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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A366532 Heinz numbers of integer partitions with at least one even and odd part.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 30, 33, 35, 36, 38, 42, 45, 48, 51, 52, 54, 56, 58, 60, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 90, 93, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 114, 116, 119, 120, 122, 123, 126, 130, 132, 135, 138, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

These partitions are counted by A006477.
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:
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Crossrefs

These partitions are counted by A006477.
Just even: A324929, counted by A047967.
Just odd: A366322, counted by A086543 (even bisection of A182616).
A031368 lists primes of odd index, even A031215.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 lists prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
A257992 counts even prime indices, distinct A324967.
Cf. A000720, A001222, A003963, A026804, A244991, A257992.

Programs

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

Formula

Intersection of A324929 and A366322.

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Comments

Or, positive integers whose prime indices include at least one 1 or prime number.
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 terms together with their prime indices begin:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Crossrefs

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

Programs

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

A341449 Heinz numbers of integer partitions into odd parts > 1.

Original entry on oeis.org

1, 5, 11, 17, 23, 25, 31, 41, 47, 55, 59, 67, 73, 83, 85, 97, 103, 109, 115, 121, 125, 127, 137, 149, 155, 157, 167, 179, 187, 191, 197, 205, 211, 227, 233, 235, 241, 253, 257, 269, 275, 277, 283, 289, 295, 307, 313, 331, 335, 341, 347, 353, 365, 367, 379, 389
Offset: 1

Views

Author

Gus Wiseman, Feb 15 2021

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 sequence of partitions together with their Heinz numbers begins:
      1: ()        97: (25)       197: (45)       307: (63)
      5: (3)      103: (27)       205: (13,3)     313: (65)
     11: (5)      109: (29)       211: (47)       331: (67)
     17: (7)      115: (9,3)      227: (49)       335: (19,3)
     23: (9)      121: (5,5)      233: (51)       341: (11,5)
     25: (3,3)    125: (3,3,3)    235: (15,3)     347: (69)
     31: (11)     127: (31)       241: (53)       353: (71)
     41: (13)     137: (33)       253: (9,5)      365: (21,3)
     47: (15)     149: (35)       257: (55)       367: (73)
     55: (5,3)    155: (11,3)     269: (57)       379: (75)
     59: (17)     157: (37)       275: (5,3,3)    389: (77)
     67: (19)     167: (39)       277: (59)       391: (9,7)
     73: (21)     179: (41)       283: (61)       401: (79)
     83: (23)     187: (7,5)      289: (7,7)      415: (23,3)
     85: (7,3)    191: (43)       295: (17,3)     419: (81)

Crossrefs

Note: A-numbers of ranking sequences are in parentheses below.
Partitions with no ones are A002865 (A005408).
The case of even parts is A035363 (A066207).
These partitions are counted by A087897.
The version for factorizations is A340101.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A056239 adds up prime indices.
A078408 counts partitions with odd parts, length, and sum (A300272).
A112798 lists the prime indices of each positive integer.
A257991/A257992 count odd/even prime indices.
Cf. A026804 (A340932), A160786 (A340931), A340386, A340604, A340607, A340785, A340854/A340855, A341446.

Programs

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

A366841 Least positive integer whose odd prime factors sum to n, starting with n = 5.

Original entry on oeis.org

5, 9, 7, 15, 27, 21, 11, 35, 13, 33, 105, 39, 17, 65, 19, 51, 195, 57, 23, 95, 171, 69, 285, 115, 29, 161, 31, 87, 483, 93, 261, 155, 37, 217, 465, 111, 41, 185, 43, 123, 555, 129, 47, 215, 387, 141, 645, 235, 53, 329, 705, 159, 987, 265, 59, 371, 61, 177
Offset: 5

Views

Author

Gus Wiseman, Oct 27 2023

Keywords

Comments

All terms are odd.
It seems that all composite terms not divisible by 3 form a supersequence of A292081. - Ivan N. Ianakiev, Oct 30 2023

Examples

			The terms together with their prime factors (which sum to n) begin:
    5 = 5
    9 = 3*3
    7 = 7
   15 = 3*5
   27 = 3*3*3
   21 = 3*7
   11 = 11
   35 = 5*7
   13 = 13
   33 = 3*11
  105 = 3*5*7

Links

Crossrefs

This is the odd case of A056240.
Positions of first appearances in A366840 (sum of odd prime factors).
The partition triangle for this statistic is A366851, even A116598.
A001414 adds up prime factors, triangle A331416.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A027746 lists prime factors, length A001222.
A087436 counts odd prime factors, even A007814.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A031368, A036349, A056239, A066208, A100005, A112798, A239261, A257991, A335657, A366839.

Programs

  • Mathematica
    nn=1000;
w=Table[Total[Times@@@DeleteCases[If[n==1,{},FactorInteger[n]],{2,_}]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[i]]];
Table[Position[w,k][[1,1]],{k,5,spnm[Join[{1,2,3,4},Take[w,nn]/.(0->1)]]}]
  • PARI
    f(n) = my(f=factor(n), j=if (n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ A366840
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Nov 02 2023
Previous Showing 81-84 of 84 results.

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