A118914 -id:A118914 - cp's OEIS frontend

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936

Offset: 1

Gus Wiseman, Jun 07 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.
Related concepts:
- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   32: {1,1,1,1,1}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  140: {1,1,3,4}
  176: {1,1,1,1,5}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}

Knapsack partitions are counted by A108917, ranked by A299702.
Complete partitions are counted by A126796, ranked by A325781.
These partitions are counted by A353865.
This is a special case of A353866, counted by A353864, complement A354583.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
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, nonprime A353834.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]

A357875 Numbers whose run-sums of prime indices are weakly increasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Oct 18 2022

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.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is not in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

These partitions are counted by A304405.
These are the indices  of rows in A354584 that are weakly increasing.
The complement is A357876.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],LessEqual@@Total/@Split[primeMS[#]]&]

A324525 Numbers divisible by prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
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 prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). 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 as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   54: {1,2,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324524, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020

A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1

Offset: 2

Matthew Vandermast, Jun 03 2012

Length of row n equals A001221(n).
The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences.  Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
Table lists exponents in the order in which they appear in the prime factorization of a member of A025487.  This ordering is common in database comments (e.g., A008966).
Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

			First rows of table read:
  1;
  1;
  2;
  1;
  1,1;
  1;
  3;
  2;
  1,1;
  1;
  2,1;
  ...
The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.

Jason Kimberley, Table of i, a(i) for i = 2..24301 (n = 2..10000)

Cf. A025487, A001221 (row lengths), A001222 (row sums). A118914 gives the nondecreasing version. A124010 lists exponents in n's prime factorization in natural order, with A124010(1) = 0.
A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688.  Other sequences determined by the exponents in the prime factorization of n include:
Multiplicative: A000005, A007425, A008683, A008836, A034444, A037445, A181819.
Additive: A001221, A001222, A056169.
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012

apply( {A212171_row(n)=vecsort(factor(n)[,2]~,,4)}, [1..40])\\ M. F. Hasler, Apr 19 2022

Row n of A118914, reversed.

Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.

A325239 Irregular triangle read by rows where row 1 is {1} and row n > 1 is the sequence starting with n and repeatedly applying A181819 until 2 is reached.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 2, 5, 2, 6, 4, 3, 2, 7, 2, 8, 5, 2, 9, 3, 2, 10, 4, 3, 2, 11, 2, 12, 6, 4, 3, 2, 13, 2, 14, 4, 3, 2, 15, 4, 3, 2, 16, 7, 2, 17, 2, 18, 6, 4, 3, 2, 19, 2, 20, 6, 4, 3, 2, 21, 4, 3, 2, 22, 4, 3, 2, 23, 2, 24, 10, 4, 3, 2, 25, 3, 2, 26, 4, 3, 2

Offset: 1

Gus Wiseman, Apr 15 2019

The function A181819 maps n = p^i*...*q^j to prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n.

			Triangle begins:
   1              26 4 3 2        51 4 3 2          76 6 4 3 2
   2              27 5 2          52 6 4 3 2        77 4 3 2
   3 2            28 6 4 3 2      53 2              78 8 5 2
   4 3 2          29 2            54 10 4 3 2       79 2
   5 2            30 8 5 2        55 4 3 2          80 14 4 3 2
   6 4 3 2        31 2            56 10 4 3 2       81 7 2
   7 2            32 11 2         57 4 3 2          82 4 3 2
   8 5 2          33 4 3 2        58 4 3 2          83 2
   9 3 2          34 4 3 2        59 2              84 12 6 4 3 2
  10 4 3 2        35 4 3 2        60 12 6 4 3 2     85 4 3 2
  11 2            36 9 3 2        61 2              86 4 3 2
  12 6 4 3 2      37 2            62 4 3 2          87 4 3 2
  13 2            38 4 3 2        63 6 4 3 2        88 10 4 3 2
  14 4 3 2        39 4 3 2        64 13 2           89 2
  15 4 3 2        40 10 4 3 2     65 4 3 2          90 12 6 4 3 2
  16 7 2          41 2            66 8 5 2          91 4 3 2
  17 2            42 8 5 2        67 2              92 6 4 3 2
  18 6 4 3 2      43 2            68 6 4 3 2        93 4 3 2
  19 2            44 6 4 3 2      69 4 3 2          94 4 3 2
  20 6 4 3 2      45 6 4 3 2      70 8 5 2          95 4 3 2
  21 4 3 2        46 4 3 2        71 2              96 22 4 3 2
  22 4 3 2        47 2            72 15 4 3 2       97 2
  23 2            48 14 4 3 2     73 2              98 6 4 3 2
  24 10 4 3 2     49 3 2          74 4 3 2          99 6 4 3 2
  25 3 2          50 6 4 3 2      75 6 4 3 2       100 9 3 2

Michael De Vlieger, Table of n, a(n) for n = 1..10581 (rows 1..2500, flattened)
Michael De Vlieger, For m in row n, plot black else plot white, n = 1..120, 4X magnification.
Michael De Vlieger, For m in row n, plot black else plot white, n = 1..2^12.

Row lengths are A182850(n) + 1.
Cf. A001221, A001222, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023, A325238, A325277.
See A353510 for a full square array version of this table.

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
Table[NestWhileList[red,n,#>2&],{n,30}]

A001222(T(n,k)) = A323023(n,k), n > 2, k <= A182850(n).

