A001222 -id:A001222 - cp's OEIS frontend

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is 5-smooth if its prime divisors are all <= 5.

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 sequence of terms together with their prime indices begins:
       1: {}            125: {3,3,3}
       3: {2}           135: {2,2,2,3}
       5: {3}           150: {1,2,3,3}
       9: {2,2}         225: {2,2,3,3}
      10: {1,3}         243: {2,2,2,2,2}
      15: {2,3}         250: {1,3,3,3}
      25: {3,3}         270: {1,2,2,2,3}
      27: {2,2,2}       300: {1,1,2,3,3}
      30: {1,2,3}       375: {2,3,3,3}
      45: {2,2,3}       405: {2,2,2,2,3}
      50: {1,3,3}       450: {1,2,2,3,3}
      75: {2,3,3}       500: {1,1,3,3,3}
      81: {2,2,2,2}     625: {3,3,3,3}
      90: {1,2,2,3}     675: {2,2,2,3,3}
     100: {1,1,3,3}     729: {2,2,2,2,2,2}

Allowing any number of parts and sum gives A051037, counted by A001399.
These are Heinz numbers of the partitions counted by A266755.
Allowing parts > 5 gives A344291, counted by A110618.
The non-3-smooth case is A344294, counted by A325691.
Requiring the sum of prime indices to be even gives A344295.
A000070 counts non-multigraphical partitions, ranked by A344292.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A300061 ranks partitions of even numbers, with 5-smooth case A344297.
Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]

Intersection of A051037 and A344291.

A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = floor(5/2) = 2.

Positions of 0's are 1 and A000040.
Positions of first appearances are A000302 = 2^(2k) for k >= 0.
Positions of 1's are A168645.
Rounding up instead of down gives A360617.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A280076, A359912, A360006, A360457, A360674.

Mathematica
```
Table[Floor[PrimeOmega[n]/2],{n,100}]
```

A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

Original entry on oeis.org

1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374

Offset: 1

Labos Elemer, Sep 24 2001

nonn

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
Partial sums of A001222, similarly to summatory A001221 increases like loglog(n), explaining small quotients.
a(25) > 10^13. - Giovanni Resta, Apr 25 2017

			Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.

A001222, A022559, A050226, A056650, A064602-A064611, A048290, A062982, A045345.

Mod[A022559(n), n]=0

a(5)-a(24) from Donovan Johnson, Nov 15 2009

A293549 Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 6, 5, 13, 12, 23, 24, 47, 47, 82, 92, 152, 167, 265, 301, 462, 532, 779, 914, 1324, 1548, 2174, 2590, 3573, 4250, 5771, 6904, 9254, 11092, 14638, 17606, 23043, 27680, 35820, 43155, 55383, 66642, 84850, 102141, 129171, 155394, 195134, 234679, 293184, 352096, 437359

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Euler transform of A001222.
Comment from R. J. Mathar, Sep 10 2021 (Start):
The triangle of the multiset transformation of A001222 looks as follows:
;1
0 ;0
1 0 ;1
1 0 0 ;1
2 1 0 0 ;3
1 1 0 0 0 ;2
2 3 1 0 0 0 ;6
1 3 1 0 0 0 0 ;5
3 6 3 1 0 0 0 0 ;13
2 5 4 1 0 0 0 0 0 ;12
2 9 8 3 1 0 0 0 0 0 ;23
1 9 9 4 1 0 0 0 0 0 0 ;24
3 14 17 9 3 1 0 0 0 0 0 0 ;47
1 12 18 11 4 1 0 0 0 0 0 0 0 ;47
2 17 29 21 9 3 1 0 0 0 0 0 0 0 ;82
...
The second column is A001222, the row sums (after the semicolons) are this sequence. (End)

N. J. A. Sloane, Transforms

Cf. A001222, A006171, A293548.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeOmega[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeOmega[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

G.f.: Product_{k>=2} 1/(1 - x^k)^b(k), where b(k) = [x^k] Sum_{p prime, j>=1} x^(p^j)/(1 - x^(p^j)).

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*bigomega(d).

A328965 Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.

Original entry on oeis.org

1, 4, 6, 16, 12, 64, 24, 256, 48, 60, 96, 4096, 120, 16384, 384, 240, 420, 262144, 480, 1048576, 840, 960, 6144, 16777216, 1680, 4620, 24576, 3840, 3360, 1073741824, 7680, 4294967296, 6720, 15360, 393216, 18480, 13440, 274877906944, 1572864, 61440, 26880, 4398046511104

Offset: 0

Gus Wiseman, Nov 02 2019

nonn

For n > 0, a(n) is of the form 2^k*primorial(d) where d is a divisor of n and k = n / d - d + 1. a(n) is never 0 since A307409(2^(n+1)) = n. - Andrew Howroyd, Nov 04 2019

			The sequence of terms together with their prime signatures begins:
      1: ()
      4: (2)
      6: (1,1)
     16: (4)
     12: (2,1)
     64: (6)
     24: (3,1)
    256: (8)
     48: (4,1)
     60: (2,1,1)
     96: (5,1)
   4096: (12)
    120: (3,1,1)
  16384: (14)
    384: (7,1)
    240: (4,1,1)
    420: (2,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Positions of first appearances in A307409.

