Martin Renner - cp's OEIS frontend

A377214 Irregular triangle T(n, k), read by rows with 1 <= k <= p = A000040(n), for the very first solution to the transversal of primes problem.

2, 3, 3, 5, 7, 5, 7, 11, 19, 23, 7, 11, 17, 23, 29, 41, 47, 11, 13, 29, 41, 53, 59, 71, 83, 89, 109, 113, 13, 17, 29, 41, 53, 71, 83, 103, 113, 127, 137, 151, 167, 17, 19, 37, 59, 73, 89, 103, 131, 151, 167, 179, 197, 211, 227, 251, 271, 283, 19, 23, 41, 59, 83, 107, 127, 139, 157, 181, 191, 227, 239, 263, 281, 293, 313, 337, 359

Offset: 1

Martin Renner, Oct 20 2024

Let p be the n-th prime number. Put 1 to p^2 into a square array in order. Choose a set of primes such that there is one and only one in each row and column. Then T(n, k) gives the first of solutions for the n-th prime according to the size of the selected prime numbers.

			Triangle starts with:
  2, 3;
  3, 5, 7;
  5, 7, 11, 19, 23;
  7, 11, 17, 23, 29, 41, 47;
  ...
For n = 4, p = 7 there are two solutions {7, 11, 17, 23, 29, 41, 47} and {7, 11, 19, 23, 31, 41, 43}, the first of which is listed in the table.

Martin Erickson, Beautiful Mathematics, Mathematical Association of America, 2011, p. 6 (Transversal of primes).

Cf. A215637.

A376541 Natural numbers whose iterated squaring modulo 1000 eventually enters the 4-cycle 201, 401, 801, 601.

Original entry on oeis.org

7, 43, 49, 51, 93, 99, 101, 107, 143, 149, 151, 157, 199, 201, 207, 243, 257, 293, 299, 301, 343, 349, 351, 357, 393, 399, 401, 407, 449, 451, 457, 493, 507, 543, 549, 551, 593, 599, 601, 607, 643, 649, 651, 657, 699, 701, 707, 743, 757, 793, 799, 801, 843

Offset: 1

Martin Renner, Sep 26 2024

nonn

The natural numbers decompose into eight categories under the operation of repeated squaring modulo 1000, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376538), 376 (cf. A376539), or 625 (cf. A017329), two of which eventually enter one of the 4-cycles 176, 976, 576, 776 (cf. A376540) or 201, 401, 801, 601 (this sequence), and two of which eventually enter one of the 20-cycles 16, 256, 536, 296, 616, 456, 936, 96, 216, 656, 336, 896, 816, 856, 736, 696, 416, 56, 136, 496 (cf. A376508) or 41, 681, 761, 121, 641, 881, 161, 921, 241, 81, 561, 721, 841, 281, 961, 521, 441, 481, 361, 321 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 36, 6, 2, 42, 6, 2, 6, 36, 6, 2, 6, 42, 2, 6, 36, 14, ...

			7^2 = 49 -> 49^2 = 401 -> 401^2 = 801 -> 801^2 = 601 -> 601^2 = 201 -> 201^2 = 401 -> ... (mod 1000).

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

Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376538, A376539, A376540.

A376540 Natural numbers whose iterated squaring modulo 1000 eventually enters the 4-cycle 176, 976, 576, 776.

Original entry on oeis.org

18, 24, 26, 32, 74, 76, 82, 118, 132, 168, 174, 176, 218, 224, 226, 232, 268, 274, 276, 282, 324, 326, 332, 368, 382, 418, 424, 426, 468, 474, 476, 482, 518, 524, 526, 532, 574, 576, 582, 618, 632, 668, 674, 676, 718, 724, 726, 732, 768, 774, 776, 782, 824

