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.

User: Patrick Devlin

Patrick Devlin's wiki page.

Patrick Devlin has authored 37 sequences. Here are the ten most recent ones:

A268541 Base in which the Fouriest transform of A268540(n) is 44.

Original entry on oeis.org

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 31, 32, 34, 37, 38, 43, 44, 45, 46, 47, 50, 52, 53, 54, 57, 58, 59, 62, 65, 67, 73, 75, 77, 79, 82, 83, 86, 94, 97, 98, 101, 102, 103, 106, 107, 108, 113, 116, 117, 118, 119, 122, 124, 126, 127, 128, 134, 137, 138, 139, 146, 158, 163, 164
Offset: 1

Views

Author

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 27 2016

Keywords

Comments

If we are ever going to understand A268236 then we need to understand this sequence first.
Based on Nathan Fox's table in A268236.

Links

Crossrefs

Cf. A268236, A268237, A268238, A268540.

Formula

a(n) = A268540(n)/4 - 1

A268540 Numbers whose Fouriest transform (see A268236) is 44.

Original entry on oeis.org

24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 68, 72, 76, 80, 84, 88, 92, 96, 108, 112, 116, 128, 132, 140, 152, 156, 176, 180, 184, 188, 192, 204, 212, 216, 220, 232, 236, 240, 252, 264, 272, 296, 304, 312, 320, 332, 336, 348, 380, 392, 396, 408, 412, 416, 428, 432, 436, 456, 468, 472, 476, 480, 492, 500, 508, 512, 516
Offset: 1

Views

Author

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 27 2016

Keywords

Comments

If we are ever going to understand A268236 then we need to understand this sequence first.
Based on Nathan Fox's extended table in A268236.
Equivalently, numbers 4k (k>5) whose representations in bases 5 through k-2 each contain at most one 4.
Equivalently, numbers 4k (k>5) whose representations in integer bases less than sqrt(4k) each contain at most one 4.
Is this sequence infinite?

Links

Crossrefs

Cf. A268236, A268237, A268238, A268541.

A268237 From the Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives the base b.

Original entry on oeis.org

4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 6, 7, 8, 9, 5, 11, 6, 13, 7, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 5, 8, 8, 8, 8, 5, 9, 9, 9, 9, 10, 5, 5, 5, 11, 5, 11, 11, 12, 7, 5, 12, 13, 12, 6, 5, 14, 6, 6, 6, 6, 6, 14, 7, 16, 5, 5, 5, 17, 5, 5, 16, 18, 7, 7, 5, 19, 7, 6, 7, 20, 9, 18, 18, 21, 5, 19, 19, 22, 19, 5, 5, 23, 5, 5, 5, 6
Offset: 0

Views

Author

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 06 2016

Keywords

Comments

If no base b gives any 4's then we take b=4.
a(n) > 4 for n >= 9 since 14 is n written in base n-4. - Chai Wah Wu, Feb 06 2016

Links

Crossrefs

Cf. A268236 (Fouriest transform of n), A268238 (number of 4's).

A268238 From the Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives the number of 4's.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2
Offset: 0

Views

Author

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 06 2016

Keywords

Comments

If no base b gives any 4's then we take b=4.
The first occurrence of any value m in this sequence is at position 5^m-1.
a(n) > 0 for n >= 9 since 14 is n written in base n-4. - Chai Wah Wu, Feb 06 2016

Links

Crossrefs

Cf. A268236 (Fouriest transform of n), A268237 (the base b).

A268236 Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives n in base b.

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 12, 13, 20, 14, 14, 14, 14, 14, 24, 14, 24, 14, 24, 34, 40, 41, 42, 43, 44, 41, 42, 43, 44, 104, 42, 43, 44, 45, 114, 43, 44, 45, 46, 124, 44, 45, 46, 47, 44, 140, 141, 142, 44, 144, 46, 47, 44, 104, 204, 47, 44, 49, 134, 214, 44, 141, 142, 143, 144, 145
Offset: 0

Views

Author

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 06 2016

Keywords

Comments

If no base b gives any 4's then we take b=4.
For n>65 "digits" greater than 9 appear in a(n) - see the first link. This explains why this sequence has no b-file: the OEIS restriction to decimal digits means that a(66) cannot be written as a single base-10 number (it would be "4,10").
The Fouriest transform pun suggests (by analogy with shaky, shakier, shakiest) investigating the Foury, Fourier, and Fouriest numbers. Three obvious candidates for the Foury numbers are A011534, A019764, and A268544, which are all "Foury" in different ways.
With respect to a fixed base b, we could say that n is Fourier than m (in base b) if the fraction [or number?] of 4's in the representation of n (base b) is greater than the analogous quantity for m. But it is not clear which definition is to be preferred. In base 10, which is Fourier, 440 or 439454?
This sequence and its companions were created during a dinner following the Experimental Mathematics Seminar at Rutgers University on Feb 04 2016.

