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.

A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

Original entry on oeis.org

2, 6, 9, 20, 50, 56, 84, 125, 126, 176, 189, 196, 240, 294, 360, 416, 441, 540, 600, 624, 686, 810, 900, 936, 968, 1029, 1040, 1088, 1215, 1350, 1404, 1500, 1560, 2025, 2106, 2250, 2340, 2401, 2432, 2600, 2704, 3159, 3375, 3510, 3648, 3750, 3900, 4056, 5265
Offset: 1

Views

Author

Gus Wiseman, Jul 02 2018

Keywords

Comments

A110295 is a subsequence.

Examples

			3750 is the Heinz number of (3,3,3,3,2,1), whose length and lcm are both 6.

Links

Crossrefs

Cf. A056239, A074761, A110295, A143773, A237984, A289508, A289509, A290103, A296150, A316413, A316428, A316430, A316431.

Programs

  • Mathematica
    Select[Range[2,200],PrimeOmega[#]==LCM@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]
  • PARI
    heinz(n) = my(f=factor(n), pr=f[,1]~,exps=f[,2], res=vector(vecsum(exps)), t=0); for(i = 1, #pr, pr[i] = primepi(pr[i]); for(j=1, exps[i],t++; res[t] = pr[i])); res
is(n) = my(h = heinz(n)); lcm(h)==#h \\ David A. Corneth, Jul 05 2018

A325761 Heinz numbers of integer partitions whose length is itself a part.

Original entry on oeis.org

1, 2, 6, 9, 15, 20, 21, 30, 33, 39, 45, 50, 51, 56, 57, 69, 70, 75, 84, 87, 93, 105, 110, 111, 123, 125, 126, 129, 130, 140, 141, 159, 165, 170, 175, 176, 177, 183, 189, 190, 195, 196, 201, 210, 213, 219, 230, 237, 245, 249, 255, 264, 267, 275, 285, 290, 291
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A002865.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   15: {2,3}
   20: {1,1,3}
   21: {2,4}
   30: {1,2,3}
   33: {2,5}
   39: {2,6}
   45: {2,2,3}
   50: {1,3,3}
   51: {2,7}
   56: {1,1,1,4}
   57: {2,8}
   69: {2,9}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   87: {2,10}

Crossrefs

Cf. A001222, A002865, A056239, A093641, A109298, A110295, A112798, A118914, A325762, A325763.

Programs

  • Mathematica
    Select[Range[100],MemberQ[PrimePi/@First/@FactorInteger[#],PrimeOmega[#]]&]

A239885 a(n) = 2^(n-2) * prime(n).

Original entry on oeis.org

1, 3, 10, 28, 88, 208, 544, 1216, 2944, 7424, 15872, 37888, 83968, 176128, 385024, 868352, 1933312, 3997696, 8781824, 18612224, 38273024, 82837504, 174063616, 373293056, 813694976, 1694498816, 3456106496, 7180648448, 14629732352, 30333206528
Offset: 1

Views

Author

Gerasimov Sergey, Mar 29 2014

Keywords

Links

Crossrefs

Cf. A000040, A000079, A110295.

Programs

  • Magma
    [2^(n-2)*NthPrime(n): n in [1..50]]; // G. C. Greubel, Jan 04 2023
  • Mathematica
    Table[Prime[n]2^(n-2),{n,30}] (* Harvey P. Dale, Aug 19 2019 *)
  • PARI
    vector(50, n, prime(n)*2^(n-2)) \\ Colin Barker, Mar 29 2014
  • SageMath
    [2^(n-2)*nth_prime(n) for n in range(1,51)] # G. C. Greubel, Jan 04 2023

Formula

a(n) = A110295(n)/2.

A367107 Numbers m not divisible by prime(bigomega(m)). Heinz numbers of integer partitions whose length is not a part (counted by A229816).

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85
Offset: 1

Views

Author

Gus Wiseman, Nov 21 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.

Crossrefs

Partitions of this type are counted by A229816.
The complement is A325761, counted by A002865.
If length is not a subset-sum: A367225, count A367213, complement A367224.
A005117 ranks strict integer partitions, counted by A000009.
A066208 ranks partitions into odd parts, also counted by A000009.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237667 counts sum-free partitions, ranks A364531.
A237668 counts sum-full partitions, sum-free A364532.
Cf. A000720, A088902, A093641, A106529, A110295, A325762.

Programs

  • Mathematica
    Select[Range[2,100],!Divisible[#,Prime[PrimeOmega[#]]]&]

A325762 Heinz numbers of integer partitions with no part greater than the number of ones.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 432, 448, 480, 512, 560, 576, 600, 640, 648, 672, 704, 720, 768, 784, 800, 832, 864, 896, 960, 972, 1000
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

After 1 and 2, first differs from A322136 in having 200.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A002865.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    24: {1,1,1,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   128: {1,1,1,1,1,1,1}
   144: {1,1,1,1,2,2}

Crossrefs

Cf. A001222, A002865, A007814, A056239, A061395, A093641, A109298, A110295, A112798, A118914, A325761, A325763.

Programs

  • Mathematica
    Select[Range[100],#==1||EvenQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=FactorInteger[#][[1,2]]&]

A265127 a(n) = prime(n) * 2^n.

Original entry on oeis.org

4, 12, 40, 112, 352, 832, 2176, 4864, 11776, 29696, 63488, 151552, 335872, 704512, 1540096, 3473408, 7733248, 15990784, 35127296, 74448896, 153092096, 331350016, 696254464, 1493172224, 3254779904, 6777995264, 13824425984, 28722593792, 58518929408, 121332826112
Offset: 1

Views

Author

Altug Alkan, Dec 02 2015

Keywords

Examples

			a(1) = 4 because prime(1) * 2^1 = 2 * 2 = 4.

Crossrefs

Cf. A000040, A000079.
Equals twice A110295. - David Radcliffe, Dec 03 2015

Programs

  • Magma
    [NthPrime(n)*2^n: n in [1..30]]; // Vincenzo Librandi, Dec 03 2015
  • Mathematica
    Array[Prime@# 2^# &, {30}] (* Michael De Vlieger, Dec 02 2015 *)
  • PARI
    a(n) = prime(n) * 2^n;

Formula

a(n) = A000079(n) * A000040(n).
Showing 1-6 of 6 results.

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