A376508 -id:A376508 - cp's OEIS frontend

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

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

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

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.

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.

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.

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.

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.

cp's OEIS Frontend

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

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

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

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

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

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

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

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