A064839 -id:A064839 - cp's OEIS frontend

A372133 Prime shadow (A181819) with shifted e_i to generate odd numbers, multiplied by 2^(A064839 - 1).

0, 1, 2, 3, 4, 7, 8, 5, 6, 14, 16, 13, 32, 28, 56, 9, 64, 26, 128, 52, 112, 224, 256, 19, 12, 448, 10, 104, 512, 25, 1024, 11, 896, 1792, 3584, 23, 2048, 7168, 14336, 38, 4096, 50, 8192, 208, 416, 28672, 16384, 31, 24, 832, 57344, 1664, 32768, 76, 114688

Offset: 1

Friedjof Tellkamp, Apr 20 2024

nonn

a(n>1) contains all positive integers exactly once.

Cf. A064839, A181819, A372134.

z = 100; prisigs = Table[If[n == 1, {0}, Sort[Last /@ FactorInteger[n]]], {n, z}];
exps = Table[Count[Take[prisigs, n], prisigs[[n]]], {n, z}]; (*A064839*)
Table[2^(exps[[n]] - 1) (Times @@ Prime[1 + prisigs[[n]]] - 2), {n, z}]

Factor n, multiplying out prime(e_i + 1) generates all odd numbers >= 3. Subtract 2 and multiply by 2^(k - 1), where k is the number of occurrences of the same prime signature of n in (1..n) (A064839).

A372134 Prime shadows (A181819) linearly transformed to generate odd numbers, multiplied by 2^(A064839 - 1).

Original entry on oeis.org

-1, 1, 2, 3, 4, 5, 8, 7, 6, 10, 16, 9, 32, 20, 40, 11, 64, 18, 128, 36, 80, 160, 256, 17, 12, 320, 14, 72, 512, 13, 1024, 19, 640, 1280, 2560, 15, 2048, 5120, 10240, 34, 4096, 26, 8192, 144, 288, 20480, 16384, 25, 24, 576, 40960, 1152, 32768, 68, 81920, 136

Offset: 1

Friedjof Tellkamp, Apr 20 2024

sign

a(n>1) contains all positive integers exactly once.

Cf. A064839, A181819, A372133.

z = 100; prisigs = Table[If[n == 1, {}, Sort[Last /@ FactorInteger[n]]], {n, z}];
exps = Table[Count[Take[prisigs, n], prisigs[[n]]], {n, z}];(* A064839 *)
Table[2^(exps[[n]] - 1) (2 Times @@ Prime[prisigs[[n]]] - 3), {n, z}]

a(n) = (2 * A181819(n) - 3) * 2^(k - 1), where k is the number of occurrences of the same prime signature of n in (1..n) (A064839).

A343133 Numbers k such that A064839(k) = A064839(k+1).

Original entry on oeis.org

1, 9, 58, 61, 73, 80, 82, 1224, 1368, 3075, 3720, 5328, 22112, 45890, 145132, 145138, 269843, 377739, 399281, 622515, 744768, 1280073, 1280437, 1280441, 1281165, 1281190, 1281241, 2961840, 33275384, 54025424, 54161775, 70695344, 91136415, 922135875, 922141772

Offset: 1

Amiram Eldar, Jul 07 2021

nonn

The corresponding values of A064839 are 1, 2, 17, 18, 21, 2, 23, 10, 12, 278, 18, 21, 150, 2842, 13434, 13435, 13547, 3654, 33805, 55229, 150, 265608, 265682, 265683, 265832, 265837, 265849, 268, 773172, 308093, 308810, 395158, 540683, 24172493, 24172646, ...

Are there numbers k such that A064839(k) = A064839(k+1) = A064839(k+2)?

			9 is a term since 9 = 3^2 = A001248(2) is the second square of a prime, and 9 + 1 = 10 = 2 * 5 = A006881(2) is the second squarefree semiprime.
58 is a term since 58 = 2*29 = A001248(17) is the 17th squarefree semiprime, and 58 + 1 = 59 = A000040(17) is the 17th prime.

Cf. A025487, A064839.