Examples

			For n=24, the base-5 representation of 24 is 44. So the Fouriest transform of 24 is a(24) = 44, which uses base b = A268237(24) = 5 and contains A268238(24) = 2 4's.
The Fouriest transform of n=66 is 4,10 in base b=14 (note the non-decimal digit) and contains a single 4.

Links

Crossrefs

Cf. A268237 (the base b), A268238 (number of 4's).
See A268540 and A268541 for the "44" entries.
See also the "Foury" numbers A011534, A019764, and A268544.
A268360 and A349031 are other Foury sequences.

A253074 Lexicographically earliest sequence of distinct numbers such that a(n-1)+a(n) is not prime.

Original entry on oeis.org

0, 1, 3, 5, 4, 2, 6, 8, 7, 9, 11, 10, 12, 13, 14, 16, 17, 15, 18, 20, 19, 21, 23, 22, 24, 25, 26, 28, 27, 29, 31, 32, 30, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 43, 44, 46, 45, 47, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 58, 59, 60, 61, 62, 63, 65, 64, 66
Offset: 1

Views

Author

N. J. A. Sloane, Feb 01 2015, based on a suggestion from Patrick Devlin

Keywords

Comments

Conjecture: this is a permutation of the nonnegative numbers. [Proof outline below due to Patrick Devlin and Semeon Artamonov]
Let x be a number that's missing.
Then eventually every term must be of the form PRIME - x. (Otherwise, x would appear as that next term.)
In particular, this means there are only finitely many multiples of x that appear in the sequence. Let Y be a multiple of x larger than all multiples of x appearing in the sequence.
Let q be a prime not dividing Y. Then since none of the terms Y, 2Y, 3Y, ..., 2qY appear, it must be that, eventually, every term in the sequence is of the form PRIME - Y and also of the form PRIME - 2Y and also of the form PRIME - 3Y, ... and also of the form PRIME - 2qY.
That means we have a prime p and a number Y such that p, p+Y, p+2Y, p + 3Y, p+4Y, ..., p+2qY are all prime. But take this sequence mod q. Since q does not divide Y, the terms 0, Y, ..., 2qY cover every residue class mod q twice. Therefore, p + kY covers each residue class mod q twice. Consequently, there are two terms congruent to 0 mod q. One can be q, but the other must be a multiple of it (contradicting its primality).
Essentially the same as A055266. - R. J. Mathar, Feb 13 2015
Simplified version of the proof: Assume x isn't in the sequence, then eventually all terms must be of the form PRIME - x, else x would appear next. In particular, no multiple of x can appear from there on. Assume k*x is the largest multiple of x in the sequence. Take a prime p not dividing x. Then m*x can't appear in the sequence for k+1 <= m <= k+p, and all terms are eventually of the form PRIME - m*x for all m in {k+1, ..., k+p}. Take one such term N > p, i.e., N + (k+1)*x, ..., N + (k+p)*x are all prime. Consider this sequence mod p. Since gcd(x,p)=1, the p terms cover each residue class mod p, so one is a multiple of p, in contradiction with their primality. - M. F. Hasler, Nov 25 2019

Links

Crossrefs

Cf. A055266, A254337, A253073, A010051.

Programs

  • Haskell
    a253074 n = a253074_list !! (n-1)
a253074_list = 0 : f 0 [1..] where
   f u vs = g vs where
     g (w:ws) | a010051' (u + w) == 1 = g ws
              | otherwise = w : f w (delete w vs)
-- Reinhard Zumkeller, Feb 02 2015
  • PARI
    A253074_upto(n=99, a, u, U)={vector(n,n, for(k=u,oo, bittest(U,k-u)|| isprime(a+k)||[a=k,break]); (a>u && U+=1<<(a-u))|| U>>=-u+u+=valuation(U+2,2); a)+if(default(debug),print([u]))} \\ additional args allow to tweak computation. If debug > 0, print least unused number at the end. - M. F. Hasler, Nov 25 2019

A253073 Lexicographically earliest sequence of distinct numbers such that neither a(n) nor a(n-1)+a(n) is prime.

Original entry on oeis.org

0, 1, 8, 4, 6, 9, 12, 10, 14, 16, 18, 15, 20, 22, 24, 21, 25, 26, 28, 27, 30, 32, 33, 35, 34, 36, 38, 39, 42, 40, 44, 46, 45, 48, 50, 49, 51, 54, 52, 56, 55, 57, 58, 60, 62, 63, 65, 64, 66, 68, 70, 72, 69, 74, 76, 77, 75, 78, 80, 81, 84, 82, 86, 85, 87, 88
Offset: 1

