A008592 -id:A008592 - cp's OEIS frontend

A169823 Multiples of 60.

0, 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1200, 1260, 1320, 1380, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1980, 2040, 2100, 2160, 2220, 2280, 2340, 2400, 2460, 2520, 2580, 2640, 2700

Offset: 0

N. J. A. Sloane, May 29 2010

Numbers that are divisible by all of 1, 2, 3, 4, 5, 6.

Erich Friedman, What's Special About This Number? (See the entry for "60")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008585, A008586, A008587, A008588, A008592, A008594.

Cf. A008597, A008602, A169825, A169827, A249674.

Range[0, 2700, 60] (* Vladimir Joseph Stephan Orlovsky, Jul 12 2011 *)

a(n)=60*n \\ Charles R Greathouse IV, Mar 19 2017

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 60*x/(x-1)^2.
E.g.f.: 60*x*exp(x).
a(n) = 60*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A249674(n) = 3*A008602(n) = 4*A008597(n) = 5*A008594(n) = 6*A008592(n) = 10*A008588(n) = 12*A008587(n) = 15*A008586(n) = 20*A008585(n) = 30*A005843(n) = 60*A001477(n) = A169827(n)/14 = A169825(n)/7. (End)

A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 0, 2, 2, 0, 6, 0, 0, 7, 2, 5, 6, 5, 2, 7, 0, 0, 8, 4, 8, 0, 0, 8, 4, 8, 0, 0, 9, 6, 1, 4, 5, 4, 1, 6, 9, 0, 0, 10, 8, 4, 8, 0, 0, 8, 4, 8, 10, 0, 0, 11, 20, 7, 2, 5, 6, 5, 2, 7, 20, 11, 0

Offset: 0

Henry Bottomley, Feb 19 2001

			Table begins:
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0 ...
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ...
  0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 20 ...
  0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 30, 33, 36, 39, 32, 35 ...
  0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40 ...
  ...
T(12, 97) = 954 since we have 12 X 97 = carryless sum of 900, (180 mod 100=)80, 70 and (14 mod 10=)4 = 954.

Stefano Spezia, First 140 antidiagonals of the table, flattened
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 10 base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.

Cf. A048720 (binary), A325820 (ternary).

len[num_]:=Length[IntegerDigits[num]]; digit[num_,d_]:=Part[IntegerDigits[num],d]; T[i_, j_] := FromDigits[Reverse[CoefficientList[PolynomialMod[Sum[digit[i,c]*x^(len[i]-c), {c, len[i]}]*Sum[digit[j,r]*x^(len[j]-r), {r, len[j]}], 10], x]]]; Flatten[Table[T[i - j, j], {i, 0, 12}, {j, 0, i}]] (* Stefano Spezia, Sep 26 2022 *)

T(n,k) = fromdigits(lift(Vec( Mod(Pol(digits(n)),10) * Pol(digits(k))))); \\ Kevin Ryde, Sep 27 2022

Minor edits by N. J. A. Sloane, Aug 24 2010

A107730 Numbers n such that prime(n+1) has the same last digit as prime(n).

Original entry on oeis.org

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 154, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 259, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 442, 443, 474, 480, 491, 497, 500, 502, 518

Offset: 1

Jonathan Vos Post, Jun 12 2007

			a(1) = 34 because prime(34) = 139, prime(35) = 149, both end with the digit 9.
a(2) = 42 because prime(42) = 181, prime(43) = 191, both end with the digit 1.
a(4) = 61 because prime(61) = 283, prime(62) = 293, both end with the digit 3.
a(5) = 68 because prime(68) = 337, prime(69) = 347, both end with the digit 7.

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Fred B. Holt, On the last digits of consecutive primes, arXiv:1604.02443 [math.NT], 2016.
Robert J. Lemke Oliver, Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
Index entries for primes, gaps between

Cf. A000040, A129750.

Union of rows r == 0 (mod 5) of A174349. Indices of multiples of 10 (A008592) in A001223.

P:=List(Filtered([1..4000],IsPrime),n->Reversed(ListOfDigits(n)));;
a:=Filtered([1..Length(P)-1],i->P[i+1][1]=P[i][1]); # Muniru A Asiru, Oct 31 2018

isA107730 := proc(n) local ldign, ldign2 ; ldign := convert(ithprime(n),base,10) ; ldign2 := convert(ithprime(n+1),base,10) ; if op(1,ldign) = op(1,ldign2) then true ; else false ; fi ; end: for n from 1 to 600 do if isA107730(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jun 15 2007

Select[Range[200],IntegerDigits[Prime[ # ]][[ -1]]==IntegerDigits[Prime[ #+1]][[ -1]]&] (* Stefan Steinerberger, Jun 14 2007 *)
Flatten[Position[Partition[Prime[Range[600]],2,1],?(Mod[#[[1]],10] == Mod[#[[2]],10]&),{1},Heads->False]] (* _Harvey P. Dale, Aug 20 2015 *)

isok(n) = (prime(n) % 10) == prime(n+1) % 10; \\ Michel Marcus, Feb 16 2017

is_A107730(n)=!((nextprime(1+n=prime(n))-n)%10) \\ This (...) is twice as fast as prime(n+1)-prime(n), and prime(n) becomes very slow for n > 41538, even with primelimit = 10^7. - M. F. Hasler, Oct 24 2018

