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.

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A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76
Offset: 1

Views

Author

Gus Wiseman, Oct 10 2023

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.
Also Heinz numbers of integer partitions containing n/2, where n is the sum of all parts.

Examples

			The prime indices of 90 are {1,2,2,3}, with sum 8 and twice maximum 6, so 90 is in the sequence.

Crossrefs

Partitions of this type are counted by A086543.
For length instead of maximum we have the complement of A340387.
The complement is A344415, counted by A035363.
A001221 counts distinct prime factors, A001222 with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A001414, A100959, A238628, A300061, A301987, A316413, A320924, A344414, A344416, A366318.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[prix[#]]!=Total[prix[#]]/2&]

A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

Original entry on oeis.org

1, 10, 39, 88, 115, 228, 259, 306, 517, 544, 620, 783, 793, 870, 1150, 1204, 1241, 1392, 1656, 1691, 1722, 1845, 2369, 2590, 2596, 2775, 2944, 3038, 3277, 3280, 3339, 3498, 3692, 3996, 4247, 4440, 4935, 5022, 5170, 5226, 5587, 5644, 5875, 5936, 6200, 6321
Offset: 1

Views

Author

Gus Wiseman, Nov 26 2021

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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their negated alternating sum.

Examples

			The terms and their prime indices begin:
     1: ()
    10: (3,1)
    39: (6,2)
    88: (5,1,1,1)
   115: (9,3)
   228: (8,2,1,1)
   259: (12,4)
   306: (7,2,2,1)
   517: (15,5)
   544: (7,1,1,1,1,1)
   620: (11,3,1,1)
   783: (10,2,2,2)
   793: (18,6)
   870: (10,3,2,1)
  1150: (9,3,3,1)
  1204: (14,4,1,1)
  1241: (21,7)
  1392: (10,2,1,1,1,1)
  1656: (9,2,2,1,1,1)
  1691: (24,8)

Crossrefs

These partitions are counted by A001523 up to 0's.
An ordered version is A349154, nonnegative A348614, reverse A349155.
The nonnegative version is A349159, counted by A000712 up to 0's.
The reverse nonnegative version is A349160, counted by A006330 up to 0's.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, A345197 count compositions by alternating sum.
A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000984, A001700, A028260, A045931, A120452, A195017, A257991, A257992, A262977, A325698, A344619, A345958.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==-2*ats[primeMS[#]]&]

Formula

A056239(a(n)) = -2*A316524(a(n)).
A346698(a(n)) = 3*A346697(a(n)).

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2022

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, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

Examples

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

Crossrefs

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Formula

Intersection of A066208 and A346635.

A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

Original entry on oeis.org

1, 3, 7, 10, 13, 16, 19, 21, 22, 27, 28, 29, 34, 36, 37, 39, 43, 46, 48, 52, 53, 55, 57, 61, 62, 64, 66, 71, 75, 76, 79, 81, 82, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 113, 115, 116, 117, 118, 120, 129, 130, 131, 133, 134, 136, 138, 139, 144
Offset: 0

Views

Author

Gus Wiseman, Oct 13 2023

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.

Examples

			The prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.
The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   10: {1,3}
   13: {6}
   16: {1,1,1,1}
   19: {8}
   21: {2,4}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   34: {1,7}
   36: {1,1,2,2}

Crossrefs

Partitions of this type are counted by A182616, strict A365828.
A066207 lists numbers with all even prime indices, odd A066208.
A086543 lists numbers with at least one odd prime index, counted by A366322.
A300063 ranks partitions of odd numbers.
A366319 ranks partitions of n not containing n/2.
A366321 ranks partitions of 2k that do not contain k.
Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[prix[#]]]&&FreeQ[prix[#],Total[prix[#]]/2]&]