Cf. A001248, A006881.

lpsv = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; lps[n_] := Module[{s = Sort[FactorInteger[n][[;; , 2]]], m}, m = Length[s]; Product[Prime[i]^s[[m - i + 1]], {i, 1, m}]]; n = 100; mx = lpsv[[n]]; c = Table[0, {n}]; v1 = 1; s = {}; Do[lps1 = lps[k]; p = Position[lpsv, lps1][[1, 1]]; c[[p]]++; v2 = c[[p]]; If[v1 == v2, AppendTo[s, k - 1]]; v1 = v2, {k, 2, mx}]; s

A101296 n has the a(n)-th distinct prime signature.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 5, 6, 2, 9, 2, 10, 4, 4, 4, 11, 2, 4, 4, 8, 2, 9, 2, 6, 6, 4, 2, 12, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 13, 2, 4, 6, 14, 4, 9, 2, 6, 4, 9, 2, 15, 2, 4, 6, 6, 4, 9, 2, 12, 7, 4, 2, 13, 4, 4, 4, 8, 2, 13, 4, 6, 4, 4, 4, 16, 2, 6, 6, 11, 2, 9, 2, 8, 9, 4, 2, 15, 2, 9, 4, 12, 2, 9, 4, 6, 6, 4, 4, 17

Offset: 1

David Wasserman, Dec 21 2004

From Antti Karttunen, May 12 2017: (Start)
Restricted growth sequence transform of A046523, the least representative of each prime signature. Thus this partitions the natural numbers to the same equivalence classes as A046523, i.e., for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j), and for that reason satisfies in that respect all the same conditions as A046523. For example, we have, for all i, j: if a(i) = a(j), then:
    A000005(i) = A000005(j), A008683(i) = A008683(j), A286605(i) = A286605(j).
So, this sequence (instead of A046523) can be used for finding sequences where a(n)'s value is dependent only on the prime signature of n, that is, only on the multiset of prime exponents in the factorization of n. (End)
This is also the restricted growth sequence transform of many other sequences, for example, that of A181819. See further comments there. - Antti Karttunen, Apr 30 2022

			From _David A. Corneth_, May 12 2017: (Start)
1 has prime signature (), the first distinct prime signature. Therefore, a(1) = 1.
2 has prime signature (1), the second distinct prime signature after (1). Therefore, a(2) = 2.
3 has prime signature (1), as does 2. Therefore, a(3) = a(2) = 2.
4 has prime signature (2), the third distinct prime signature after () and (1). Therefore, a(4) = 3. (End)
From _Antti Karttunen_, May 12 2017: (Start)
Construction of restricted growth sequences: In this case we start with a(1) = 1 for A046523(1) = 1, and thereafter, for all n > 1, we use the least so far unused natural number k for a(n) if A046523(n) has not been encountered before, otherwise [whenever A046523(n) = A046523(m), for some m < n], we set a(n) = a(m).
For n = 2, A046523(2) = 2, which has not been encountered before (first prime), thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n = 3, A046523(2) = 2, which was already encountered as A046523(1), thus we set a(3) = a(2) = 2.
For n = 4, A046523(4) = 4, not encountered before (first square of prime), thus we allot for a(4) the least so far unused number, which is 3, thus a(4) = 3.
For n = 5, A046523(5) = 2, as for the first time encountered at n = 2, thus we set a(5) = a(2) = 2.
For n = 6, A046523(6) = 6, not encountered before (first semiprime pq with distinct p and q), thus we allot for a(6) the least so far unused number, which is 4, thus a(6) = 4.
For n = 8, A046523(8) = 8, not encountered before (first cube of a prime), thus we allot for a(8) the least so far unused number, which is 5, thus a(8) = 5.
For n = 9, A046523(9) = 4, as for the first time encountered at n = 4, thus a(9) = 3.
(End)
From _David A. Corneth_, May 12 2017: (Start)
(Rough) description of an algorithm of computing the sequence:
Suppose we want to compute a(n) for n in [1..20].
We set up a vector of 20 elements, values 0, and a number m = 1, the minimum number we haven't checked and c = 0, the number of distinct prime signatures we've found so far.
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
We check the prime signature of m and see that it's (). We increase c with 1 and set all elements up to 20 with prime signature () to 1. In the process, we adjust m. This gives:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. The least number we haven't checked is m = 2. 2 has prime signature (1). We increase c with 1 and set all elements up to 20 with prime signature (1) to 2. In the process, we adjust m. This gives:
[1, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0]
We check the prime signature of m = 4 and see that its prime signature is (2). We increase c with 1 and set all numbers up to 20 with prime signature (2) to 3. This gives:
[1, 2, 2, 3, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0]
Similarily, after m = 6, we get
[1, 2, 2, 3, 2, 4, 2, 0, 3, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0], after m = 8 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0], after m = 12 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 0, 2, 6, 2, 0], after m = 16 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 0], after m = 20 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 8]. Now, m > 20 so we stop. (End)
The above method is inefficient, because the step "set all elements a(n) up to n = Nmax with prime signature s(n) = S[c] to c" requires factoring all integers up to Nmax (or at least comparing their signature, once computed, with S[c]) again and again. It is much more efficient to run only once over each m = 1..Nmax, compute its prime signature s(m), add it to an ordered list in case it did not occur earlier, together with its "rank" (= new size of the list), and assign that rank to a(m). The list of prime signatures is much shorter than [1..Nmax]. One can also use m'(m) := the smallest n with the prime signature of m (which is faster to compute than to search for the signature) as representative for s(m), and set a(m) := a(m'(m)). Then it is sufficient to have just one counter (number of prime signatures seen so far) as auxiliary variable, in addition to the sequence to be computed. - _M. F. Hasler_, Jul 18 2019