Cf. A001221, A001222, A002110, A113901, A124010, A320632, A323023, A328956, A328958, A328959, A328962, A328963, A328964.

dat=Table[(PrimeOmega[n]-1)*PrimeNu[n],{n,1000}];
Table[Position[dat,i][[1,1]],{i,First[Split[Union[dat],#2==#1+1&]]}]

a(n)={if(n<1, 1, my(m=oo); fordiv(n, d, if(d<=n/d+1, m=min(m, 2^(n/d-d+1)*vecprod(primes(d))))); m)} \\ Andrew Howroyd, Nov 04 2019

From Andrew Howroyd, Nov 03 2019: (Start)
a(p) = 2^(p + 1) for odd prime p.
a(n) = min_{d|n, d<=n/d+1} 2^(n/d-d+1)*A002110(d) for n > 0. (End)

Terms a(23) and beyond from Andrew Howroyd, Nov 03 2019

A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

Original entry on oeis.org

5, 10, 15, 25, 30, 45, 50, 75, 90, 100, 125, 135, 150, 225, 250, 270, 300, 375, 405, 450, 500, 625, 675, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050, 4500, 5000, 5625, 6075, 6250

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is d-smooth iff its prime divisors are all <= d.

A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).

			The sequence of terms together with their prime indices begins:
       5: {3}           270: {1,2,2,2,3}
      10: {1,3}         300: {1,1,2,3,3}
      15: {2,3}         375: {2,3,3,3}
      25: {3,3}         405: {2,2,2,2,3}
      30: {1,2,3}       450: {1,2,2,3,3}
      45: {2,2,3}       500: {1,1,3,3,3}
      50: {1,3,3}       625: {3,3,3,3}
      75: {2,3,3}       675: {2,2,2,3,3}
      90: {1,2,2,3}     750: {1,2,3,3,3}
     100: {1,1,3,3}     810: {1,2,2,2,2,3}
     125: {3,3,3}       900: {1,1,2,2,3,3}
     135: {2,2,2,3}    1000: {1,1,1,3,3,3}
     150: {1,2,3,3}    1125: {2,2,3,3,3}
     225: {2,2,3,3}    1215: {2,2,2,2,2,3}
     250: {1,3,3,3}    1250: {1,3,3,3,3}

Allowing any number of parts and sum gives A080193, counted by A069905.
The partitions with these Heinz numbers are counted by A325691.
Relaxing the smoothness conditions gives A344291, counted by A110618.
Allowing 3-smoothness gives A344293, counted by A266755.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n whose length is n/2, ranked by A340387.
A051037 lists 5-smooth numbers (complement: A279622).
A056239 adds up prime indices, row sums of A112798.
A257993 gives the least gap of the partition with Heinz number n.
A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]==5&]

Intersection of A080193 and A344291.

A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 7, 9, 8, 9, 11, 11, 15, 16, 16, 18, 20, 22, 26, 28, 31, 32, 36, 40, 45, 46, 46, 50, 59, 64, 70, 75, 78, 83, 89, 94, 108, 106, 104, 120, 137, 142, 147, 150, 161, 174, 190, 200, 220, 226, 224, 248, 274, 274, 287, 301, 320, 340, 351, 361

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(11) = 7 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (41)  (42)  (43)   (62)   (54)   (82)   (74)
                              (51)  (61)   (71)   (63)   (91)   (65)
                                    (421)  (431)  (81)   (451)  (83)
                                                  (621)  (631)  (92)
                                                                (A1)
                                                                (821)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

The weakly decreasing version is A358909 (complement A358910).
The version not counting multiplicity is A358903, weakly decreasing A358902.
For equal numbers of prime factors we have A319169, compositions A358911.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A056239, A129519, A141199, A218482, A300335, A319071, A320324, A358335, A358831, A358908.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeOmega/@#&]],{n,0,60}]

a(61) and beyond from Lucas A. Brown, Dec 14 2022

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

Original entry on oeis.org

2, 6, 7, 8, 9, 10, 11, 13, 15, 19, 24, 28, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 54, 57, 58, 59, 60, 61, 65, 67, 70, 73, 76, 77, 79, 85, 86, 90, 95, 96, 97, 98, 103, 106, 107, 109, 110, 111, 112, 117, 119, 123, 124, 126, 127, 128, 129