Numbers n such that A000040(n)==A000040(n+1) mod 10, or A000040(n+1) - A000040(n) = 10*k for some integer k, or n such that A129750(n) = 0. [Corrected and edited by M. F. Hasler, Oct 24 2018]
A107730 = A001223^(-1)(A008592) = { i > 0 | A001223(i) == 0 (mod 10)} = U_{k>0} {A174349(5k,j); j >= 1}. - M. F. Hasler, Oct 24 2018
Union of A320703, A320708, A320713, A320718, ... A116493,..., A116496 ... etc. - R. J. Mathar, Apr 30 2024

More terms from Stefan Steinerberger and R. J. Mathar, Jun 14 2007

A291639 Numbers k such that 0 is the smallest decimal digit of k^3.

Original entry on oeis.org

10, 16, 20, 22, 30, 34, 37, 40, 42, 43, 47, 48, 50, 52, 59, 60, 63, 67, 69, 70, 73, 74, 79, 80, 84, 86, 87, 89, 90, 93, 94, 99, 100, 101, 102, 103, 106, 107, 109, 110, 112, 115, 116, 117, 118, 120, 123, 124, 126, 127, 128, 130, 131, 134, 135, 138, 140, 141

Offset: 1

Colin Barker, Aug 28 2017

The sequence is infinite. For example, A062397(i) is in the sequence for any i > 1, since A168575(i) contains the digit 0 for any i > 1. - Felix Fröhlich, Aug 28 2017

Also contains A008592, and has asymptotic density 1. - Robert Israel, Aug 29 2017

			16 is in the sequence because 16^3 = 4096, the smallest decimal digit of which is 0.

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

Cf. A008592, A062397, A168575, A291640, A291641, A291642, A291643, A291644.

select(n -> min(convert(n^3,base,10))=0, [$1..1000]); # Robert Israel, Aug 29 2017

Select[Range[150],DigitCount[#^3,10,0]>0&] (* Harvey P. Dale, Feb 03 2025 *)

select(k->vecmin(digits(k^3))==0, vector(500, k, k))

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

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

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.

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.

A166730 Positive integers with English names ending in "y".

Original entry on oeis.org

20, 30, 40, 50, 60, 70, 80, 90, 120, 130, 140, 150, 160, 170, 180, 190, 220, 230, 240, 250, 260, 270, 280, 290, 320, 330, 340, 350, 360, 370, 380, 390, 420, 430, 440, 450, 460, 470, 480, 490, 520, 530, 540, 550, 560, 570, 580, 590, 620, 630, 640, 650, 660

Offset: 1

Rick L. Shepherd, Oct 20 2009

			Twenty (20) is a term. Ten (10) is not a term (but is a term of A060228); one hundred (100) is not a term (but is a term of A166731); one million (1000000) is not a term (but is in A060228).

Cf. A008592, A166726, A166727, A166728, A166729, A166731, A059093, A060228.

A008592 MINUS {n | n = 0 mod 100 or n = 10 mod 100}.

A349278 a(n) is the product of the sum of the last i digits of n, with i going from 1 to the total number of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 0, 4, 10, 18, 28, 40, 54, 70, 88, 108, 0, 5, 12, 21, 32, 45, 60, 77, 96, 117, 0, 6, 14, 24, 36, 50, 66, 84, 104, 126, 0, 7, 16, 27, 40, 55, 72, 91, 112, 135, 0

Offset: 1

Michel Marcus, Nov 13 2021

This is similar to A349194 but with digits taken in reversed order.
The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Dec 04 2021
The positive terms form a subsequence of A349194. - Bernard Schott, Dec 19 2021

			For n=256, a(256) = 6*(6+5)*(6+5+2) = 858.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A349194, A349279 (fixed points).

Cf. A349864, A349865.

a[n_] := Times @@ Accumulate @ Reverse @ IntegerDigits[n]; Array[a, 70] (* Amiram Eldar, Nov 13 2021 *)

a(n) = my(d=Vecrev(digits(n))); prod(i=1, #d, sum(j=1, i, d[j]));

from math import prod
from itertools import accumulate
def a(n): return 0 if n%10==0 else prod(accumulate(map(int, str(n)[::-1])))
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Nov 13 2021

From Bernard Schott, Dec 04 2021: (Start)
a(n) = 0 iff n is a multiple of 10 (A008592).
a(n) = 1 iff n = 1.
a(n) = 2 (resp. 3, 4, 5, 7, 9) iff n = 10^k+1 (A000533) (resp. 2*10^k+1 (A199682), 3*10^k+1 (A199683), 4*10^k+1 (A199684), 6*10^k+1 (A199686), 8*10^k+1 (A199689)).
a(R_n) = n! where R_n = A002275(n) is repunit > 0, and n! = A000142(n).
a(n) = A349194(n) if n is palindrome (A002113). (End)

cp's OEIS Frontend

A169823 Multiples of 60.

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A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

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A107730 Numbers n such that prime(n+1) has the same last digit as prime(n).

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A291639 Numbers k such that 0 is the smallest decimal digit of k^3.

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

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