Michel Marcus (terms 1..10000) & Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A025487, A046523, A064839 (ordinal transform of this sequence), A181819, and arrays A095904, A179216.
Cf. A000005, A008683.
Sequences that are unions of finite number (>= 2) of equivalence classes determined by the values that this sequence obtains (i.e., sequences mentioned in David A. Corneth's May 12 2017 formula): A001358 (A001248 U A006881, values 3 & 4), A007422 (values 1, 4, 5), A007964 (2, 3, 4, 5), A014612 (5, 6, 9), A030513 (4, 5), A037143 (1, 2, 3, 4), A037144 (1, 2, 3, 4, 5, 6, 9), A080258 (6, 7), A084116 (2, 4, 5), A167171 (2, 4), A217856 (6, 9).
Cf. also A077462, A305897 (stricter variants, with finer partitioning) and A254524, A286603, A286605, A286610, A286619, A286621, A286622, A286626, A286378 for other similarly constructed sequences.

A101296 := proc(n)
    local a046523, a;
    a046523 := A046523(n) ;
    for a from 1 do
        if A025487(a) = a046523 then
            return a;
        elif A025487(a) > a046523 then
            return -1 ;
        end if;
    end do:
end proc: # R. J. Mathar, May 26 2017

With[{nn = 120}, Function[s, Table[Position[Keys@s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}] ] (* Michael De Vlieger, May 12 2017, Version 10 *)

find(ps, vps) = {for (k=1, #vps, if (vps[k] == ps, return(k)););}
lisps(nn) = {vps = []; for (n=1, nn, ps = vecsort(factor(n)[,2]); ips = find(ps, vps); if (! ips, vps = concat(vps, ps); ips = #vps); print1(ips, ", "););} \\ Michel Marcus, Nov 15 2015; edited by M. F. Hasler, Jul 16 2019

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(100000,n,A046523(n))),"b101296.txt");
\\ Antti Karttunen, May 12 2017

