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 1-9 of 9 results.

A316524 Signed sum over the prime indices of n.

Original entry on oeis.org

0, 1, 2, 0, 3, -1, 4, 1, 0, -2, 5, 2, 6, -3, -1, 0, 7, 1, 8, 3, -2, -4, 9, -1, 0, -5, 2, 4, 10, 2, 11, 1, -3, -6, -1, 0, 12, -7, -4, -2, 13, 3, 14, 5, 3, -8, 15, 2, 0, 1, -5, 6, 16, -1, -2, -3, -6, -9, 17, -1, 18, -10, 4, 0, -3, 4, 19, 7, -7, 2, 20, 1, 21, -11, 2, 8, -1, 5, 22, 3, 0, -12, 23, -2, -4, -13, -8, -4, 24
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2018

Keywords

Comments

If n = prime(x_1) * prime(x_2) * prime(x_3) * ... * prime(x_k) then a(n) = x_1 - x_2 + x_3 - ... + (-1)^(k-1) x_k, where the x_i are weakly increasing positive integers.
The value of a(n) depends only on the squarefree part of n, A007913(n). - Antti Karttunen, May 06 2022

Links

Crossrefs

Cf. A000040, A000607, A007913, A071321, A100118, A175352, A316313, A316523.
Cf. A027746, A112798, A119899 (positions of negative terms).
Cf. A344616 (absolute values), A344617 (signs).

Programs

  • Mathematica
    Table[Sum[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]][[k]]*(-1)^(k-1),{k,PrimeOmega[n]}],{n,100}]
  • PARI
    a(n) = {my(f = factor(n), vp = []); for (k=1, #f~, for( j=1, f[k,2], vp = concat (vp, primepi(f[k,1])));); sum(k=1, #vp, vp[k]*(-1)^(k+1));} \\ Michel Marcus, Jul 06 2018
  • Python
    from sympy import factorint, primepi
def A316524(n):
    fs = [primepi(p) for p in factorint(n,multiple=True)]
    return sum(fs[::2])-sum(fs[1::2]) # Chai Wah Wu, Aug 23 2021

Formula

a(n) = A344616(n) * A344617(n) = a(A007913(n)). - Antti Karttunen, May 06 2022

Extensions

More terms from Antti Karttunen, May 06 2022

A359912 Numbers whose prime indices do not have integer median.

Original entry on oeis.org

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 60, 65, 69, 74, 77, 84, 86, 93, 95, 106, 119, 122, 123, 132, 141, 142, 143, 145, 150, 156, 158, 161, 177, 178, 185, 196, 201, 202, 204, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 276, 278
Offset: 1

Views

Author

Gus Wiseman, Jan 24 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.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The terms together with their prime indices begin:
   1: {}
   6: {1,2}
  14: {1,4}
  15: {2,3}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  51: {2,7}
  58: {1,10}
  60: {1,1,2,3}

Crossrefs

For prime factors instead of indices we have A072978, complement A359913.
These partitions are counted by A307683.
For mean instead of median: A348551, complement A316413, counted by A349156.
The complement is A359908, counted by A325347.
Positions of odd terms in A360005.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A051293, A067538, A175352, A175761, A289509, A359890, A359905, A360006, A359907, A360009.

Programs

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

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752
Offset: 0

Views

Author

Gus Wiseman, Nov 14 2021

Keywords

Comments

Equivalently, partitions whose length does not divide their sum.
By conjugation, also the number of integer partitions of n with greatest part not dividing n.

Examples

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Crossrefs

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

Formula

a(n > 0) = A000041(n) - A067538(n).

