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.

A276088 The least significant nonzero digit in primorial base representation of n: a(n) = A276094(n) / A002110(A276084(n)) (with a(0) = 0).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 4
Offset: 0

Views

Author

Antti Karttunen, Aug 22 2016

Keywords

Comments

For any n >= 1, start from k = n and repeatedly try to divide as many successive primes as possible out of k, iterating as k/2 -> k, k/3 -> k, k/5 -> k, until a nonzero remainder is encountered, which is then the value of a(n). (See the last example).
Note that the sequence has been defined so that it will eventually include also "digits" (actually: value holders) > 9 that occur as the least significant nonzero digits in primorial base representation. Thus any eventual decimal corruption of A049345 will not affect these values.
The sums of the first 10^k terms (starting from n=1), for k = 1, 2, ..., are 12, 138, 1441, 14565, 145950, 1459992, 14600211, 146002438, 1460025336, 14600254674, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.460025... . - Amiram Eldar, Sep 10 2022
The asymptotic density of the occurrences of k = 1, 2, ..., is d(1) = c = A064648 for k = 1, and d(k) = c - Sum_{i = 1..PrimePi(k)} 1/primorial(i) for k >= 2, where primorial(i) = A002110(i). The asymptotic mean of this sequence is Sum_{k>=1} k * d(k) = c + Sum_{i>=1} (prime(i) * gap(i) + t(gap(i)-1)) * (c - Sum_{j = 1..i} 1/primorial(i)) = 1.460025488658067356046281458556..., where t(i) = A000217(i) and gap(i) = A001223(i). - Amiram Eldar, Feb 20 2025

Examples

			   n   A049345  the rightmost nonzero = a(n)
---------------------------------------------------------
   0       0             0
   1       1             1
   2      10             1
   3      11             1
   4      20             2
   5      21             1
   6     100             1
   7     101             1
   8     110             1
   9     111             1
  10     120             2
  11     121             1
  12     200             2
  13     201             1
  14     210             1
  15     211             1
  16     220             2
.
For n=48 according to the iteration interpretation, we obtain first 48/2 = 24, and the remainder is zero, so we continue: 24/3 = 8 and here the remainder is zero as well, so we try next 8/5, but this gives the nonzero remainder 3, thus a(48)=3.
For n=2100, which could be written "A0000" in primorial base (where A stands for digit "ten", as 2100 = 10*A002110(4)), the least significant nonzero value holder (also the most significant) is thus 10 and a(2100) = 10. (The first point where this sequence attains a value larger than 9).

Links

Crossrefs

Cf. A000040, A002110, A049345, A067029, A276084, A276086, A276094, A328569.
Cf. A000217, A000720, A001223, A064648.

Programs

  • Mathematica
    nn = 120; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[Last[IntegerDigits[n, b] /. 0 -> Nothing, 0], {n, 0, nn}] (* Version 11, 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]]]; {0}~Join~Table[Last@ DeleteCases[f@ n, d_ /; d == 0], {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)
  • PARI
    A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); }; \\ Antti Karttunen, Oct 29 2019
  • Python
    from sympy import nextprime, primepi, primorial
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def a257993(n): return primepi(a053669(n))
def a002110(n): return 1 if n<1 else primorial(n)
def a276094(n): return 0 if n==0 else n%a002110(a257993(n))
def a(n): return 0 if n==0 else a276094(n)//a002110(a257993(n) - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017
  • Scheme
    (define (A276088 n) (if (zero? n) n (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) d (loop (/ (- n d) p) (+ 1 i)))))))
  • Scheme
    (define (A276088 n) (if (zero? n) n (/ (A276094 n) (A002110 (A276084 n)))))

Formula

a(0) = 0, and for n >= 1, a(n) = A276094(n) / A002110(A276084(n)).
From Antti Karttunen, Oct 29 2019: (Start)
a(n) = A067029(A276086(n)).
a(A276086(n)) = A328569(n).
(End).

A276093 The least significant nonzero digit in primorial base is replaced with zero: a(n) = n - A276094(n), a(0) = 0.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 6, 6, 8, 6, 10, 0, 12, 12, 14, 12, 16, 0, 18, 18, 20, 18, 22, 0, 24, 24, 26, 24, 28, 0, 30, 30, 32, 30, 34, 30, 36, 36, 38, 36, 40, 30, 42, 42, 44, 42, 46, 30, 48, 48, 50, 48, 52, 30, 54, 54, 56, 54, 58, 0, 60, 60, 62, 60, 64, 60, 66, 66, 68, 66, 70, 60, 72, 72, 74, 72, 76, 60, 78, 78, 80, 78, 82, 60, 84, 84, 86, 84, 88, 0, 90
Offset: 0

Views

Author

Antti Karttunen, Aug 22 2016

Keywords

Examples

			   n   A049345  with the rightmost      converted back
                nonzero replaced by 0   to decimal = a(n)
---------------------------------------------------------
   0       0             0                     0
   1       1             0                     0
   2      10            00                     0
   3      11            10                     2
   4      20            00                     0
   5      21            20                     4
   6     100           000                     0
   7     101           100                     6
   8     110           100                     6
   9     111           110                     8
  10     120           100                     6
  11     121           120                    10
  12     200           000                     0
  13     201           200                    12
  14     210           200                    12
  15     211           210                    14
  16     220           200                    12

Links

Crossrefs

Cf. A276094, A276086.