Offset: 1

Martin Renner, Sep 26 2024

nonn

The natural numbers decompose into eight categories under the operation of repeated squaring modulo 1000, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376538), 376 (cf. A376539), or 625 (cf. A017329), two of which eventually enter one of the 4-cycles 176, 976, 576, 776 (this sequence) or 201, 401, 801, 601 (cf. A376541), and two of which eventually enter one of the 20-cycles 16, 256, 536, 296, 616, 456, 936, 96, 216, 656, 336, 896, 816, 856, 736, 696, 416, 56, 136, 496 (cf. A376508) or 41, 681, 761, 121, 641, 881, 161, 921, 241, 81, 561, 721, 841, 281, 961, 521, 441, 481, 361, 321 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 6, 2, 6, 42, 2, 6, 36, 14, 36, 6, 2, 42, 6, 2, 6, 36, ...

			18^2 = 324 -> 324^2 = 976 -> 976^2 = 576 -> 576^2 = 776 -> 776^2 = 176 -> 176^2 = 976 -> ... (mod 1000).

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

Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376538, A376539, A376541.

A376539 Natural numbers whose iterated squaring modulo 1000 eventually settles at the attractor 376.

Original entry on oeis.org

68, 124, 126, 182, 318, 374, 376, 432, 568, 624, 626, 682, 818, 874, 876, 932, 1068, 1124, 1126, 1182, 1318, 1374, 1376, 1432, 1568, 1624, 1626, 1682, 1818, 1874, 1876, 1932, 2068, 2124, 2126, 2182, 2318, 2374, 2376, 2432, 2568, 2624, 2626, 2682, 2818, 2874

Offset: 1

Martin Renner, Sep 26 2024

The natural numbers decompose into eight categories under the operation of repeated squaring modulo 1000, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376538), 376 (this sequence), or 625 (cf. A017329), two of which eventually enter one of the 4-cycles 176, 976, 576, 776 (cf. A376540) or 201, 401, 801, 601 (cf. A376541), and two of which eventually enter one of the 20-cycles 16, 256, 536, 296, 616, 456, 936, 96, 216, 656, 336, 896, 816, 856, 736, 696, 416, 56, 136, 496 (cf. A376508) or 41, 681, 761, 121, 641, 881, 161, 921, 241, 81, 561, 721, 841, 281, 961, 521, 441, 481, 361, 321 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 56, 2, 56, 136, ...

			68^2 = 624 -> 624^2 = 376 -> 376^2 = 376 -> ... (mod 1000).

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

Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376538, A376540, A376541.

A376538 Natural numbers whose iterated squaring modulo 1000 eventually settles at the attractor 1.

Original entry on oeis.org

1, 57, 193, 249, 251, 307, 443, 499, 501, 557, 693, 749, 751, 807, 943, 999, 1001, 1057, 1193, 1249, 1251, 1307, 1443, 1499, 1501, 1557, 1693, 1749, 1751, 1807, 1943, 1999, 2001, 2057, 2193, 2249, 2251, 2307, 2443, 2499, 2501, 2557, 2693, 2749, 2751, 2807

Offset: 1

Martin Renner, Sep 26 2024

The natural numbers decompose into eight categories under the operation of repeated squaring modulo 1000, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (this sequence), 376 (cf. A376539), or 625 (cf. A017329), two of which eventually enter one of the 4-cycles 176, 976, 576, 776 (cf. A376540) or 201, 401, 801, 601 (cf. A376541), and two of which eventually enter one of the 20-cycles 16, 256, 536, 296, 616, 456, 936, 96, 216, 656, 336, 896, 816, 856, 736, 696, 416, 56, 136, 496 (cf. A376508) or 41, 681, 761, 121, 641, 881, 161, 921, 241, 81, 561, 721, 841, 281, 961, 521, 441, 481, 361, 321 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 56, 136, 56, 2, ...

			57^2 = 249 -> 249^2 = 1 -> 1^2 = 1 -> ... (mod 1000).

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

Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376539, A376540, A376541.

A376509 Natural numbers whose iterated squaring modulo 100 eventually enters the 4-cycle 21, 41, 81, 61.

