Kyle Stern - cp's OEIS frontend

A215414 Unix epoch timestamp for start of year, beginning with 1970.

0, 31536000, 63072000, 94694400, 126230400, 157766400, 189302400, 220924800, 252460800, 283996800, 315532800, 347155200, 378691200, 410227200, 441763200, 473385600, 504921600, 536457600, 567993600, 599616000, 631152000, 662688000, 694224000, 725846400, 757382400

Offset: 1

Kyle Stern, Aug 09 2012

This is based on a naive multiplication of A033172 with a fixed number of seconds per year, 24*3600 = 86400. It ignores that leap years are not regularly occurring after 4 years (but after 400 years, note the formula that relates a(n+4) to a(n) and also the simple Mma implementation), ignores leap seconds, and any other influences that align the slowing down of the Earth rotation in an astronomical fixed coordinate system measured relative to atomic clocks. In summary, the use of "year" in the definition is not commensurate with years in standard astronomical or earth observational terms. - R. J. Mathar, Aug 21 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Unix time
Wikipedia, Leap second
Wikipedia, Revised Julian Calendar
Wikipedia, International Earth Rotation and Reference Systems Service

Cf. A033172, A193910.

lst = {}; t = 86400; Do[e = t*(365*(n - 1) + Ceiling[n/4]); If[! Mod[n, 4] == 0, e = e - t]; AppendTo[lst, e], {n, 25}]; lst (* Arkadiusz Wesolowski, Aug 20 2012 *)
CoefficientList[Series[86400*(365*x + 365*x^2 + 366*x^3 + 365*x^4)/((x - 1)^2*(1 +x +x^2 +x^3)), {x,0,50}], x] (* G. C. Greubel, Feb 26 2017 *)

x='x+O('x^50); Vec(86400*(365*x +365*x^2 +366*x^3 +365*x^4)/((1-x)^2*(1+x+x^2+x^3))) \\ G. C. Greubel, Feb 26 2017

From Alexander R. Povolotsky, Aug 20 2012: (Start)
a(n) = 10800*(2922*n + (-1)^n + (1+i)*(-i)^n + (1-i)*i^n - 2923).
a(n+4) = a(n) + 126230400.
G.f.: 86400*(365*x +365*x^2 +366*x^3 +365*x^4)/((1-x)^2*(1+x+x^2+x^3)). (End)

a(11)-a(25) from Arkadiusz Wesolowski, Aug 20 2012

A190760 Product of digits is divisible by number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 94, 96, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 113, 116, 119

Offset: 1

Kyle Stern, May 18 2011

Almost all numbers are in this sequence: there are at least n - 1.125 n^0.95... elements up to n, where the exponent is log(9)/log(10). - Charles R Greathouse IV, May 20 2011

			3*8*2 = 48 and 48 is divisible by the number of digits, 3, so 382 is included.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A007954, A055642, A061383.

A190760 := proc(n) option remember: local k: if(n=1)then return 0: fi: for k from procname(n-1)+1 do if(mul(d,d=convert(k,base,10)) mod length(k) = 0)then return k: fi: od: end: seq(A190760(n),n=1..100); # Nathaniel Johnston, May 19 2011

A178968 Numbers that are represented in Roman numerals by exactly four letters.

Original entry on oeis.org

8, 13, 17, 22, 24, 26, 29, 31, 35, 42, 44, 46, 49, 53, 57, 62, 64, 66, 69, 71, 75, 80, 92, 94, 96, 99, 103, 107, 112, 114, 116, 119, 121, 125, 130, 141, 145, 152, 154, 156, 159, 161, 165, 170, 191, 195, 202, 204, 206, 209, 211, 215, 220, 240, 251, 255, 260

Offset: 1

Kyle Stern, Jan 01 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..215 (complete up to 3999)

Cf. A142958.

for n from 1 to 3999 do if(length(convert(n, roman)) = 4)then printf("%d, ", n): fi: od: # Nathaniel Johnston, May 18 2011

Select[Range[300],StringLength[RomanNumeral[#]]==4&] (* Harvey P. Dale, Aug 20 2021 *)

Extended by Nathaniel Johnston, May 18 2011

A180961 Numbers such that the American English name of the number has four syllables.