A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

Original entry on oeis.org

4, 10, 12, 16, 22, 25, 28, 30, 34, 36, 40, 46, 48, 52, 55, 62, 64, 66, 70, 75, 76, 82, 84, 85, 88, 90, 94, 100, 102, 108, 112, 115, 116, 118, 120, 121, 130, 134, 136, 138, 144, 146, 148, 154, 155, 156, 160, 165, 166, 172, 175, 184, 186, 187, 190, 192, 194, 196
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices are the following. Each multiset has even sum and at least one odd part.
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   46: {1,9}
   48: {1,1,1,1,2}
   52: {1,1,6}
   55: {3,5}
   62: {1,11}
   64: {1,1,1,1,1,1}

Crossrefs

These partitions are counted by A182616, even bisection of A086543.
Not requiring at least one odd part gives A300061.
Allowing partitions of odd numbers gives A366322.
A031368 lists primes of odd index.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A047967, A257992, A324929, A358137.

Programs

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

A353188 Number of partitions of n that contain at least one composite part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 8, 12, 19, 27, 41, 56, 80, 109, 150, 199, 268, 350, 461, 596, 771, 984, 1258, 1589, 2007, 2514, 3145, 3905, 4846, 5973, 7356, 9010, 11020, 13418, 16315, 19756, 23890, 28788, 34639, 41548, 49767, 59441, 70899, 84354, 100221, 118803, 140645, 166153, 196035, 230853, 271512
Offset: 0

Views

Author

Omar E. Pol, Jun 22 2022

Keywords

Examples

			For n = 6 the partitions of 6 that contain at least one composite parts are [6], [4, 2] and [4, 1, 1]. There are three of these partitions so a(6) = 3.

Crossrefs

Cf. A000041, A002096, A002808, A023895, A034891, A047967, A085642, A086543, A116449, A144300, A204389.

Programs

  • PARI
    a(n) = my(nb=0); forpart(p=n, if (#select(x->((x>1) && !isprime(x)), Vec(p)) >=1, nb++);); nb; \\ Michel Marcus, Jun 23 2022

Formula

a(n) = A000041(n) - A034891(n).

A365826 Number of strict integer partitions of n that are not of length 2 and do not contain n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 20, 20, 30, 31, 45, 46, 66, 68, 93, 97, 130, 136, 179, 188, 242, 256, 325, 344, 432, 459, 568, 606, 742, 793, 963, 1031, 1240, 1331, 1589, 1707, 2026, 2179, 2567, 2766, 3240, 3493, 4072, 4393, 5094, 5501, 6351
Offset: 0

Views

Author

Gus Wiseman, Sep 20 2023

Keywords

Comments

Also the number of strict integer partitions of n without two parts (allowing parts to be re-used) summing to n.

Examples

			The a(6) = 1 through a(12) = 7 strict partitions:
  (6)  (7)      (8)      (9)      (10)       (11)       (12)
       (4,2,1)  (5,2,1)  (4,3,2)  (6,3,1)    (5,4,2)    (5,4,3)
                         (5,3,1)  (7,2,1)    (6,3,2)    (7,3,2)
                         (6,2,1)  (4,3,2,1)  (6,4,1)    (7,4,1)
                                             (7,3,1)    (8,3,1)
                                             (8,2,1)    (9,2,1)
                                             (5,3,2,1)  (5,4,2,1)

Crossrefs

The second condition alone has bisections A078408 and A365828.
The complement is counted by A365659.
The non-strict version is A365825, complement A238628.
The first condition alone is A365827, complement A140106.
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A004526, A005408, A008967, A035363, A058984, A086543, A100959, A344415, A365376, A365377, A365543.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Tuples[#,2],n]&]], {n,0,30}]

A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 21, 22, 25, 26, 30, 33, 34, 35, 38, 39, 40, 46, 49, 51, 55, 57, 58, 62, 63, 65, 69, 70, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159
Offset: 1

