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.

Previous Showing 21-30 of 30 results.

A317590 Heinz numbers of integer partitions that are not uniformly normal.

Original entry on oeis.org

10, 14, 15, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110
Offset: 1

Views

Author

Gus Wiseman, Aug 01 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

Examples

			Sequence of all non-uniformly normal integer partitions begins: (31), (41), (32), (311), (42), (51), (2111), (61), (411), (52), (71), (43), (81), (62), (3111), (421), (511), (322), (91), (21111), (331).

Crossrefs

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317493, A317588, A317589.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],!uninrmQ[primeMS[#]]&]

A325331 Number of integer partitions of n whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 7, 10, 14, 18, 30, 34, 44, 65, 73, 88, 110, 127, 155, 183, 202, 231, 277, 301, 339, 382, 430, 461, 551, 579, 681, 762, 896, 1010, 1255, 1406, 1752, 2061, 2555, 3001, 3783, 4437, 5512, 6611, 8056, 9539, 11668, 13692, 16515, 19435, 23098
Offset: 0

Views

Author

Gus Wiseman, May 01 2019

Keywords

Comments

Partitions with distinct multiplicities that cover an initial interval of positive integers are counted by A320348, with Heinz numbers A325337. Partitions whose multiplicities appear with distinct multiplicities are counted by A325329, with Heinz numbers A325369. Partitions whose multiplicities appear with multiplicities that cover an initial interval of positive integers of counted by A325330, with Heinz numbers A325370.
The Heinz numbers of these partitions are given by A325371.

Examples

			The a(0) = 1 through a(8) = 7 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (3211)     (44)
                        (1111)           (222)     (1111111)  (2222)
                                         (111111)             (3221)
                                                              (4211)
                                                              (32111)
                                                              (11111111)
For example, the partition p = (5,5,4,3,3,3,2,2) has multiplicities (2,3,1,2), which appear with multiplicities (1,2,1), which cover an initial interval but are not distinct, so p is not counted under a(27). The partition q = (5,5,5,4,4,4,3,3,2,2,1,1) has multiplicities (3,3,2,2,2), which appear with multiplicities (3,2), which are distinct but do not cover an initial interval, so q is not counted under a(39). The partition r = (3,3,2,1,1) has multiplicities (2,1,2), which appear with multiplicities (1,2), which are distinct and cover an initial interval, so r is counted under a(10).

Crossrefs

Cf. A098859, A130091, A317081, A317090, A320348, A325329, A325330, A325337, A325369, A325370, A325371.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&&UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]

A325371 Numbers whose prime signature has multiplicities of its parts all distinct and covering an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 126, 127, 128, 131, 132, 137, 139, 140, 149, 150, 151, 156, 157, 163
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

The first term that is not 1 or a prime power is 60.
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
Numbers whose prime signature has distinct parts that cover an initial interval are given by A325337.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325331.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Crossrefs

Cf. A055932, A056239, A098859, A112798, A118914, A130091, A317090, A325329, A325330, A325331, A325337, A325369, A325370.

Programs

  • Mathematica
    normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&&UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]

A325372 Totally abnormal numbers. Heinz numbers of totally abnormal integer partitions.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 225, 227
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.
The enumeration of totally abnormal integer partitions by sum is given by A325332.

Examples

			The sequence of terms together with their prime indices are the following. See also the example at A325373.
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

Crossrefs

Cf. A055932, A056239, A112798, A181819, A317089, A317090, A317246 (supernormal), A317492 (fully normal), A317589 (uniformly normal), A319151, A325332, A325373.

Programs

  • Mathematica
    normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[100],totabnQ]

A317082 Number of integer partitions of n whose multiplicities are weakly decreasing and span an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 9, 13, 17, 22, 26, 35, 42, 53, 66, 81, 96, 122, 143, 174, 210, 251, 293, 358, 417, 493, 582, 686, 793, 941, 1087, 1267, 1471, 1709, 1961, 2285, 2615, 3013, 3460, 3976, 4523, 5204, 5914, 6747, 7681, 8745, 9884, 11262, 12714, 14393, 16261
Offset: 0

Views

Author

Gus Wiseman, Jul 21 2018

Keywords

Examples

			The a(7) = 8 integer partitions are (7), (61), (52), (511), (43), (421), (322), (3211).

Crossrefs

Cf. A000041, A000837, A055932, A304678, A317081, A317084, A317085, A317087, A317090.

Programs

  • Mathematica
    normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[Length/@Split[#]],OrderedQ[Length/@Split[#]]]&]],{n,60}]

A317493 Heinz numbers of integer partitions that are not fully normal.

Original entry on oeis.org

9, 24, 25, 27, 36, 40, 48, 49, 54, 56, 72, 80, 81, 88, 96, 100, 104, 108, 112, 120, 121, 125, 135, 136, 144, 152, 160, 162, 168, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 264, 270, 272, 280, 288, 289, 296, 297, 304, 312
Offset: 1

Views

Author

Gus Wiseman, Jul 30 2018

Keywords

Comments

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

Examples

			Sequence of all integer partitions that are not fully normal begins: (22), (2111), (33), (222), (2211), (3111), (21111), (44), (2221), (4111), (22111), (31111), (2222), (5111), (211111), (3311).