Offset: 1

Gus Wiseman, May 14 2024

nonn

The odd version is A372590.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
          {2}   2  (1)
        {2,3}   6  (2,1)
      {1,2,3}   7  (4)
          {4}   8  (1,1,1)
        {1,4}   9  (2,2)
        {2,4}  10  (3,1)
      {1,2,4}  11  (5)
      {1,3,4}  13  (6)
    {1,2,3,4}  15  (3,2)
      {1,2,5}  19  (8)
        {4,5}  24  (2,1,1,1)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
          {6}  32  (1,1,1,1,1)
        {1,6}  33  (5,2)
        {2,6}  34  (7,1)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
    {1,2,3,6}  39  (6,2)
        {4,6}  40  (3,1,1,1)
      {1,4,6}  41  (13)
      {2,4,6}  42  (4,2,1)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
Positions of even terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372590.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],EvenQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 9, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 10, 25, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 13, 1, 2, 4, 7, 14, 49, 1, 3, 5, 9, 15, 25, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 19, 1, 2, 4, 5, 8, 10, 20, 25, 50, 125

Offset: 1

Michael De Vlieger, Oct 09 2024

Analogous to A162306 regarding m such that rad(m) | n, but instead of taking m <= n, we take m such that bigomega(m) <= bigomega(n).
Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k <= bigomega(n).
For prime power n = p^k, k >= 0 (i.e., n in A000961), row p^k of this sequence is the same as row p^k of A027750 and A162306. Therefore, for prime p, row p of this sequence is the same as row p of A027750 and A162306: {1, p}.
For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n.

			Triangle begins:
   n    row n of this sequence:
  -------------------------------------------
   1:   1;
   2:   1,  2;
   3:   1,  3;
   4:   1,  2   4;
   5:   1,  5;
   6:   1,  2,  3,  4,  6,  9;
   7:   1,  7;
   8:   1,  2,  4,  8;
   9:   1,  3,  9;
  10:   1,  2,  4,  5, 10, 25;
  11:   1, 11;
  12:   1,  2,  3,  4,  6,  8, 9, 12, 18, 27;
        ...
Row n = 10 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 5^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m and the parenthetic 8 are in row 10 of A162306:
   1   2   4  (8)
   5  10
  25*
Row n = 12 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 3^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m are in row 12 of A162306:
   1   2   4   8
   3   6  12
   9  18*
  27*

Michael De Vlieger, Table of n, a(n) for n = 1..17475 (rows n = 1..1000, flattened)
Michael De Vlieger, Log log scatterplot of rows n = 1..2^16 of this sequence, flattened, (3153752 terms).

Cf. A000961, A001221, A001222, A006530, A007947, A010846, A024619, A027750, A162306, A376567.

Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]]; k = PrimeOmega[n]; w = PrimeNu[n]; Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &, Select[Tuples[Range[0, k], w], Total[#] <= k &] ], {n, 120}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) <= bigomega(n) }.

A376567(n) = binomial(bigomega(n) + omega(n)) = Length of row n, where omega = A001221.

A064800 n plus the number of its prime factors: a(n) = n + A001222(n).

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 8, 11, 11, 12, 12, 15, 14, 16, 17, 20, 18, 21, 20, 23, 23, 24, 24, 28, 27, 28, 30, 31, 30, 33, 32, 37, 35, 36, 37, 40, 38, 40, 41, 44, 42, 45, 44, 47, 48, 48, 48, 53, 51, 53, 53, 55, 54, 58, 57, 60, 59, 60, 60, 64, 62, 64, 66, 70, 67, 69, 68, 71, 71, 73, 72

Offset: 1

Reinhard Zumkeller, Oct 21 2001

nonn

Prime factors counted with multiplicity. - Harvey P. Dale, Aug 19 2015

			a(42) = 45 = 42 + 3 (as 42 = 2 * 3 * 7)

Harry J. Smith, Table of n, a(n) for n=1,...,1000

Cf. A001222, A229109.

a064800 n = a001222 n + n  -- Reinhard Zumkeller, Oct 27 2012

A064800 := proc(n)
    n+numtheory[bigomega](n) ;
end proc: # R. J. Mathar, Oct 21 2012

Table[n + PrimeOmega[n],{n,80}] (* Harvey P. Dale, Aug 19 2015 *)

{ for (n=1, 1000, write("b064800.txt", n, " ", n + bigomega(n)) ) } \\ Harry J. Smith, Sep 26 2009

cp's OEIS Frontend

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

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A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

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A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

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A293549 Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

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A328965 Smallest k such that (bigomega(k) - 1) * omega(k) = n, and 0 if none exists, where omega = A001221, bigomega = A001222.

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A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

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A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

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A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

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