Original entry on oeis.org

27, 37, 47, 57, 67, 71, 72, 73, 74, 75, 76, 78, 79, 87, 97, 101, 102, 103, 104, 105, 106, 108, 109, 110, 112, 201, 202, 203, 204, 205, 206, 208, 209, 210, 212, 301, 302, 303, 304, 305, 306, 308, 309, 310, 312, 401, 402, 403, 404, 405, 406, 408, 409, 410, 412

Offset: 1

Kyle Stern, Sep 28 2010

There are a finite number of terms, considering all terms up to 10^66 using English names of large numbers and various conventional extensions thereof (see Wikipedia link). - Michael S. Branicky, May 27 2024

			27 = "twen-ty sev-en", 101 = "one hun-dred one"

Kyle Stern, Table of n, a(n) for n = 1..100
Wikipedia, Names of Large Numbers

Cf. A075774.

# uses function in A075774
print([k for k in range(500) if A075774(k) == 4]) # Michael S. Branicky, May 27 2024

A075774(a(n)) = 4. - Michael S. Branicky, May 27 2024

Corrected by Kyle Stern, Sep 30 2010

A160666 Numbers whose distance to the closest prime number is a prime number.

Original entry on oeis.org

0, 9, 15, 21, 25, 26, 27, 33, 34, 35, 39, 45, 49, 50, 51, 55, 56, 57, 63, 64, 65, 69, 75, 76, 77, 81, 85, 86, 87, 91, 92, 94, 95, 99, 105, 111, 115, 116, 118, 120, 122, 124, 125, 129, 133, 134, 135, 141, 142, 144, 146, 147, 153, 154, 155, 159, 160, 161, 165, 169, 170

Offset: 1

Kyle Stern, May 22 2009

nonn

Terms n=2..31 are identical to terms n=1..30 of A079364.

K. Stern, Table of n, a(n) for n=1..10000

Cf. A000040, A051699.

isA160666 := proc(n) local ppl,pmi ; if isprime(n) then RETURN(false): elif n =0 then RETURN(true): elif n =1 then RETURN(false): fi; ppl := nextprime(n)-n ; pmi := n-prevprime(n) ; RETURN (isprime(min(ppl,pmi)) ) ; end: for n from 0 to 200 do if isA160666(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, May 25 2009

fQ[n_] := PrimeQ[ Min[ NextPrime[n] - n, n - NextPrime[n, -1]]]; Select[ Range[0, 174], !PrimeQ@ # && fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

A161750 Numbers n such that the decimal digits of 123456789*n are all distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 20, 22, 23, 25, 26, 31, 32, 34, 35, 40, 41, 43, 44, 50, 52, 53, 61, 62, 70, 71, 80

Offset: 1

Kyle Stern, Jun 17 2009

David Wolfe, When Multiplication Mixes Up Digits, Mathematics Magazine, Volume 80, Number 5, December 2007, pp.380-383

Cf. A053654. [From Zak Seidov, Nov 04 2009]

a := proc (n) local nn, nnn: nn := convert(123456789*n, base, 10): nnn := convert(nn, set): if nops(nn) = nops(nnn) then n else end if end proc: seq(a(n), n = 1 .. 80); # Emeric Deutsch, Jun 21 2009

m=123456789;se=Select[Range[0,99],Sort[id=IntegerDigits[m*# ]]==Union[id]&] (* Zak Seidov, Nov 04 2009 *)
Select[Range[0,100],Max[DigitCount[123456789#]]==1&] (* Harvey P. Dale, Jun 02 2017 *)

a(1)=0 added by Zak Seidov, Nov 04 2009

Edited by Charles R Greathouse IV, Aug 02 2010

A165723 The (d+1)th digit after the decimal point in the decimal representation of 1/n, where d is the number of digits of n.

Original entry on oeis.org

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

Offset: 1

Kyle Stern, Sep 24 2009

			1/1 = 1.000 so a(1) = 0;
1/4 = 0.250 so a(4) = 5;
1/14 = 0.0714... so a(14) = 1;
1/114 = 0.00877... so a(114) = 7.

K. Stern, Table of n, a(n) for n = 1..500

A055642 := proc(n) max(1,ilog10(n)+1) ; end: A165723 := proc(n) local d; d := A055642(n) ; floor(10^(d+1)/n) mod 10 ; end: seq(A165723(n),n=1..100) ; R. J. Mathar, Sep 26 2009

More terms from R. J. Mathar, Sep 26 2009

A160180 Largest proper divisor of the n-th composite number.