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Nov 14 2021

Keywords

Comments

Equivalently, partitions whose length does not divide their sum.
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 and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

Crossrefs

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

Programs

  • Maple
    q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021
  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A359913 Numbers whose multiset of prime factors has integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81
Offset: 1

Views

Author

Gus Wiseman, Jan 25 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}

Crossrefs

Prime factors are listed by A027746.
The complement is A072978, for prime indices A359912.
For mean instead of median we have A078175, for prime indices A316413.
For prime indices instead of factors we have A359908, counted by A325347.
Positions of even terms in A360005.
A067340 lists numbers whose prime signature has integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, strict A359907.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103
Offset: 1

Views

Author

Gus Wiseman, Feb 16 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

Crossrefs

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

A175397 Numbers such that both arithmetic means of distinct and all prime factors are not integers.

Original entry on oeis.org

1, 6, 10, 12, 14, 18, 22, 24, 26, 28, 30, 34, 36, 38, 40, 46, 48, 52, 54, 56, 58, 62, 66, 70, 72, 74, 76, 80, 82, 86, 88, 90, 94, 96, 98, 100, 102, 104, 106, 108, 118, 120, 122, 124, 130, 132, 134, 136, 138, 142, 144, 146, 148, 150, 152, 154, 158, 160, 162, 165, 166, 172, 174, 176, 178, 182, 184
Offset: 1

Views

Author

Jaroslav Krizek, May 01 2010

Keywords

Comments

Contains all even semiprimes. - Robert Israel, Nov 10 2024

Examples

			For a(13) = 36: 36 = 2^2*3^3; both (2+2+3+3)/4 and (2+3)/2 are not integers.

Links

Crossrefs

Subsequence of A176552, A175352 and A176587. Complement of A175418. Cf. A174894.

Programs

  • Maple
    filter:= proc(n) local F,t,m;
  F:= ifactors(n)[2]; m:= nops(F);
  not (add(t[1],t=F)/m)::integer and not (add(t[1]*t[2],t=F)/add(t[2],t=F))::integer
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Nov 10 2024

Extensions

a(27) corrected, and more terms from Robert Israel, Nov 10 2024

A175761 Odd nonprimes such that the arithmetic mean of all prime factors is not an integer.

Original entry on oeis.org

1, 45, 63, 75, 99, 117, 135, 147, 153, 165, 171, 175, 207, 245, 255, 261, 273, 279, 315, 325, 333, 345, 351, 363, 369, 375, 385, 387, 399, 405, 423, 435, 455, 459, 475, 477, 495, 507, 531, 539, 549, 561, 567, 595, 603, 615, 639, 651, 657, 665, 675, 705, 711, 715, 735
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 29 2010

Keywords

Comments

The presence of 1 (which has no prime factors) is for compatibility with A175352.

Examples

			a(6) = 117 because 117 = 3*3*13 and (3 + 3 + 13)/3 is not an integer.

Links

Crossrefs

Cf. A141468, A175352.

Programs

  • Mathematica
    fQ[n_] := Block[{fi = Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, len = Length@ fi; len != 1 && ! IntegerQ[ Plus @@ fi/len]]; Join[{1},Select[1 + 2 Range@ 356, fQ]] (* Robert G. Wilson v, Aug 31 2010 *)

Formula

Equals: Intersection of A175352 and A014076.

Extensions

Corrected (315, 345 inserted, 355 removed) by R. J. Mathar, Aug 30 2010

A175418 Complement of A175397, where A175397 = numbers such that both arithmetic means of distinct and all prime factors are not integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 44, 45, 47, 49, 50, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2010

Keywords

Comments

For these numbers hold that both arithmetic means of distinct and all prime factors are integers or only one of these means is an integer.
Includes all prime powers and odd semiprimes. - Robert Israel, Nov 10 2024

Links

Crossrefs

Cf. A174894, A176552, A175352, A175397, A176587.

Programs

  • Maple
    filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  (add(t[1],t=F)/nops(F))::integer or (add(t[1]*t[2],t=F)/add(t[2],t=F))::integer
end proc:
select(filter, [$2..100]); # Robert Israel, Nov 10 2024

Extensions

a(49) corrected by Robert Israel, Nov 10 2024
Showing 1-9 of 9 results.

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