cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 30, 32, 36, 38, 42, 44, 60, 62, 66, 68, 72, 74, 90, 92, 96, 98, 102, 104, 120, 122, 126, 128, 132, 134, 210, 212, 216, 218, 222, 224, 240, 242, 246, 248, 252, 254, 270, 272, 276, 278, 282, 284, 300, 302, 306, 308, 312, 314, 330, 332, 336, 338, 342, 344, 420, 422, 426, 428, 432, 434, 450, 452, 456, 458, 462, 464, 480, 482, 486, 488
Offset: 0

Views

Author

Antti Karttunen, Aug 24 2016

Keywords

Examples

			   n   A049345  with one zero           converted back
                appended to the right   to decimal = a(n)
---------------------------------------------------------
   0       0            00                     0
   1       1            10                     2
   2      10           100                     6
   3      11           110                     8
   4      20           200                    12
   5      21           210                    14
   6     100          1000                    30
   7     101          1010                    32
   8     110          1100                    36
   9     111          1110                    38
  10     120          1200                    42
  11     121          1210                    44
  12     200          2000                    60
  13     201          2010                    62
  14     210          2100                    66
  15     211          2110                    68
  16     220          2200                    72

Links

Crossrefs

Complement: A276155.
Cf. A002110, A003961, A049345, A276085, A276086, A276151, A276152, A286629 [= a(A061720(n-1))], A324384 [= gcd(n, a(n))], A323879, A328770 (a subsequence).
Cf. also A276156, A328461, A328464.
Dispersion array and its transpose: A276943, A276945, with primorials divided out: A286623, A286625.
Analogous to A153880.

Programs

  • Mathematica
    nn = 75; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[#] <= nn &]]; Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, 75}] (* Michael De Vlieger, Aug 26 2016 *)
  • PARI
    A276154(n) = A276085(A003961(A276086(n))); \\ Antti Karttunen, Mar 15 2021
  • PARI
    A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n)))); \\ Antti Karttunen, Mar 15 2021
  • Scheme
    (definec (A276154 n) (if (zero? n) n (+ (A276152 n) (A276154 (A276151 n)))))

Formula

a(0) = 0; for n >= 1, a(n) = A276152(n) + a(A276151(n)).
a(n) = A276085(A003961(A276086(n))). - Antti Karttunen, Mar 15 2021

A061720 First differences of sequence of primorials.

Original entry on oeis.org

1, 4, 24, 180, 2100, 27720, 480480, 9189180, 213393180, 6246600360, 194090796900, 7220177644680, 296829525392400, 12778511068142820, 601807021256821380, 31974268694601553320, 1890171191677022594340, 115365621009252758344200, 7741033169720860084895820
Offset: 0

Views

Author

Labos Elemer, Jun 20 2001

Keywords

Comments

Largest number below primorial(n + 1) to be divisible by the first n primes. - Alonso del Arte, Dec 13 2014

Examples

			a(2) = primorial(3) - primorial(2) = 30 - 6 = 24.
a(3) = primorial(4) - primorial(3) = 210 - 30 = 180.

Links

Crossrefs

Cf. A002110, A006093, A286629.
Subsequence of A276155.

Programs

  • Maple
    p:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*p(n-1)) end:
a:= n-> p(n+1)-p(n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 23 2022
  • Mathematica
    Differences[FoldList[Times, 1, Prime[Range[20]]]] (* Alonso del Arte, Dec 13 2014 *)
  • PARI
    { n=-1; r=q=1; forprime (p=2, prime(101), r*=p; write("b061720.txt", n++, " ", r-q); q=r ) } \\ Harry J. Smith, Jul 27 2009

Formula

a(n) = A002110(n+1) - A002110(n) = A002110(n)*A006093(n+1).

Extensions

Since primorial(0) = 1, term a(0) = 1 added by Harry J. Smith, Jul 27 2009

A286630 a(0) = 1; for n >= 1, a(n) = A000040(n) * A002110(n).

Original entry on oeis.org

1, 4, 18, 150, 1470, 25410, 390390, 8678670, 184294110, 5131136010, 187621103670, 6217375194030, 274567310987970, 12474260804615610, 562558737261811290, 28899819781659096270, 1727225399291072370690, 113442860659098545705130, 7154591262923825229979470, 526507543922377892743899030, 39613798938995626228686492690
Offset: 0

Views

Author

Antti Karttunen, Jul 07 2017

Keywords

Comments

The terms a(0) .. a(5), when viewed in primorial base (A049345) look as: 1, 20, 300, 5000, 70000, E00000, where "E" stands for the digit eleven.

Links

Crossrefs

Cf. A000040, A001248, A002110, A286629.
Subsequence of A276155.

Programs

  • Mathematica
    Table[If[n==0, 1, Prime[n] Product[Prime[k], {k, n}]], {n, 0, 100}] (* Indranil Ghosh, Jul 07 2017 *)
  • PARI
    a(n) = if (n==0, 1, prime(n)*prod(k=1, n, prime(k))); \\ Michel Marcus, Jul 07 2017
  • Python
    from sympy import prime, primorial
def a002110(n): return 1 if n<1 else primorial(n)
def a(n): return 1 if n==0 else prime(n)*a002110(n)
print([a(n) for n in range(41)]) # Indranil Ghosh, Jul 07 2017
  • Scheme
    (define (A286630 n) (if (zero? n) 1 (* (A000040 n) (A002110 n))))

Formula

a(0) = 1; for n >= 1, a(n) = A000040(n) * A002110(n).
For n >= 1, a(n) = A001248(n) * A002110(n-1) = A002110(n) + A286629(n).
Showing 1-3 of 3 results.

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