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.

A330507 a(n) is the smallest number k having for every prime p <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0 (see comments).

Original entry on oeis.org

2, 6, 10, 18, 24, 30, 51, 57, 69, 60, 99, 111, 123, 143, 147, 159, 177, 189, 201, 213, 225, 245, 255, 267, 291, 303, 309, 321, 345, 357, 381, 393, 411, 427, 447, 465, 471, 493, 507, 519, 537, 553, 573, 583, 623, 621, 633, 669, 681, 695, 707, 729, 749, 753, 783
Offset: 1

Views

Author

David James Sycamore, Mar 01 2020

Keywords

Comments

Alternatively, a(n) is the smallest number whose product of distinct least part primes from all partitions of n into prime parts, is equal to primorial(n).
2 is the only prime term.
a(n) = 0 for n = 90, 151, 349, 352, 444, ... . - Alois P. Heinz, Mar 12 2020

Examples

			a(1) = 2 because [2] is the only prime partition of prime(1) = 2.
a(2) = 6 because [2,2,2] and [3,3] are the only possible prime partitions of 6, namely with prime(1) and prime(2) the only least parts.

Links

Crossrefs

Cf. A000040, A051034, A331634, A332861, A002110, A333238.

Programs

  • Maple
    b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= proc(n) option remember; local f, k, p; p:= ithprime(n);
      for k to 4*p do f:= b(k, 2, x); if degree(f)<= p and andmap(
        h->0Alois P. Heinz, Mar 12 2020
  • Mathematica
    With[{s = Array[Union@ Select[IntegerPartitions[#], AllTrue[#, PrimeQ] &][[All, -1]] &, 70]}, TakeWhile[Map[FirstPosition[s, #][[1]] &, Rest@ NestList[Append[#, Prime[Length@ # + 1]] &, {}, 12]], IntegerQ]] (* Michael De Vlieger, Mar 06 2020 *)
(* Second program: *)
Block[{a, m = 125, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], Last@ s], a = ReplacePart[a, # -> Union@ Append[a[[#]], Last@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; TakeWhile[Map[FirstPosition[a, #][[1]] &, Rest@ NestList[Append[#, Prime[Length@ # + 1]] &, {}, Max[Length /@ a]]], IntegerQ]] (* Michael De Vlieger, Mar 11 2020 *)

Extensions

a(19)-a(21) from Michael De Vlieger, Mar 12 2020
a(22)-a(55) from Alois P. Heinz, Mar 12 2020

A330433 Numbers k such that if there is a prime partition of k with least part p, then there exists at least one other prime partition of k with least part p.

Original entry on oeis.org

63, 161, 195, 235, 253, 425, 513, 581, 611, 615, 635, 667, 767, 779, 791, 803, 959, 1001, 1015, 1079, 1095, 1121, 1127, 1251, 1253, 1265, 1267, 1547, 1557, 1595, 1617, 1625, 1647, 1649, 1681, 1683, 1687, 1771, 1817, 1829, 1915, 1921, 2071, 2125, 2159, 2185
Offset: 1

Views

Author

David James Sycamore, Mar 01 2020

Keywords

Comments

If k is prime then [k] is the only prime partition of k with least part k, and therefore k cannot be in this sequence. If k > 2 is even, then (assuming the validity of Goldbach's conjecture) there is a prime partition [p,q] of k (p <= q) in which p is the greatest possible least part and therefore no other partition of k is possible with least part p, so k is not a term. Therefore all terms of this sequence are odd composites.

Examples

			9 is not a term because [3,3,3] is the only prime partition of 9 having 3 as least part.
63 is a term because every possible prime partition is accounted for as follows, where (m,p) means m partitions of 63 with least part p: (2198,2), (323,3), (60,5), (15,7), (5,11), (2,13), (2,17), (sum of m values = 2605 = A000607(63)). 63 must be in the sequence because (1,p) does not appear in this list, and is the smallest such number because every odd composite < 63 has at least one prime partition with unique least part (as for 9 above).

Crossrefs

Cf. A000040, A000607, A051034, A331634, A332861, A333365.

Programs

  • Maple
    b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= proc(n) option remember; local k; for k from a(n-1)+1
      while 1 in {coeffs(b(k, 2, x))} do od; k
    end: a(0):=1:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 21 2020
  • Mathematica
    b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, Function[q, Sum[b[n - p j, q, 1], {j, 1, n/p}] t^p + b[n, q, t]][NextPrime[p]]]];
a[0] = 1;
a[n_] := a[n] = Module[{k}, For[k = a[n-1]+1, MemberQ[CoefficientList[b[k, 2, x], x], 1], k++]; k];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

A333417 a(n) is the greatest number k having for every prime <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0.

Original entry on oeis.org

4, 9, 16, 27, 35, 49, 63, 65, 85, 95, 105, 121, 135, 145, 169, 175, 187, 203, 207, 221, 253, 265, 273, 289, 301, 305, 319, 351, 369, 387, 403, 407, 425, 445, 473, 485, 495, 517, 529, 545, 551, 567, 611, 615, 629, 637, 671, 679, 693, 697, 725, 747, 781, 793, 799
Offset: 1

Views

Author

David James Sycamore, Mar 20 2020

Keywords

Comments

Alternatively a(n) is the greatest number whose product of distinct least part primes from all prime partitions of n, is equal to primorial(n). Companion sequence to A330507.
From Michael De Vlieger, Mar 20 2020: (Start)
a(n) = 0 for n = {90, 151, 349, 352, 444, ...}, cf. the comment from Alois P. Heinz at A330507.
Index m of last instance of A002110(n) in A333129 as m increases.
Last row n in A333238 that contains the consecutive primes (1...n).
Last index of the occurrence of 2^n - 1 in A333259, which is the decimal value of the characteristic function of primes in A333238 interpreted as a binary number. (End)

Examples

			a(1) = 4 because [2,2] is the only prime partition of 4, and no greater number n has only 2 as least part in any partition of n into primes.
From _Michael De Vlieger_, Mar 20 2020: (Start)
Looking at this sequence as the first position of 2^n - 1 in A333259, which in binary is a k-bit repunit, we look for the last occasion of such in A333259, indicated by the arrows. a(k) = n for rows n that have an arrow. In the chart, we reverse the portrayal of the binary rendition of A333259(n), replacing zeros with "." for clarity:
   n   A333259(n)            k
------------------------------
   2   1                     1
   3   . 1
   4   1                  -> 1
   5   1 . 1
   6   1 1                   2
   7   1 . . 1
   8   1 1                   2
   9   1 1                -> 2
  10   1 1 1                 3
  11   1 1 . . 1
  12   1 1 1                 3
  13   1 1 . . . 1
  14   1 1 . 1
  15   1 1 1                 3
  16   1 1 1              -> 3
  17   1 1 1 . . . 1
  18   1 1 1 1               4
  19   1 1 1 . . . . 1
  20   1 1 1 1               4
  ... (End)

Crossrefs

Cf. A000040, A002110, A051034, A331634, A332861, A330507, A333129, A333238, A333259, A333365.

Programs

  • Mathematica
    With[{s = TakeWhile[Import["https://oeis.org/A333259/b333259.txt", "Data"], Length@ # > 0 &][[All, -1]]}, Array[If[Length[#] == 0, 0, #[[-1, 1]] - 1] &@ Position[s, 2^# - 1] &, 55]] (* Michael De Vlieger, Mar 20 2020, using the b-file at A333259 *)

Extensions

More terms from Michael De Vlieger, Mar 20 2020
Showing 1-3 of 3 results.

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