Original entry on oeis.org

2, 3, 4, 3, 5, 6, 7, 5, 8, 9, 10, 7, 11, 12, 5, 13, 9, 14, 15, 16, 11, 17, 7, 18, 19, 13, 20, 21, 22, 15, 23, 24, 7, 25, 17, 26, 27, 11, 28, 19, 29, 30, 31, 21, 32, 13, 33, 34, 23, 35, 36, 37, 25, 38, 11, 39, 40, 27, 41, 42, 17, 43, 29, 44, 45, 13, 46, 31, 47, 19, 48, 49, 33, 50, 51, 52

Offset: 1

Kyle Stern, May 03 2009, May 04 2009

Old name: The n-th positive composite number divided by its lowest nontrivial factor.

			a(1) = 4/2 = 2, a(2) = 6/2 = 3, a(3) = 8/2 = 4, a(4) = 9/3 = 3, a(5) = 10/2 = 5.

K. Stern, Table of n, a(n) for n = 1..9999

Cf. A002808, A032742, A056608, A163870, A144925.

a160180 = a032742 . a002808  -- Reinhard Zumkeller, Mar 29 2014

function [a] = A160180(k) j = 0; n = 1; while j < k if isprime(n) == 1 skip elseif isprime(n) == 0 j = j + 1; factors = factor(n); lowfactor = factors(1,1); a(j,1) = n/lowfactor; end n = n + 1; end - Kyle Stern, May 04 2009

f[n_] := Block[{k = n + PrimePi@ n + 1}, While[k != n + PrimePi@ k + 1, k++ ]; k/FactorInteger[k][[1, 1]]]; Array[f, 75] (* Robert G. Wilson v, May 11 2012 *)
Divisors[#][[-2]]&/@Select[Range[200],CompositeQ] (* Harvey P. Dale, Dec 06 2021 *)
(# / FactorInteger[#][[1, 1]])& /@ Select[Range[300], CompositeQ] (* Amiram Eldar, Jun 18 2022 *)

a(n) = A032742(A002808(n)) = A002808(n) / A056608(n) = A163870(n,A144925(n)). - Reinhard Zumkeller, Mar 29 2014

Indices of b-file corrected, more terms added using b-file. - N. J. A. Sloane, Aug 31 2009
New name from Reinhard Zumkeller, Mar 29 2014
Incorrect formula removed by Ridouane Oudra, Oct 15 2021

A160522 The n-th odd composite number minus the n-th even composite number.

Original entry on oeis.org

5, 9, 13, 15, 15, 19, 19, 21, 25, 27, 27, 29, 29, 33, 33, 35, 39, 39, 41, 43, 43, 45, 45, 45, 47, 51, 55, 57, 57, 57, 57, 57, 57, 59, 61, 61, 65, 65, 65, 65, 69, 69, 71, 71, 73, 75, 75, 77, 77, 81, 81, 81, 81, 85, 89, 89, 89, 89, 89, 91, 91, 91, 91, 91, 93, 97, 99, 99, 103, 103

Offset: 1

Kyle Stern, May 16 2009

K. Stern, Table of n, a(n) for n = 1..1000

Cf. A002808, A005843, A071904, A047846.

composite function [a] = A160522(k) j = 1; n = 1; even = 4; while j < k n = n + 1; if isprime(n) == 1 else if mod(n,2) == 0 else a(j,1) = n - even; even = even + 2; j = j + 1; end end end

Last[t = GatherBy[Select[Range[4, 245], ! PrimeQ[#] &], OddQ]] - Take[First[t], Length[Last[t]]] (* Jayanta Basu, Aug 11 2013 *)

m=70; v=vector(m); k=4; n=0; while(n0&&!isprime(k), n++; v[n]=k-2*(n+1)); k++); v \\ Klaus Brockhaus, May 22 2009

from sympy import primepi
def A160522(n):
    if n == 1: return 5
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return m-(n+1<<1) # Chai Wah Wu, Aug 01 2024

a(n) = A071904(n) - A005843(n+1).

Extended and formula edited by Klaus Brockhaus, May 22 2009

cp's OEIS Frontend

User: Kyle Stern

A215414 Unix epoch timestamp for start of year, beginning with 1970.

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A190760 Product of digits is divisible by number of digits.

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Maple

A178968 Numbers that are represented in Roman numerals by exactly four letters.

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A180961 Numbers such that the American English name of the number has four syllables.

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Python

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A160666 Numbers whose distance to the closest prime number is a prime number.

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A161750 Numbers n such that the decimal digits of 123456789*n are all distinct.

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Maple

Mathematica

Extensions

A165723 The (d+1)th digit after the decimal point in the decimal representation of 1/n, where d is the number of digits of n.

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Programs

Maple

Extensions

A160180 Largest proper divisor of the n-th composite number.

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