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 1-6 of 6 results.

A239737 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part or the number of numbers having multiplicity > 1 is a part.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 7, 10, 13, 21, 28, 38, 54, 77, 99, 137, 180, 236, 306, 398, 504, 644, 807, 1018, 1278, 1599, 1972, 2458, 3039, 3743, 4592, 5659, 6884, 8436, 10235, 12445, 15021, 18204, 21842, 26334, 31501, 37746, 44956, 53707, 63657, 75738, 89536, 106057
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 7 partitions:  42, 411, 321, 3111, 2211, 21111, 111111.

Crossrefs

Cf. A241413, A241414, A241415, A241416, A241417, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A241417(n) = A000041(n) for n >= 0.

A241413 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part of p.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 5, 8, 10, 17, 21, 29, 38, 59, 68, 100, 124, 170, 214, 288, 351, 470, 576, 743, 921, 1176, 1430, 1816, 2214, 2753, 3364, 4176, 5015, 6215, 7478, 9120, 10966, 13351, 15916, 19301, 22982, 27618, 32846, 39354, 46515, 55570, 65598, 77842, 91730
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 5 partitions:  42, 411, 321, 3111, 21111; e.g., 411 is counted because 1 part of 411 has multiplicity 1, and 1 is a part of 411.

Crossrefs

Cf. A241414, A241415, A241416, A241417, A239737.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

A241414 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part and the number of numbers having multiplicity > 1 is a part.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 7, 8, 14, 17, 25, 30, 45, 52, 72, 91, 123, 153, 205, 253, 339, 419, 542, 673, 864, 1051, 1336, 1625, 2023, 2461, 3040, 3642, 4490, 5383, 6527, 7837, 9481, 11291, 13624, 16208, 19403, 23087, 27541, 32619, 38832, 45923, 54327, 64150, 75737
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 3 partitions:  411, 3111, 21111.

Crossrefs

Cf. A241413, A241415, A241416, A241417, A239737, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A241415(n) + A241416(n) = A239737(n) for n >= 0.

A241415 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is a part.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 18, 31, 37, 56, 66, 92, 110, 153, 174, 231, 275, 357, 423, 542, 642, 825, 990, 1228, 1483, 1869, 2221, 2757, 3325, 4055, 4853, 5926, 7033, 8519, 10128, 12110, 14353, 17142, 20168, 23938, 28215, 33243, 39019, 45968
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 2 partitions:  2211, 111111.

Crossrefs

Cf. A241413, A241414, A241416, A241417, A239737, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A241415(n) + A241416(n) = A239737(n) for n >= 0.

A241417 Number of partitions p of n such that the number of numbers p having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is not a part.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 4, 5, 9, 9, 14, 18, 23, 24, 36, 39, 51, 61, 79, 92, 123, 148, 195, 237, 297, 359, 464, 552, 679, 822, 1012, 1183, 1465, 1707, 2075, 2438, 2956, 3433, 4173, 4851, 5837, 6837, 8218, 9554, 11518, 13396, 16022, 18697, 22300, 25923, 30873, 35838
Offset: 0

Views

Author

Clark Kimberling, Apr 23 2014

Keywords

Examples

			a(6) counts these 4 partitions:  6, 51, 33, 222.

Crossrefs

Cf. A241413, A241414, A241415, A241416, A239737, A000041.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Formula

a(n) + A239737(n) = A000041(n) for n >= 0.

A251415 Values of A098550 where A098550(k)/k reaches a record high.

Original entry on oeis.org

1, 9, 15, 25, 35, 329, 581, 679, 3443, 5753, 6941, 9229, 10417, 11561, 14963, 30043, 45071, 120107, 135187, 150137, 255221, 1786819, 2552567, 2807737, 3063077, 4849921, 14549573, 33948953, 38798741, 43648643, 63048061, 72747599, 82447327, 87297191, 111546389
Offset: 1

Views

Author

N. J. A. Sloane, Dec 02 2014

Keywords

Comments

The prime factorizations of these numbers are 1, 3^2, 3*5, 5^2, 5*7, 7*47, 7*83, 7*97, 11*313, 11*523, 11*631, 11*839, 11*947, 11*1051, 13*1151, 13*2311, 13*3467, 13*9239, 13*10399, 13*11549, 17*15013, 17*105107, ... It would be nice to know what this is trying to tell us!

Crossrefs

Cf. A098550, A241416, A250127.

Programs

  • Python
    from math import gcd
A251415_list, l1, l2, s, u, l, b = [1], 3, 2, 4, 1, 1, {}
for n in range(4,10**4):
    i = s
    while True:
        if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:
            l2, l1, b[i] = l1, i, 1
            while s in b:
                b.pop(s)
                s += 1
            if u*n < i*l:
                A251415_list.append(i)
                u, l = i, n
            break
        i += 1 # Chai Wah Wu, Dec 06 2014

Extensions

a(23)-a(26) added and typo in definition corrected by Chai Wah Wu, Dec 06 2014
a(27)-a(34) from David Applegate, Dec 18 2014
a(35) from Jinyuan Wang, Jan 26 2025
Showing 1-6 of 6 results.

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

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