A008592 -id:A008592 - cp's OEIS frontend

A317633 Numbers congruent to {1, 7, 9} mod 10.

1, 7, 9, 11, 17, 19, 21, 27, 29, 31, 37, 39, 41, 47, 49, 51, 57, 59, 61, 67, 69, 71, 77, 79, 81, 87, 89, 91, 97, 99, 101, 107, 109, 111, 117, 119, 121, 127, 129, 131, 137, 139, 141, 147, 149, 151, 157, 159, 161, 167, 169

Offset: 1

Paul Curtz, Aug 02 2018

When multiplied by 10, one gets the numbers ending in "dix" in French (10, 70, 90, 110, ...).

			G.f. = x + 7*x^2 + 9*x^3+ 11*x^4 + 17*x^5 + 19*x^6 + 21*x^7 + 27*x^8 + ... - _Michael Somos_, Aug 19 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A008592, A010692.

[n: n in [0..170]|n mod 10 in {1, 7, 9}]; // Vincenzo Librandi, Aug 05 2018

Table[2 n + 4 Floor[(n + 1)/3] - 1, {n, 1, 60}] (* Bruno Berselli, Jul 02 2018 *)
Select[Range[0, 250], MemberQ[{1, 7, 9}, Mod[#, 10]]&] (* Vincenzo Librandi, Aug 05 2018 *)
CoefficientList[ Series[(x^3 + 2x^2 + 6x + 1)/((x - 1)^2 (x^2 + x + 1)), {x, 0, 60}], x] (* or *)
LinearRecurrence[{1, 0, 1, -1}, {1, 7, 9, 11}, 61] (* Robert G. Wilson v, Aug 08 2018 *)

x='x+O('x^60); Vec(x*(1+6*x+2*x^2+x^3)/((1-x)^2*(1+x+x^2))) \\ G. C. Greubel, Aug 08 2018

a(n) = a(n-3) + 10, a(1) = 1, a(2) = 7, a(3) = 9.
From Bruno Berselli, Jul 02 2018: (Start)
G.f.: x*(1 + 6*x + 2*x^2 + x^3)/((1 - x)^2*(1 + x + x^2)).
a(n) = 2*n + 4*floor((n+1)/3) - 1. (End)

Definition from Jianing Song, Aug 02 2018

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

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.

A059691 Carryless product 12 X n base 10.

Original entry on oeis.org

0, 12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144, 156, 168, 170, 182, 194, 106, 118, 240, 252, 264, 276, 288, 290, 202, 214, 226, 238, 360, 372, 384, 396, 308, 310, 322, 334, 346, 358, 480, 492, 404, 416, 428, 430, 442, 454, 466, 478, 500, 512, 524, 536

Offset: 0

Henry Bottomley, Feb 19 2001

			a(97)=954 since we have 12 X 97 = carryless sum of 900, 80, 70 and 4 = 954

Seiichi Manyama, Table of n, a(n) for n = 0..9999
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X n base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.

A104045 Numbers k such that k9 is prime and k is a multiple of ten.

Original entry on oeis.org

10, 40, 50, 70, 80, 100, 110, 140, 160, 170, 230, 260, 290, 310, 320, 370, 440, 490, 500, 520, 530, 670, 710, 730, 800, 820, 860, 910, 920, 1000, 1070, 1090, 1190, 1210, 1240, 1280, 1300, 1310, 1330, 1370, 1400, 1580, 1720, 1750, 1760, 1790, 1900, 1930, 1960, 1970, 2050, 2080, 2210

Offset: 1

Parthasarathy Nambi, Mar 01 2005

			If k =  10, then k9 =  109 (prime).
If k = 160, then k9 = 1609 (prime).
If k = 320, then k9 = 3209 (prime).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A030433, A008592, A102700, A166560 (resulting primes).

select(n-> isprime(10*n+9), [10*i$i=1..300])[];  # Alois P. Heinz, Jan 19 2024

Select[Range[10,2210,10],PrimeQ[FromDigits[Prepend[{9},#]]]&] (* James C. McMahon, Jan 19 2024 *)

from sympy import isprime
from itertools import count, islice
def agen(): yield from (k for k in count(10, 10) if isprime(10*k+9))
print(list(islice(agen(), 53))) # Michael S. Branicky, Jan 19 2024

A121719 Strings of digits which are composite regardless of the base in which they are interpreted. Exclude bases in which numbers are not interpretable.

