8 of A354940.
Cf. A017077, A017101, A344005, A345992.

Programs

  • Mathematica
    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);

A354939 Row 9 of A354940: Numbers k for which A345992(k) = 9, divided by 9.

Original entry on oeis.org

5, 7, 10, 14, 16, 19, 23, 25, 28, 32, 37, 41, 43, 46, 50, 59, 61, 64, 68, 73, 79, 82, 86, 91, 97, 100, 109, 113, 118, 122, 127, 131, 136, 145, 149, 151, 158, 163, 167, 169, 172, 181, 185, 194, 199, 212, 221, 223, 226, 235, 239, 241, 244, 253, 257, 262, 271, 277, 289, 293, 298, 302, 307, 311, 313, 316, 325, 331, 334
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Comments

Apparently, all terms are either of the form 9k+1 (in A017173), or 9k+5 (in A017221), or 9k+7 (in A017245).

Crossrefs

Row 9 of A354940.
Cf. A017173, A017221, A017245, A344005, A345992.

Programs

  • Mathematica
    q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 9*n], m++]; GCD[9*n, m] == 9]; Select[Range[335], q] (* Amiram Eldar, Jun 17 2022 *)
  • PARI
    A354939(n) = A354940sq(9,n); \\ See the program in A354940.

A137827 Prime powers (A246655) congruent to 1 (mod 3).

Original entry on oeis.org

4, 7, 13, 16, 19, 25, 31, 37, 43, 49, 61, 64, 67, 73, 79, 97, 103, 109, 121, 127, 139, 151, 157, 163, 169, 181, 193, 199, 211, 223, 229, 241, 256, 271, 277, 283, 289, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499
Offset: 1

Views

Author

Eric W. Weisstein, Feb 12 2008

Keywords

Comments

Numbers k not being powers of 3 such that x^2 + x + 1 (or x^2 - x + 1) is reducible over GF(k). - Jianing Song, Sep 24 2019

Links

Crossrefs

Row 3 of A354940 (conjectured).
Intersection of A016777 and A246655.
Cf. A354982 (characteristic function), A354984 (terms * 3).
Cf. also A327752, A327753.

Programs

  • Mathematica
    Select[ Range[4, 500], (Mod[#, 3] == 1 && Mod[#, # - EulerPhi[#]] == 0)& ] (* Jean-François Alcover, Oct 26 2012 *)
  • PARI
    isok(n) = isprimepower(n) && ((n % 3) == 1); \\ Michel Marcus, Sep 24 2019

A354930 Square array where the row n lists all nonnegative numbers k for which A345992(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 2, 6, 3, 10, 12, 4, 14, 21, 20, 5, 18, 39, 36, 15, 7, 22, 48, 52, 30, 42, 8, 26, 57, 68, 40, 78, 28, 9, 34, 75, 84, 55, 114, 35, 24, 11, 38, 93, 100, 65, 150, 56, 72, 45, 13, 46, 111, 116, 80, 186, 77, 88, 63, 110, 16, 50, 129, 148, 115, 222, 105, 136, 90, 310, 33, 17, 54, 147, 164, 130, 258, 133, 152, 126, 410, 44, 156
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2022

Keywords

Comments

Array is read by descending antidiagonals with (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ... where A(n,k) is the k-th solution x to A345992(x) = n.

Examples

			The top left 15x9 corner of the array:
n\k |  1   2    3    4    5    6    7    8    9   10   11   12   13   14   15
----+--------------------------------------------------------------------------
  1 |  1,  2,   3,   4,   5,   7,   8,   9,  11,  13,  16,  17,  19,  23,  25,
  2 |  6, 10,  14,  18,  22,  26,  34,  38,  46,  50,  54,  58,  62,  74,  82,
  3 | 12, 21,  39,  48,  57,  75,  93, 111, 129, 147, 183, 192, 201, 219, 237,
  4 | 20, 36,  52,  68,  84, 100, 116, 148, 164, 196, 212, 228, 244, 292, 324,
  5 | 15, 30,  40,  55,  65,  80, 115, 130, 155, 180, 205, 215, 230, 255, 265,
  6 | 42, 78, 114, 150, 186, 222, 258, 294, 366, 402, 438, 474, 582, 618, 654,
  7 | 28, 35,  56,  77, 105, 133, 154, 175, 203, 224, 273, 301, 329, 350, 371,
  8 | 24, 72,  88, 136, 152, 200, 216, 264, 328, 344, 392, 456, 472, 536, 584,
  9 | 45, 63,  90, 126, 144, 171, 207, 225, 252, 288, 333, 369, 387, 414, 450,

Links

Crossrefs

Cf. A344005, A345992.
Column 1: A354931.
Rows 1..3: A000961, A278568 (without its initial 2, conjectured), A354984 (conjectured).
See also array A354940 where the entries are divided by their row index.

Programs

  • PARI
    up_to = 105;
A345992(n) = for(m=1, oo, if((m*(m+1))%n==0, return(gcd(n,m))));
memoA354930sq = Map();
A354930sq(n, k) = { my(v=0); if(!mapisdefined(memoA354930sq,[n,k-1],&v),if(1==k, v=0, v = A354930sq(n, k-1))); for(i=1+v,oo,if(A345992(i)==n,mapput(memoA354930sq,[n,k],i); return(i))); };
A354930list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A354930sq(col,(a-(col-1))))); (v); };
v354930 = A354930list(up_to);
A354930(n) = v354930[n];

A354932 a(n) = 1/n * {the least k for which A345992(k) = n}.

Original entry on oeis.org

1, 3, 4, 5, 3, 7, 4, 3, 5, 11, 3, 13, 7, 5, 4, 7, 3, 19, 4, 7, 11, 9, 3, 7, 13, 9, 4, 17, 3, 31, 4, 3, 17, 5, 9, 31, 5, 11, 5, 9, 3, 17, 4, 5, 13, 31, 3, 7, 5, 17, 4, 19, 3, 5, 7, 19, 23, 9, 3, 61, 8, 7, 8, 5, 11, 19, 4, 23, 7, 47, 3, 29, 7, 5, 19, 9, 13, 47, 4, 7, 13, 11, 3, 17, 16, 23, 4, 23, 3, 13, 4, 31, 11, 5
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2022

Keywords

Comments

The first four terms that are not powers of primes are: a(112) = 15, a(122) = 35, a(145) = 33, a(155) = 12.
Question: Are 2, 3, 4, 6, 10, 12, 18, 30, 60 the only n such that a(n) = 1+n?

Links

Crossrefs

Column 1 of A354940.
Cf. A344005, A345992, A354931.

Programs

  • Mathematica
    s[n_] := Module[{m = 1}, While[!Divisible[m*(m+1), n], m++]; GCD[n, m]]; a[n_] := Module[{k = n}, While[s[k] != n, k+=n]; k/n]; Array[a, 100] (* Amiram Eldar, Jun 15 2022 *)
  • PARI
    A345992(n) = gcd(n,A344005(n));
A354932(n) = for(k=1,oo,if(A345992(k)==n,return(k/n)));

Formula

a(n) = A354931(n) / n.
Showing 1-9 of 9 results.

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