A071625 -id:A071625 - cp's OEIS frontend

A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947

Offset: 1

Gus Wiseman, Feb 14 2025

nonn

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.

All terms are odd.

			The first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
      1: {}
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    169: {6,6}
    289: {7,7}
     81: {2,2,2,2}
    125: {3,3,3}
    841: {10,10}
    961: {11,11}
    675: {2,2,2,3,3}
   1681: {13,13}
   1849: {14,14}
   2209: {15,15}
    243: {2,2,2,2,2}
   3481: {17,17}
   1125: {2,2,3,3,3}

For factors instead of indices we have A064549 (sorted A001694), firsts of A003557.
The additive version for factors is A280286 (sorted A381075), firsts of A280292.
Position of first appearance of n in A290106.
The additive version is A380956 (sorted A380957), firsts of A380955.
For difference instead of quotient see A380986.
The sorted version is A380988.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,10000}];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

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.

All terms are odd.

			The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    81: {2,2,2,2}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   243: {2,2,2,2,2}
   289: {7,7}
   625: {3,3,3,3}
   675: {2,2,2,3,3}
   729: {2,2,2,2,2,2}
   841: {10,10}
   961: {11,11}
  1125: {2,2,3,3,3}
  1331: {5,5,5}
  1681: {13,13}
  1849: {14,14}
  2025: {2,2,2,2,3,3}

For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
Sorted firsts of A290106.
The additive version is A380957 (sorted A380956), firsts of A380955.
For difference instead of quotient see A380986.
The unsorted version is A380987.
The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  240: {1,1,1,1,2,3}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  420: {1,1,2,3,4}
  432: {1,1,1,1,2,2,2}
  480: {1,1,1,1,1,2,3}
  576: {1,1,1,1,1,1,2,2}
  768: {1,1,1,1,1,1,1,1,2}
  840: {1,1,1,2,3,4}
  864: {1,1,1,1,1,2,2,2}
  960: {1,1,1,1,1,1,2,3}

Cf. A001221, A001222, A001358, A005117, A007774, A007916, A055932, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304636, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}],2]

A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

Original entry on oeis.org

8, 27, 30, 42, 66, 70, 78, 102, 105, 110, 114, 125, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 343, 345, 354, 357, 360, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
     8: {1,1,1}
    30: {1,2,3}
   360: {1,1,1,2,2,3}
   720: {1,1,1,1,2,2,3}
   900: {1,1,2,2,3,3}
  1440: {1,1,1,1,1,2,2,3}
  2160: {1,1,1,1,2,2,2,3}
  2880: {1,1,1,1,1,1,2,2,3}
  4320: {1,1,1,1,1,2,2,2,3}
  5760: {1,1,1,1,1,1,1,2,2,3}
  8640: {1,1,1,1,1,1,2,2,2,3}
Starting with A112798(1801800) and repeatedly taking the multiset of multiplicities we have {1,1,1,2,2,3,3,4,5,6} -> {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so 1801800 belongs to the sequence.

Cf. A001221, A001222, A005117, A007916, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A304464, A304465, A304634, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,200}],3]

A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).

These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]

A325247 Numbers whose omega-sequence is strict (no repeated parts).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

First differs from A323306 in having 216.
We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).
Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    11: {5}
    13: {6}
    16: {1,1,1,1}
    17: {7}
    19: {8}
    23: {9}
    25: {3,3}
    27: {2,2,2}
    29: {10}
    31: {11}
    32: {1,1,1,1,1}
    36: {1,1,2,2}

Positions of squarefree numbers in A325248.
Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.
Omega-sequence statistics: A001221 (second omega), A001222 (first omega), A071625 (third omega), A304465 (second-to-last omega), A182850 or A323014 (depth), A323022 (fourth omega), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#1]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],UnsameQ@@omseq[#]&]

A325259 Numbers with one fewer distinct prime exponents than distinct prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with one fewer distinct multiplicities than distinct parts. The enumeration of these partitions by sum is given by A325244.

			The sequence of terms together with their prime indices begins:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}
   51: {2,7}
   55: {3,5}
   57: {2,8}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}

Cf. A056239, A060687, A090858, A112798, A116608, A118914, A130091, A323023, A325241, A325242, A325244, A325270, A325281.

Select[Range[100],PrimeNu[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A001221(a(n)) = A071625(a(n)) + 1.

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 133

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also Heinz numbers of integer partitions with 1 fewer distinct multiplicities than parts, where 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 A117571.

Also numbers whose sorted prime signature is (1,1), (2), or (1,2). - Gus Wiseman, Jul 03 2019

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   44: {1,1,5}

Cf. A001221, A001222, A000961, A005117, A060687, A062770, A071625, A072774, A090858, A117571, A118914, A130091, A325244, A325259.

Select[Range[100],PrimeOmega[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 7, 13, 14, 15, 7, 17, 14, 19, 15, 21, 22, 23, 15, 13, 26, 13, 21, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 13, 41, 42, 43, 33, 35, 46, 47, 21, 19, 26, 51, 39, 53, 26, 55, 35, 57, 58, 59, 35, 61, 62, 19, 13, 65, 66, 67, 51

Offset: 1

Gus Wiseman, May 25 2022

nonn

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.

The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The partition run-sum trajectory of 87780 is: 87780 -> 65835 -> 51205 -> 19855 -> 2915, so a(87780) = 2915.

The fixed points and image are A005117.
For run-lengths instead of sums we have A304464/A304465, counted by A325268.
These are the row-ends of A353840.
Other sequences pertaining to partition trajectory are A353841-A353846.
The version for compositions is A353855, run-ends of A353853.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A182850 and A323014 give frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A071625, A073093, A181819, A325239, A325277, A353834, A353838, A353847, A353865, A353867.

Table[NestWhile[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&],{n,100}]

A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 84, 134, 213, 338, 536, 850, 1349, 2136, 3389, 5367, 8509, 13480, 21362, 33843, 53624, 84957, 134600, 213251, 337850, 535251, 847987, 1343440, 2128372, 3371895, 5341977, 8463051, 13407689, 21241181, 33651507, 53312538, 84460690

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (24)
                 (111)  (31)    (32)     (33)
                        (211)   (41)     (42)
                        (1111)  (221)    (51)
                                (311)    (222)
                                (2111)   (231)
                                (11111)  (321)
                                         (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..5004 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358903.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A116608 counts partitions by sum and number of distinct parts.
A334028 counts distinct parts in standard compositions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A056239, A071625, A129519, A300335, A358831, A358908, A358911.

p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t<=i, b(n-j, t), 0))(p(j)), j=1..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 14 2024

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeNu/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

cp's OEIS Frontend

A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

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A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

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A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

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A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

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A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

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A325247 Numbers whose omega-sequence is strict (no repeated parts).

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A325259 Numbers with one fewer distinct prime exponents than distinct prime factors.

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Formula

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

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A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

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