Crossrefs

Cf. A055932, A056239, A181819, A182850, A296150, A305733, A317089, A317090, A317245, A317246, A317491, A317492.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Select[Range[100],!fulnrmQ[Reverse[primeMS[#]]]&]

A381632 Numbers such that (greatest prime exponent) = (sum of distinct prime indices).

Original entry on oeis.org

2, 9, 24, 54, 72, 80, 108, 125, 216, 224, 400, 704, 960, 1215, 1250, 1568, 1664, 2000, 2401, 2500, 2688, 2880, 4352, 4800, 5000, 5103, 6075, 7290, 7744, 8064, 8448, 8640, 8960, 9375, 9728, 10000, 10976, 14400, 14580, 18816, 19968, 21632, 23552, 24000, 24057
Offset: 1

Views

Author

Gus Wiseman, Mar 24 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The terms together with their prime indices begin:
      2: {1}
      9: {2,2}
     24: {1,1,1,2}
     54: {1,2,2,2}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    108: {1,1,2,2,2}
    125: {3,3,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    400: {1,1,1,1,3,3}
    704: {1,1,1,1,1,1,5}
    960: {1,1,1,1,1,1,2,3}

Crossrefs

For (length) instead of (sum of distinct) we have A000961.
Including number of parts gives A062457 (degenerate).
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
Partitions of this type are counted by A381079.
A001222 counts prime factors, distinct A001221.
A047993 counts partitions with max part = length, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, complement A351293.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A000720, A048767, A130091, A246655, A317090, A380955, A381437, A381439.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Last/@FactorInteger[#]==Total[Union[prix[#]]]&]

Formula

A051903(a(n)) = A066328(a(n)).

A317084 Number of integer partitions of n whose multiplicities are weakly increasing and span an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 17, 23, 27, 34, 38, 50, 54, 70, 79, 97, 107, 135, 148, 180, 205, 243, 270, 328, 360, 429, 480, 561, 625, 738, 810, 949, 1057, 1219, 1349, 1571, 1723, 1986, 2206, 2515, 2776, 3188, 3496, 3983, 4408, 4980, 5485, 6228, 6826
Offset: 0

Views

Author

Gus Wiseman, Jul 21 2018

Keywords

Examples

			The a(7) = 6 integer partitions are (7), (61), (52), (43), (421), (331).

Crossrefs

Cf. A000041, A000837, A055932, A242031, A317081, A317082, A317085, A317087, A317090.

Programs

  • Mathematica
    normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[Length/@Split[#]],OrderedQ[Reverse[Length/@Split[#]]]]&]],{n,60}]

A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3780, 3800, 3816, 3960, 3996, 4056, 4116, 4200, 4248, 4312, 4392, 4428
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2024

Keywords

Comments

First differs from its subsequence A163569 at n = 25: a(25) = 2520 = 2^3 * 3^2 * 5 * 7 is not a term of A163569.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 2, 3}.
The asymptotic densities of this sequence and A375074 are equal (0.0156712..., see A375074 for a formula), since the terms in A375074 that are not in this sequence (A375073) have a density 0.

Links

Crossrefs

Intersection of A375072 and A317090.
Equals A375074 \ A375073.
Subsequence of A046100 and A176297.
A163569 is a subsequence.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A136568, A215267.

Programs

  • Mathematica
    Select[Range[4500], Union[FactorInteger[#][[;; , 2]]] == {1, 2, 3} &]
  • PARI
    is(k) = Set(factor(k)[,2]) == [1, 2, 3];

A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

Original entry on oeis.org

9, 25, 27, 49, 81, 100, 121, 125, 169, 196, 225, 243, 289, 343, 361, 441, 484, 529, 625, 676, 729, 841, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2187, 2197, 2209, 2401, 2601, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481
Offset: 1

Views

Author

Gus Wiseman, May 02 2019

Keywords

Comments

The first term that is not a perfect power (A001597) is 11880, with prime indices {1,1,1,2,2,2,3,5} and prime signature {1,1,3,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.
The enumeration of totally abnormal integer partitions by sum is given by A325332.

Examples

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    49: {4,4}
    81: {2,2,2,2}
   100: {1,1,3,3}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   243: {2,2,2,2,2}
   289: {7,7}
   343: {4,4,4}
   361: {8,8}
   441: {2,2,4,4}
   484: {1,1,5,5}
   529: {9,9}
   625: {3,3,3,3}
   676: {1,1,6,6}

Crossrefs

Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.

Programs

  • Mathematica
    normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
Select[Range[10000],!PrimeQ[#]&&totabnQ[#]&]
Previous Showing 21-30 of 30 results.

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

Content is available under The OEIS End-User License Agreement.