Views

Author

Gus Wiseman, Oct 08 2023

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
     4: {1,1}      38: {1,8}         77: {4,5}
     6: {1,2}      39: {2,6}         82: {1,13}
     9: {2,2}      40: {1,1,1,3}     84: {1,1,2,4}
    10: {1,3}      46: {1,9}         85: {3,7}
    12: {1,1,2}    49: {4,4}         86: {1,14}
    14: {1,4}      51: {2,7}         87: {2,10}
    15: {2,3}      55: {3,5}         91: {4,6}
    21: {2,4}      57: {2,8}         93: {2,11}
    22: {1,5}      58: {1,10}        94: {1,15}
    25: {3,3}      62: {1,11}        95: {3,8}
    26: {1,6}      63: {2,2,4}      106: {1,16}
    30: {1,2,3}    65: {3,6}        111: {2,12}
    33: {2,5}      69: {2,9}        112: {1,1,1,1,4}
    34: {1,7}      70: {1,3,4}      115: {3,9}
    35: {3,4}      74: {1,12}       118: {1,17}

Crossrefs

The first condition alone is A001358, counted by A004526.
The complement of the first condition is A100959, counted by A058984.
The partitions with these Heinz numbers are counted by A238628.
The second condition alone is A344415, counted by A035363.
The complement of the second condition is A366319, counted by A086543.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344296 solves for k in A001222(k) >= A056239(k)/2, counted by A025065.
Cf. A001414, A316413, A320924, A325044, A340387, A344414, A344416.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[prix[#]]==2||MemberQ[prix[#],Total[prix[#]]/2]&]

Formula

Union of A001358 and A344415.

A366527 Number of integer partitions of 2n containing at least one even part.

Original entry on oeis.org

0, 1, 3, 7, 16, 32, 62, 113, 199, 339, 563, 913, 1453, 2271, 3496, 5308, 7959, 11798, 17309, 25151, 36225, 51748, 73359, 103254, 144363, 200568, 277007, 380437, 519715, 706412, 955587, 1286762, 1725186, 2303388, 3063159, 4058041, 5356431, 7045454, 9235841
Offset: 0

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

Also partitions of 2n with even product.

Examples

			The a(1) = 1 through a(4) = 16 partitions:
  (2)  (4)    (6)      (8)
       (22)   (42)     (44)
       (211)  (222)    (62)
              (321)    (332)
              (411)    (422)
              (2211)   (431)
              (21111)  (521)
                       (611)
                       (2222)
                       (3221)
                       (4211)
                       (22211)
                       (32111)
                       (41111)
                       (221111)
                       (2111111)

Crossrefs

This is the even bisection of A047967.
For odd instead of even parts we have A182616, ranks A366321 or A366528.
These partitions have ranks A366529, subset of A324929.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions w/ at least one odd and even part, ranks A366532.
A086543 counts partitions of n not containing n/2, ranks A366319.
A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]

Formula

a(n) = A000041(2n) - A000009(2n).

A366529 Heinz numbers of integer partitions of even numbers with at least one even part.

Original entry on oeis.org

3, 7, 9, 12, 13, 19, 21, 27, 28, 29, 30, 36, 37, 39, 43, 48, 49, 52, 53, 57, 61, 63, 66, 70, 71, 75, 76, 79, 81, 84, 87, 89, 90, 91, 101, 102, 107, 108, 111, 112, 113, 116, 117, 120, 129, 130, 131, 133, 138, 139, 144, 147, 148, 151, 154, 156, 159, 163, 165
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
   3: {2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  13: {6}
  19: {8}
  21: {2,4}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  36: {1,1,2,2}
  37: {12}
  39: {2,6}
  43: {14}
  48: {1,1,1,1,2}

Crossrefs

The complement is counted by A047967.
For all even parts we have A066207, counted by A035363, odd A066208.
Not requiring an even part gives A300061.
For odd instead of even we have A300063.
Not requiring even sum gives A324929.
Partitions of this type are counted by A366527.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
A257992 counts even prime indices, distinct A324967.
Cf. A000720, A001222, A003963, A033844, A086543, A324927, A358137, A366530.

Programs

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