Original entry on oeis.org

4, 6, 8, 9, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 42, 44, 46, 48, 50, 55, 60, 62, 63, 64, 66, 68, 69, 70, 77, 80, 82, 84, 86, 88, 90, 93, 96, 99, 100, 110, 112, 114, 116, 118, 120, 121, 130, 132, 134, 136, 138, 140, 143, 144

Offset: 1

Tanya Khovanova, Sep 08 2006

Comments from Franklin T. Adams-Watters:
"Think of these as polynomials. E.g. 121 is the polynomial n^2+2n+1. There are three cases:
"(1) If the coefficients (digits) all have a common factor, the result will be divisible by that factor.
"(2) If the polynomial can be factored, the numbers will be composite. n^2+2n+1 = (n+1)^2, so it is always composite.
"(3) Otherwise, look at the polynomial modulo primes up to its degree. For example, 112 (n^2+n+2, degree 2) modulo 2 is always 0, so it is always divisible by 2.
"Note that condition (1) is really a special case of condition (2), where one of the factors is a constant.
"If none of the above conditions apply, the polynomial will (probably) have prime values."
From Iain Fox, Sep 02 2020: (Start)
lim_{k->infinity} (1/k)*Sum_{i=1..k} a_c(i) > .3 if it exists, where a_c(n) is the characteristic function of a(n) (1 if n is in a(n), otherwise 0).
If the Bunyakovsky conjecture is true, the list of reasons a number is in this sequence detailed by Franklin T. Adams-Watters above is a complete list.
If the Bunyakovsky conjecture and the Extended Riemann Hypothesis are true, the above limit equals 4340435807/13235512500 = 0.3279386... (proof by Ravi Fernando in link by Iain Fox).
All members of A008592 except 1 and 10 are in this sequence.
(End)

			String 55 in every base in which it is interpretable is divisible by 5. String 1001 in base a is divisible by a+1. Hence 55 and 1001 both belong to this sequence.

Iain Fox, Table of n, a(n) for n = 1..10000
Iain Fox, What percentage of positive integers, written in base 10, are composite regardless of what base they are interpreted in?, Math StackExchange, September 2020.
Ed Pegg, Jr., Bouniakowsky Conjecture.
Eric W. Weisstein, Extended Riemann Hypothesis.
Wikipedia, Bunyakovsky conjecture.
Wikipedia, Generalized Riemann Hypothesis.

Supersequence: A002808.

Cf. A008592, A267509.

is(n)=if(n<10, return(!isprime(n)&&n>1)); if(content(n=digits(n))>1, return(1)); if(vecsum(factor(n*=vectorv(#n, i, x^(#n-i)))[,2])>1, return(1)); forprime(p=2, #n-1, for(x=1, p, if(eval(n)%p, next(2))); return(1)); for(x=vecmax(Vec(n))+1, +oo, if(isprime(eval(n)), return(0))) \\ Iain Fox, Aug 31 2020

More terms from Franklin T. Adams-Watters, Sep 12 2006

A166150 a(n) = 5n^2 + 5n - 9.

Original entry on oeis.org

1, 21, 51, 91, 141, 201, 271, 351, 441, 541, 651, 771, 901, 1041, 1191, 1351, 1521, 1701, 1891, 2091, 2301, 2521, 2751, 2991, 3241, 3501, 3771, 4051, 4341, 4641, 4951, 5271, 5601, 5941, 6291, 6651, 7021, 7401, 7791, 8191, 8601, 9021, 9451, 9891, 10341

Offset: 1

Vincenzo Librandi, Oct 08 2009

First differences are in A008592.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008592.

[5*n^2+5*n-9: n in [1..45]]; // Vincenzo Librandi, Sep 13 2013

A166150:=n->5*n^2+5*n-9: seq(A166150(n), n=1..100); # Wesley Ivan Hurt, May 01 2016

Table[(5 n^2 + 5  n - 9), {n, 50}] (* or *) CoefficientList[Series[(1 + 18 x - 9 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 13 2013 *)
LinearRecurrence[{3,-3,1},{1,21,51}, 50] (* G. C. Greubel, May 01 2016 *)

a(n)=5*n*(n+1)-9 \\ Charles R Greathouse IV, Jan 11 2012

a(n) = a(n-1) + 10*n (with a(1)=1).
G.f.: x*(1+18*x-9*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 13 2013
From G. C. Greubel, May 01 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (5*x^2 + 10*x - 9)*exp(x) + 9. (End)
Sum_{n>=1} 1/a(n) = 1/9 + (Pi/sqrt(205))*tan(sqrt(41/5)*Pi/2). - Amiram Eldar, Feb 20 2023

a(29)-a(45) corrected by Charles R Greathouse IV, Jan 11 2012

New name from Charles R Greathouse IV, Jan 11 2012

cp's OEIS Frontend

A317633 Numbers congruent to {1, 7, 9} mod 10.

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

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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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A059691 Carryless product 12 X n base 10.

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A104045 Numbers k such that k9 is prime and k is a multiple of ten.

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A166150 a(n) = 5n^2 + 5n - 9.