A025487(a(n)) = A046523(n).
Indices of records give A025487. - Michel Marcus, Nov 16 2015
From David A. Corneth, May 12 2017: (Start) [Corresponding characteristic function in brackets]
a(A000012(n)) = 1 (sig.: ()).      [A063524]
a(A000040(n)) = 2 (sig.: (1)).     [A010051]
a(A001248(n)) = 3 (sig.: (2)).     [A302048]
a(A006881(n)) = 4 (sig.: (1,1)).   [A280710]
a(A030078(n)) = 5 (sig.: (3)).
a(A054753(n)) = 6 (sig.: (1,2)).   [A353472]
a(A030514(n)) = 7 (sig.: (4)).
a(A065036(n)) = 8 (sig.: (1,3)).
a(A007304(n)) = 9 (sig.: (1,1,1)). [A354926]
a(A050997(n)) = 10 (sig.: (5)).
a(A085986(n)) = 11 (sig.: (2,2)).
a(A178739(n)) = 12 (sig.: (1,4)).
a(A085987(n)) = 13 (sig.: (1,1,2)).
a(A030516(n)) = 14 (sig.: (6)).
a(A143610(n)) = 15 (sig.: (2,3)).
a(A178740(n)) = 16 (sig.: (1,5)).
a(A189975(n)) = 17 (sig.: (1,1,3)).
a(A092759(n)) = 18 (sig.: (7)).
a(A189988(n)) = 19 (sig.: (2,4)).
a(A179643(n)) = 20 (sig.: (1,2,2)).
a(A189987(n)) = 21 (sig.: (1,6)).
a(A046386(n)) = 22 (sig.: (1,1,1,1)).
a(A162142(n)) = 23 (sig.: (2,2,2)).
a(A179644(n)) = 24 (sig.: (1,1,4)).
a(A179645(n)) = 25 (sig.: (8)).
a(A179646(n)) = 26 (sig.: (2,5)).
a(A163569(n)) = 27 (sig.: (1,2,3)).
a(A179664(n)) = 28 (sig.: (1,7)).
a(A189982(n)) = 29 (sig.: (1,1,1,2)).
a(A179666(n)) = 30 (sig.: (3,4)).
a(A179667(n)) = 31 (sig.: (1,1,5)).
a(A179665(n)) = 32 (sig.: (9)).
a(A189990(n)) = 33 (sig.: (2,6)).
a(A179669(n)) = 34 (sig.: (1,2,4)).
a(A179668(n)) = 35 (sig.: (1,8)).
a(A179670(n)) = 36 (sig.: (1,1,1,3)).
a(A179671(n)) = 37 (sig.: (3,5)).
a(A162143(n)) = 38 (sig.: (2,2,2)).
a(A179672(n)) = 39 (sig.: (1,1,6)).
a(A030629(n)) = 40 (sig.: (10)).
a(A179688(n)) = 41 (sig.: (1,3,3)).
a(A179689(n)) = 42 (sig.: (2,7)).
a(A179690(n)) = 43 (sig.: (1,1,2,2)).
a(A189991(n)) = 44 (sig.: (4,4)).
a(A179691(n)) = 45 (sig.: (1,2,5)).
a(A179692(n)) = 46 (sig.: (1,9)).
a(A179693(n)) = 47 (sig.: (1,1,1,4)).
a(A179694(n)) = 48 (sig.: (3,6)).
a(A179695(n)) = 49 (sig.: (2,2,3)).
a(A179696(n)) = 50 (sig.: (1,1,7)).
(End)

Data section extended to 120 terms by Antti Karttunen, May 12 2017

Minor edits/corrections by M. F. Hasler, Jul 18 2019

A065036 Product of the cube of a prime (A030078) and a different prime.

Original entry on oeis.org

24, 40, 54, 56, 88, 104, 135, 136, 152, 184, 189, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 459, 472, 488, 513, 536, 568, 584, 621, 632, 664, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029, 1048, 1096, 1107, 1112

Offset: 1

Alford Arnold, Nov 04 2001

This sequence appears on row 8 of the list illustrated in A064839 and is similar to A054753 which appears on row 6. Previous rows are generated by A000007, A000040, A001248, A006881, A030078 respectively.

Or, the numbers n such that 20=number of perfect partitions of n. - Juri-Stepan Gerasimov, Sep 26 2009

			a(4)= 56 since 56 = 2*2*2*7.

Cf. A000007, A000040, A001248, A002033, A006881, A030078, A054753, A064839, A089233.