Programs

  • Mathematica
    nn = 91; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; FromDigits[#, b] & /@ Join[{{0}}, Table[Function[w, Join[Take[w, Length@ w - # - 1], ConstantArray[0, # + 1]] &@ Length@ TakeWhile[Reverse@ w, # == 0 &]]@ IntegerDigits[n, b], {n, 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]]]; g[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; g /@ Join[{{0}}, Table[Function[w, Join[Take[w, Length@ w - # - 1], ConstantArray[0, # + 1]] &@ Length@ TakeWhile[Reverse@ w, # == 0 &]]@ f@ n, {n, 91}]] (* Michael De Vlieger, Aug 30 2016 *)
  • Scheme
    (define (A276093 n) (- n (A276094 n)))

Formula

a(n) = n - A276094(n).

A257993 Least gap in the partition having Heinz number n; index of the least prime not dividing n.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3
Offset: 1

Views

Author

Emeric Deutsch, May 18 2015

Keywords

Comments

The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
Sum of least gaps of all partitions of m = A022567(m).
From Antti Karttunen, Aug 22 2016: (Start)
Index of the least prime not dividing n. (After a formula given by Heinz.)
Least k such that A002110(k) does not divide n.
One more than the number of trailing zeros in primorial base representation of n, A049345.
(End)
The least gap is also called the mex (minimal excludant) of the partition. - Gus Wiseman, Apr 20 2021

Examples

			a(18) = 3 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having least gap equal to 3.

References

  • G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
  • Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Links

Crossrefs

Cf. A215366, A002110, A022567, A049345, A053669, A055396, A064648, A276086, A276094, A276152.
Positions of 1's are A005408.
Positions of 2's are A047235.
The number of gaps is A079067.
The version for crank is A257989.
The triangle counting partitions by this statistic is A264401.
One more than A276084.
The version for greatest difference is A286469 or A286470.
A maximal instead of minimal version is A339662.
Positions of even terms are A342050.
Positions of odd terms are A342051.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339737 counts partitions by sum and greatest gap.
Cf. A001223, A001511, A005117, A026794, A029707, A072233, A079068, A098743, A124010, A279945, A325351, A325352.

Programs

  • Maple
    with(numtheory): a := proc (n) local B, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: for q while member(q, B(n)) = true do  end do: q end proc: seq(a(n), n = 1 .. 150);
# second Maple program:
a:= n-> `if`(n=1, 1, (s-> min({$1..(max(s)+1)} minus s))(
        {map(x-> numtheory[pi](x[1]), ifactors(n)[2])[]})):
seq(a(n), n=1..100);  # Alois P. Heinz, May 09 2016
# faster:
A257993 := proc(n) local p, c; c := 1; p := 2;
while n mod p = 0 do p := nextprime(p); c := c + 1 od: c end:
seq(A257993(n), n=1..100); # Peter Luschny, Jun 04 2017
  • Mathematica
    A053669[n_] := For[p = 2, True, p = NextPrime[p], If[CoprimeQ[p, n], Return[p]]]; a[n_] := PrimePi[A053669[n]]; Array[a, 100] (* Jean-François Alcover, Nov 28 2016 *)
Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; k, {n, 100}] (* Michael De Vlieger, Jun 22 2017 *)
  • PARI
    a(n) = forprime(p=2,, if (n % p, return(primepi(p)))); \\ Michel Marcus, Jun 22 2017
  • Python
    from sympy import nextprime, primepi
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def a(n): return primepi(a053669(n)) # Indranil Ghosh, May 12 2017
  • Scheme
    (define (A257993 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) i (loop (/ (- n d) p) (+ 1 i))))))
;; Antti Karttunen, Aug 22 2016

Formula

a(n) = A000720(A053669(n)). - Alois P. Heinz, May 18 2015
From Antti Karttunen, Aug 22-30 2016: (Start)
a(n) = 1 + A276084(n).
a(n) = A055396(A276086(n)).
A276152(n) = A002110(a(n)).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} 1/A002110(k) = 1.705230... (1 + A064648). - Amiram Eldar, Jul 23 2022
a(n) << log n/log log n. - Charles R Greathouse IV, Dec 03 2022

Extensions

A simpler description added to the name by Antti Karttunen, Aug 22 2016
Showing 1-3 of 3 results.

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