Original entry on oeis.org

3, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 47, 53, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 97, 103, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 147, 153, 159, 161, 163, 167, 169, 171, 173, 177, 179, 181

Offset: 1

Martin Renner, Sep 25 2024

nonn

The natural numbers decompose into six categories under the operation of repeated squaring modulo 100, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376506), 25 (cf. A017329), or 76 (cf. A376507), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (cf. A376508) or 21, 41, 81, 61 (this sequence).

The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 6, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 6, 6, ...

			3^2 = 9 -> 9^2 = 81 -> 81^2 = 61 -> 61^2 = 21 -> 21^2 = 41 -> 41^2 = 81 -> ... (mod 100)

Alexander K. Dewdney, Computer-Kurzweil. Mit einem Computer-Mikroskop untersuchen wir ein Objekt von faszinierender Struktur in der Ebene der komplexen Zahlen. In: Spektrum der Wissenschaft, Oct 1985, p. 8-14, here p. 11-13 (Iterations on a finite set), 14 (Iteration diagram).

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

Cf. A008592, A017329, A376506, A376507, A376508.

A376508 Natural numbers whose iterated squaring modulo 100 eventually enters the 4-cycle 16, 56, 36, 96.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 16, 22, 28, 34, 36, 38, 42, 44, 46, 48, 52, 54, 56, 58, 62, 64, 66, 72, 78, 84, 86, 88, 92, 94, 96, 98, 102, 104, 106, 108, 112, 114, 116, 122, 128, 134, 136, 138, 142, 144, 146, 148, 152, 154, 156, 158, 162, 164, 166, 172, 178, 184, 186

Offset: 1

Martin Renner, Sep 25 2024

nonn

The natural numbers decompose into six categories under the operation of repeated squaring modulo 100, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376506), 25 (cf. A017329), or 76 (cf. A376507), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (this sequence) or 21, 41, 81, 61 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 2, 2, 2, 4, 2, 2, 6, 6, 6, 2, 2, 4, 2, 2, 2, 4, ...

			2^2 = 4 -> 4^2 = 16 -> 16^2 = 56 -> 56^2 = 36 -> 36^2 = 96, 96^2 = 16 -> ... (mod 100).

Alexander K. Dewdney, Computer-Kurzweil. Mit einem Computer-Mikroskop untersuchen wir ein Objekt von faszinierender Struktur in der Ebene der komplexen Zahlen. In: Spektrum der Wissenschaft, Oct 1985, p. 8-14, here p. 11-13 (Iterations on a finite set), 14 (Iteration diagram).

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

Cf. A008592, A017329, A376506, A376507, A376509.

A376507 Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 76.

Original entry on oeis.org

18, 24, 26, 32, 68, 74, 76, 82, 118, 124, 126, 132, 168, 174, 176, 182, 218, 224, 226, 232, 268, 274, 276, 282, 318, 324, 326, 332, 368, 374, 376, 382, 418, 424, 426, 432, 468, 474, 476, 482, 518, 524, 526, 532, 568, 574, 576, 582, 618, 624, 626, 632, 668, 674

Offset: 1

Martin Renner, Sep 25 2024

The natural numbers decompose into six categories under the operation of repeated squaring modulo 100, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376506), 25 (cf. A017329), or 76 (this sequence), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (cf. A376508) or 21, 41, 81, 61 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 6, 2, 6, 36, ...

			18^2 = 24 -> 24^2 = 76 -> 76^2 = 76 -> ... (mod 100).

Alexander K. Dewdney, Computer-Kurzweil. Mit einem Computer-Mikroskop untersuchen wir ein Objekt von faszinierender Struktur in der Ebene der komplexen Zahlen. In: Spektrum der Wissenschaft, Oct 1985, p. 8-14, here p. 11-13 (Iterations on a finite set), 14 (Iteration diagram).

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

Cf. A008592, A017329, A376506, A376508, A376509.

G.f.: 2*x*(9 + 3*x + x^2 + 3*x^3 + 9*x^4)/((1 - x)^2*(1 + x + x^2 + x^3)). - Stefano Spezia, Sep 26 2024

A376506 Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 1.