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, -1, 1, -3, 0, -9, 3, 0, -2, -21, 2, -51, -10, -3, 9, -111, 3, -237, 0, -15, -26, -489, 10, -2, -70, 2, -12, -995, 0, -2017, 21, -39, -158, -19, 15, -4059, -346, -105, 12, -8151, -18, -16341, -36, -5, -722, -32721, 26, -32, 5, -237, -108, -65483, 19, -53

Offset: 1

Gus Wiseman, Mar 20 2022

sign

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.

Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...

			The partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, so a(196) = 196 - 189 = 7.

Positions of zeros are A088902, counted by A000700.
A similar sequence is A175508.
Positions of nonzero terms are A352486, counted by A330644.
Positions of negative terms are A352487, counted by A000701.
Positions of nonnegative terms are A352488, counted by A046682.
Positions of nonpositive terms are A352489, counted by A046682.
Positions of positive terms are A352490, counted by A000701.
A000041 counts integer partitions, strict A000009.
A003963 is product of prime indices, conjugate A329382.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A238744 is partition conjugate of prime signature, ranked by A238745.
Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

a(n) = n - A122111(n).

A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Gus Wiseman, Nov 26 2023

nonn

As an operation on multisets, this is represented by A367579.

			90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so a(90) = 12.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
Agrees with A181819 at positions A367683, counted by A367682.
Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
Positions of first appearances are A367584, sorted A367585.
Positions of powers of 2 are A367586.
Divides n at positions A367685, counted by A367684.
The opposite version (cokernel) is A367859.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n,100}]

a(n^k) = a(n) for all positive integers n and k.
A001221(a(n)) = A071625(n).
A001222(a(n)) = A001221(n).
If n is squarefree, a(n) = A020639(n)^A001222(n).
A056239(a(n)) = A367581(n).

A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 162, 250, 256, 288, 324, 500, 512, 576, 648, 729, 1000, 1024, 1125, 1152, 1296, 1458, 2000, 2048, 2250, 2304, 2401, 2592, 2916, 4000, 4096, 4500, 4608, 4802, 5184, 5832, 6561, 8000, 8192, 9000, 9216

Offset: 1

Gus Wiseman, Mar 07 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
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 prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A062457.

			The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  162: {1,2,2,2,2}
  250: {1,3,3,3}
  256: {1,1,1,1,1,1,1,1}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588.
Range of values of A090884.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
select(q, [$1..10000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[k,PrimePi[p]]]&]
v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* Amiram Eldar, Sep 30 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020

A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.

Original entry on oeis.org

1, 2, 9, 12, 40, 112, 125, 352, 360, 675, 832, 1008, 2176, 2401, 3168, 3969, 4864, 7488, 11776, 14000, 19584, 29403, 29696, 43776, 44000, 63488, 75600, 104000, 105984, 123201, 151552, 161051, 214375, 237600, 267264, 272000, 335872, 496125, 561600, 571392, 608000

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679). The increasing case is A109298.
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 ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base.
Also Heinz numbers of the integer partitions counted by A324572. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Each finite set of positive integers determines a unique term with those prime indices. For example, corresponding to {1,2,4,5} is 1397088 = prime(1)^5 * prime(2)^4 * prime(4)^2 * prime(5)^1.

			The sequence of terms together with their prime indices begins as follows. For example, we have 40: {1,1,1,3} because 40 = prime(1) * prime(1) * prime(1) * prime(3).
      1: {}
      2: {1}
      9: {2,2}
     12: {1,1,2}
     40: {1,1,1,3}
    112: {1,1,1,1,4}
    125: {3,3,3}
    352: {1,1,1,1,1,5}
    360: {1,1,1,2,2,3}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
   1008: {1,1,1,1,2,2,4}
   2176: {1,1,1,1,1,1,1,7}
   2401: {4,4,4,4}
   3168: {1,1,1,1,1,2,2,5}
   3969: {2,2,2,2,4,4}
   4864: {1,1,1,1,1,1,1,1,8}
   7488: {1,1,1,1,1,1,2,2,6}
  11776: {1,1,1,1,1,1,1,1,1,9}
  14000: {1,1,1,1,3,3,3,4}
  19584: {1,1,1,1,1,1,1,2,2,7}

Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078.
Cf. A324524, A324525, A324572.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360, A304679.

Select[Range[1000],Reverse[PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==Last/@If[#==1,{},FactorInteger[#]]&]

A329138 Numbers whose prime signature is a necklace.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A304678 in having 1350 = 2^1 * 3^3 * 5^2. First differs from A316529 in having 150 = 2^1 * 3^1 * 5^2.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   6: (1,1)
   7: (1)
   8: (3)
   9: (2)
  10: (1,1)
  11: (1)
  13: (1)
  14: (1,1)
  15: (1,1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  21: (1,1)
  22: (1,1)

Complement of A329142.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is aperiodic are A329139.
Cf. A001037, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329140.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],neckQ[Last/@FactorInteger[#]]&]

cp's OEIS Frontend

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

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A357875 Numbers whose run-sums of prime indices are weakly increasing.

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A324525 Numbers divisible by prime(k)^k for each prime index k.

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Maple

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A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

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A325239 Irregular triangle read by rows where row 1 is {1} and row n > 1 is the sequence starting with n and repeatedly applying A181819 until 2 is reached.

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A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

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A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

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