Views

Author

N. J. A. Sloane, Feb 01 2015, based on a suggestion from Patrick Devlin

Keywords

Comments

Conjecture: this is a permutation of the nonprimes. [Proof outline given below by Semeon Artamonov and Pat Devlin.]
Let x be a number that's missing.
Then eventually every term must be of the form PRIME - x. (Otherwise, x would appear as that next term.)
In particular, this means there are only finitely many multiples of x that appear in the sequence. To make this cleaner, let Y be a multiple of x larger than all multiples of x appearing in the sequence.
Let q be a prime not dividing Y. Then since none of the terms Y, 2Y, 3Y, ..., 2qY appear, it must be that, eventually, every term in the sequence is of the form PRIME - Y and also of the form PRIME - 2Y and also of the form PRIME - 3Y, ... and also of the form PRIME - 2qY.
That means we have a prime p and a number Y such that p, p+Y, p+2Y, p + 3Y, p+4Y, ..., p+2qY are all prime. But take this sequence mod q. Since q does not divide Y, the terms 0, Y, ..., 2qY cover every residue class mod q twice. Therefore, p + kY covers each residue class mod q twice. Consequently, there are two terms congruent to 0 mod q. One can be q, but the other must be a multiple of it (contradicting its primality).

Links

Crossrefs

Cf. A254337, A002808, A253074.
Cf. A018252, A010051.

Programs

  • Haskell
    a253073 n = a253073_list !! (n-1)
a253073_list = 0 : f 0 a018252_list where
   f u vs = g vs where
     g (w:ws) | a010051' (u + w) == 1 = g ws
              | otherwise = w : f w (delete w vs)
-- Reinhard Zumkeller, Feb 02 2015

A229167 a(n) = smallest index i such that A010062(i) >= 2^n.

Original entry on oeis.org

0, 1, 3, 5, 8, 14, 22, 40, 71, 126, 224, 397, 721, 1318, 2431, 4531, 8493, 15999, 30234, 57281, 108744, 206868, 394293, 752985, 1440705, 2761606, 5302954, 10200322, 19652438, 37921447, 73277740, 141783308, 274656312, 532615214, 1033834916, 2008466475, 3905027107
Offset: 0

Views

Author

Patrick Devlin and N. J. A. Sloane, Sep 27 2013

Keywords

Examples

			The first time A010062(i) is >= 16 is A010062(8) = 17, so a(4) = 8.

Links

Crossrefs

Cf. A000120, A010062.

Programs

  • Maple
    # uses wt() from A000120
t1:=[0]; a:=1; l:=1;
for i from 1 to 1000000 do
a:=a+wt(a);
if a >= 2^l then l:=l+1; t1:=[op(t1),i]; fi;
od:
t1;

Extensions

a(24)-a(36) from Donovan Johnson, Sep 27 2013

A224878 Number T(n,k) of partitions of n into distinct parts with boundary size k (where one part of size 0 is allowed).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 5, 2, 0, 1, 4, 5, 0, 1, 4, 6, 1, 0, 1, 6, 8, 1, 0, 1, 7, 9, 3, 0, 1, 6, 13, 4, 0, 1, 7, 15, 7, 0, 1, 7, 18, 10, 0, 1, 8, 20, 14, 1, 0, 1, 11, 23, 17, 2, 0, 1, 8, 28, 24, 3, 0, 1, 9, 31, 30, 5, 0, 1
Offset: 0

Views

Author

Patrick Devlin, Jul 23 2013

Keywords

Comments

Boundary size of a partition (or set) is the number of parts (elements) having fewer than 2 neighbors.
T(n,k) is also the number of subsets of {0, 1, 2, ...} whose elements sum to n and that have k elements in its boundary.

Examples

			T(9,1) = 1: [9].
T(9,2) = 6: [0,9], [1,8], [2,7], [3,6], [4,5], [2,3,4].
T(9,3) = 8: [1,2,6], [1,3,5], [0,1,8], [0,2,7], [0,3,6], [0,4,5], [0,2,3,4], [0,1,2,6].
T(9,4) = 1: [0,1,3,5].
Triangle T(n,k) begins:
1, 1; (namely, the empty set and the set {0})
0, 1, 1;
0, 1, 1;
0, 1, 3;
0, 1, 2,  1;
0, 1, 3,  2;
0, 1, 5,  2;
0, 1, 4,  5;
0, 1, 4,  6, 1;
0, 1, 6,  8, 1;
0, 1, 7,  9, 3;
0, 1, 6, 13, 4;
0, 1, 7, 15, 7;

Links

Crossrefs

Cf. A227551 (no parts of size 0 are allowed).
Row sums are twice A000009.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0 and i<0, `if`(t>1, x, 1),
      expand(`if`(i<0, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..30);  # Alois P. Heinz, Jul 23 2013
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n==0 && i<0, If[t>1, x, 1], Expand[If[i<0, 0, If[t>1, x, 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t==2, x, 1] * b[n-i, i-1, Quotient[t, 2]+2]]]]]; T[n_] := Function[p, Table[ Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

A225366 Grundy value of an n X n rectangle in the game of Chomp.

Original entry on oeis.org

0, 2, 5, 6, 6, 13, 20, 13, 20, 19, 23, 41
Offset: 1

Views

Author

Patrick Devlin, May 06 2013

Keywords

References

  • E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways vol. 2, 598-600.

Links

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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