Original entry on oeis.org

1, 7, 43, 49, 51, 57, 93, 99, 101, 107, 143, 149, 151, 157, 193, 199, 201, 207, 243, 249, 251, 257, 293, 299, 301, 307, 343, 349, 351, 357, 393, 399, 401, 407, 443, 449, 451, 457, 493, 499, 501, 507, 543, 549, 551, 557, 593, 599, 601, 607, 643, 649, 651, 657

Offset: 1

Martin Renner, Sep 25 2024

The natural numbers decompose into six categories under the operation of repeated squaring modulo 100, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (this sequence), 25 (cf. A017329), or 76 (cf. A376507), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (cf. A376508) or 21, 41, 81, 61 (cf. A376509).

The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 6, 36, 6, 2, ...

			7^2 = 49 -> 49^2 = 1 -> 1^2 = 1 -> ... (mod 100).

Alexander K. Dewdney, Computer-Kurzweil. Mit einem Computer-Mikroskop untersuchen wir ein Objekt von faszinierender Struktur in der Ebene der komplexen Zahlen. In: Spektrum der Wissenschaft, Oct 1985, p. 8-14, here p. 11-13 (Iterations on a finite set), 14 (Iteration diagram).

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

Cf. A008592, A017329, A376507, A376508, A376509.

G.f.: x*(1 + 6*x + 36*x^2 + 6*x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3)). - Stefano Spezia, Sep 26 2024

A367296 Numbers k such that 8 is the first digit of 2^k.

Original entry on oeis.org

3, 13, 23, 33, 43, 106, 116, 126, 136, 146, 199, 209, 219, 229, 239, 302, 312, 322, 332, 342, 395, 405, 415, 425, 435, 498, 508, 518, 528, 538, 591, 601, 611, 621, 631, 684, 694, 704, 714, 724, 787, 797, 807, 817, 827, 880, 890, 900, 910, 920, 983, 993, 1003

Offset: 1

Martin Renner, Nov 12 2023

The asymptotic density of this sequence is log_10(9/8) = 0.051152...

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

Cf. A000079, A067497, A067469, A172404, A367294, A363060, A367295, A330243, A097415.

x := 1:
L := []:
for n from 0 to 10^3 do
  if 8 <= x and x < 9 then
    L := [op(L), n]
  fi;
  x := 2*x;
  if x > 10 then
    x := (1/10)*x fi;
od:
L;
# alternative:
select(t -> floor(2^t/10^ilog10(2^t))=8, [$1..10^4]); # Robert Israel, Nov 12 2024

Select[Range[1010], IntegerDigits[2^#][[1]] == 8 &] (* Amiram Eldar, Nov 12 2023 *)

from itertools import islice
def A367296_gen(): # generator of terms
    a, b, c, l = 8, 9, 1, 0
    while True:
        if a<=c:
            if cA367296_list = list(islice(A367296_gen(),30)) # Chai Wah Wu, Nov 13 2023

cp's OEIS Frontend

User: Martin Renner

A377214 Irregular triangle T(n, k), read by rows with 1 <= k <= p = A000040(n), for the very first solution to the transversal of primes problem.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A376541 Natural numbers whose iterated squaring modulo 1000 eventually enters the 4-cycle 201, 401, 801, 601.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376540 Natural numbers whose iterated squaring modulo 1000 eventually enters the 4-cycle 176, 976, 576, 776.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376539 Natural numbers whose iterated squaring modulo 1000 eventually settles at the attractor 376.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376538 Natural numbers whose iterated squaring modulo 1000 eventually settles at the attractor 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376509 Natural numbers whose iterated squaring modulo 100 eventually enters the 4-cycle 21, 41, 81, 61.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A376508 Natural numbers whose iterated squaring modulo 100 eventually enters the 4-cycle 16, 56, 36, 96.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A376507 Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 76.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A376506 Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula