A017329 -id:A017329 - cp's OEIS frontend

A094884 Decimal expansion of phi/sqrt(2), where phi = (1+sqrt(5))/2.

1, 1, 4, 4, 1, 2, 2, 8, 0, 5, 6, 3, 5, 3, 6, 8, 5, 9, 5, 2, 0, 0, 1, 4, 5, 5, 6, 7, 1, 6, 0, 6, 0, 4, 1, 5, 3, 0, 7, 2, 3, 0, 6, 7, 5, 3, 6, 7, 5, 5, 4, 1, 2, 2, 5, 0, 0, 8, 5, 4, 6, 1, 4, 7, 6, 9, 5, 8, 3, 1, 7, 2, 9, 2, 7, 5, 3, 3, 6, 3, 1, 5, 0, 4, 8, 6, 5, 8, 9, 1, 0, 6, 7, 6, 7, 3, 5, 4, 6

Offset: 1

N. J. A. Sloane, Jun 15 2004

An algebraic number with minimal polynomial 4*x^4 - 6*x^2 + 1. - Charles R Greathouse IV, Mar 25 2014

			1.144122805635368595200145567160604153072306753675541225...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A001622 (phi), A002193 (sqrt(2)), A017329, A094887, A239798.

SetDefaultRealField(RealField(100)); R:= RealField(); (1+Sqrt(5) )/(2*Sqrt(2)); // G. C. Greubel, Sep 27 2018

RealDigits[GoldenRatio/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Feb 11 2015 *)

sqrt(sqrt(5)+3)/2 \\ Charles R Greathouse IV, Mar 25 2014

Equals Product_{k>=0} (1 + (-1)^k/(10*k+5)). - Amiram Eldar, Nov 23 2024

Equals A094887/2 = sqrt(A239798). - Hugo Pfoertner, Nov 23 2024

A139619 a(n) = 171*n + 19.

Original entry on oeis.org

19, 190, 361, 532, 703, 874, 1045, 1216, 1387, 1558, 1729, 1900, 2071, 2242, 2413, 2584, 2755, 2926, 3097, 3268, 3439, 3610, 3781, 3952, 4123, 4294, 4465, 4636, 4807, 4978, 5149, 5320, 5491, 5662, 5833, 6004, 6175, 6346, 6517, 6688

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 19th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 19th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016957, A017329, A057145, A139600, A139606-A139618, A139620.

Cf. A017173, A051682.

[19*(9*n + 1): n in [0..50]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 100]*171 + 19 (* Paolo Xausa, Apr 17 2024 *)

a(n)=171*n+19 \\ Charles R Greathouse IV, Oct 05 2011

From Chai Wah Wu, Apr 14 2017: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (152*x + 19)/(x - 1)^2. (End)
From Elmo R. Oliveira, Apr 11 2024: (Start)
E.g.f.: 19*exp(x)*(1 + 9*x).
a(n) = 19*A017173(n) = 19*(A051682(n+1) - A051682(n)). (End)

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 3, 0, 1, 2, 4, 1, 0, 1, 3, 5, 4, 6, 0, 1, 4, 6, 7, 9, 3, 0, 1, 5, 7, 10, 12, 9, 10, 0, 1, 6, 8, 13, 15, 15, 16, 6, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, 0, 1, 8, 10, 19, 21, 27, 28, 26, 25, 10, 0, 1, 9, 11, 22, 24, 33, 34, 36, 35

Offset: 0

Omar E. Pol, Feb 05 2012

Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

			Array begins:
(A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
(A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
Cf. A139600.

A322489 Numbers k such that k^k ends with 4.

Original entry on oeis.org

2, 18, 22, 38, 42, 58, 62, 78, 82, 98, 102, 118, 122, 138, 142, 158, 162, 178, 182, 198, 202, 218, 222, 238, 242, 258, 262, 278, 282, 298, 302, 318, 322, 338, 342, 358, 362, 378, 382, 398, 402, 418, 422, 438, 442, 458, 462, 478, 482, 498, 502, 518, 522, 538, 542, 558

Offset: 1

Bruno Berselli, Dec 12 2018

Also numbers k == 2 (mod 4) such that 2^k and k^2 end with the same digit.

Numbers congruent to {2, 18} mod 20. - Amiram Eldar, Feb 27 2023

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

Cf. A004526, A056849.
Subsequence of A139544, A235700.
Numbers k such that k^k ends with d: A008592 (d=0), A017281 (d=1), A067870 (d=3), this sequence (d=4), A017329 (d=5), A271346 (d=6), A322490 (d=7), A017377 (d=9).

GAP
```
List([1..70], n -> 10*n+3*(-1)^n-5);
```

[10*n+3*(-1)^n-5 for n in 1:70] |> println

Magma
```
[10*n+3*(-1)^n-5: n in [1..70]];
```

select(n->n^n mod 10=4,[$1..558]); # Paolo P. Lava, Dec 18 2018

Mathematica
```
Table[10 n + 3 (-1)^n - 5, {n, 1, 60}]
```
Maxima
```
makelist(10*n+3*(-1)^n-5, n, 1, 70);
```

apply(A322489(n)=10*n+3*(-1)^n-5, [1..70]) \\ M. F. Hasler, Dec 14 2018

Vec(2*x*(1 + 8*x + x^2) / ((1 - x)^2*(1 + x)) + O(x^70)) \\ Colin Barker, Dec 13 2018

[10*n+3*(-1)**n-5 for n in range(1, 70)]

Sage
```
[10*n+3*(-1)^n-5 for n in (1..70)]
```

O.g.f.: 2*x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
E.g.f.: 2 + 3*exp(-x) + 5*(2*x - 1)*exp(x).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 10*n + 3*(-1)^n - 5. Therefore:
a(n) = 10*n - 8 for odd n;
a(n) = 10*n - 2 for even n.
a(n+2*k) = a(n) + 20*k.
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(2*Pi/5)*Pi/20 = sqrt(5+2*sqrt(5))*Pi/20. - Amiram Eldar, Feb 27 2023

A332300 The least prime factor of the numerator of Bernoulli(2*n), or 1 if the numerator is 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 691, 7, 3617, 43867, 283, 11, 103, 13, 7, 5, 37, 17, 26315271553053477373, 19, 137616929, 1520097643918070802691, 11, 23, 653, 5, 13, 39409, 7, 29, 2003, 31, 1226592271, 11, 17, 5, 3112655297839, 37, 19, 13, 631, 41, 233, 43, 11, 5, 23, 47, 7823741903

Offset: 0

Amiram Eldar, Feb 09 2020

nonn

a(n)=5 if and only if n is in A017329. - Robert Israel, Feb 09 2020
From Chai Wah Wu, Feb 10 2020: (Start)
For n > 1, clearly if a(n) = n, then n is prime. However, the converse is not true. Prime numbers p such that a(p) != p are: 2, 3, 109, 167, 211, 227, 271, ...
Conjecture: for prime p > 3, p is a prime factor of the numerator of Bernoulli(2*p), thus the conjecture implies that a(p) <= p for prime p.
(End)

			a(10) = 283, since Bernoulli(2*10) = -174611/330, and 283 is the least prime factor of its numerator, 174611 = 283 * 617.

Chai Wah Wu, Table of n, a(n) for n = 0..191 (n = 0..103 from Amiram Eldar)
S. S. Wagstaff, Jr., Factors of Bernoulli numbers.

Cf. A000367, A017329, A020639, A079294, A090947, A242193, A326727.

[n le 4 select 1 else Min(PrimeDivisors(Abs(Numerator(Bernoulli(2*n))))):n in [0..48]]; // Marius A. Burtea, Feb 09 2020

Array[FactorInteger[Abs @ Numerator @  BernoulliB[2*#]][[1, 1]] &, 30, 0]

a(n) = my(x=abs(numerator(bernfrac(2*n)))); if (x==1, 1, vecmin(factor(x)[,1])); \\ Michel Marcus, Feb 09 2020

from sympy import bernoulli, primefactors
def A332300(n):
    x = abs(bernoulli(2*n).p)
    return 1 if x == 1 else min(primefactors(x)) # Chai Wah Wu, Feb 10 2020

a(n) = A020639(abs(A000367(n))).

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.

A017333 a(n) = (10*n + 5)^5.

Original entry on oeis.org

3125, 759375, 9765625, 52521875, 184528125, 503284375, 1160290625, 2373046875, 4437053125, 7737809375, 12762815625, 20113571875, 30517578125, 44840334375, 64097340625, 89466096875, 122298103125, 164130859375, 216699865625, 281950621875, 362050628125, 459401384375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A013663, A016757, A017329.

Cf. A272914 (first comment). [Bruno Berselli, May 26 2016]

[(10*n+5)^5: n in [0..25]]; // Vincenzo Librandi, Aug 02 2011

(10*Range[0,20]+5)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{3125,759375,9765625,52521875,184528125,503284375},20] (* Harvey P. Dale, May 15 2018 *)

G.f.: 3125*(x+1)*(x^4+236*x^3+1446*x^2+236*x+1)/(x-1)^6. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^5.
a(n) = 5^5 * A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/100000.
Sum_{n>=0} (-1)^n/a(n) = Pi^5/960000. (End)

cp's OEIS Frontend

A094884 Decimal expansion of phi/sqrt(2), where phi = (1+sqrt(5))/2.

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Magma

Mathematica

PARI

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A139619 a(n) = 171*n + 19.

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Mathematica

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A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

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A322489 Numbers k such that k^k ends with 4.

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GAP

Julia

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Sage

Formula

A332300 The least prime factor of the numerator of Bernoulli(2*n), or 1 if the numerator is 1.

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Magma

Mathematica

PARI

Python

Formula

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

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Keywords

Comments

Examples

Links

Crossrefs

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

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Author

Keywords

Comments

Examples

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.