Select[ Range[1500], Sort[ Transpose[ FactorInteger[ # ]] [[2]]] == {1, 3} & ]
Module[{upto=1200},Select[(Union[Flatten[{#[[1]]^3 #[[2]],#[[1]]#[[2]]^3}&/@Subsets[Prime[Range[upto/8]],{2}]]]),#<=upto&]] (* Harvey P. Dale, May 23 2015 *)

list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/3),t=p^3; forprime(q=2,lim\t,if(p==q,next);listput(v,t*q)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

is(n)=my(f=factor(n)[,2]); f==[3,1]~||f==[1,3]~ \\ Charles R Greathouse IV, Oct 15 2015

from sympy import primepi, primerange, integer_nthroot
def A065036(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,4)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A002033(a(n)) = 20. - Juri-Stepan Gerasimov, Sep 26 2009
A089233(a(n)) = 3. - Reinhard Zumkeller, Sep 04 2013
A000005(a(n)) = 8. - Altug Alkan, Nov 11 2015

A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, 14641, 56, 66, 243, 36, 31, 841, 34, 6859, 50, 28561, 88, 70

Offset: 0

Alford Arnold, Jul 10 2004

The unit, 1, has the empty prime signature { } (thus not in triangle).
Downwards diagonals:
* Rightmost diagonal: smallest numbers of a given prime signature in increasing order (A025487). This defines the order of signatures used.
  This special ordering of prime signatures (by increasing smallest numbers of a given prime signature, A181087) is unrelated to any of the 8 variants of graded lexicographic or colexicographic orderings (based on the exponents only) since it depends on the magnitudes of the prime numbers. It is not even graded by Omega(n).
* Second rightmost diagonal: second smallest numbers of a given prime signature (A077560). (They are not increasing anymore.)
Upwards diagonals:
* Leftmost diagonal: primes.                           {1}     (A000040)
* 2nd leftmost diagonal: squares of primes.            {2}     (A001248)
* 3rd leftmost diagonal: squarefree biprimes.          {1,1}   (A006881)
* 4th leftmost diagonal: cubes of primes.              {3}     (A030078)
* 5th leftmost diagonal: signature (Achilles numbers)  {1,2}   (A054753)
* 6th leftmost diagonal: fourth powers of primes.      {4}     (A030514)
* 7th leftmost diagonal: signature (Achilles numbers)  {1,3}   (A065036)
* 8th leftmost diagonal: squarefree triprimes.         {1,1,1} (A007304)
  The Achilles numbers are nonsquarefree while not perfect powers.
Prime signatures are often expressed in increasing order of exponents. The decreasing order of exponents (as on the Wiki page, see links) has the advantage of listing the exponents in the same order (with the canonical factorization convention) as the smallest number of a given prime signature.

			343 is in the 4th left- and 4th rightmost diagonal, because it is the 4th value with the 4th prime signature {3}.
First 8 rows of triangular array (Cf. table link for this sequence):
                                   2
                              3         4
                         5         9         6
                    7        25        10         8
               11       49        14        27        12
          13      121        15       125        18        16
     17       169       21       343        20        81        24
19       289       22       1331       28       625        40        30

OEIS Wiki, Prime signature.
OEIS Wiki, Orderings.

Cf. A083140, A064839, A181087.

Extended by Ray Chandler, Jul 31 2004
Corrected (minor) by Daniel Forgues, Jan 21 2011
Example, comments by Daniel Forgues, Jan 21 2011
Edited by Alois P. Heinz, Jan 23 2011
Edited by Daniel Forgues, Jan 23 2011

A325365 Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 4, 8, 4, 20, 4, 12, 12, 48, 8, 56, 16, 64, 48, 48, 36, 324, 162, 81, 567, 378, 168, 1680, 168, 1848, 264, 264, 264, 2640, 240, 288, 288, 3456, 576, 7488, 1152, 4032, 4032, 2016, 1872, 28080, 9360, 6240, 3360, 6720, 3584, 28672, 6144

Offset: 1

Gus Wiseman, Jul 06 2019

nonn

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

			The a(1) = 1 through a(12) = 20 subsets (A = 10, B = 11, C = 12) are the following. The common cardinality of sets in column n is A085089(n).
  1  12  13  124  145  1246  1467  12468  12689  1248A  1468B  12468C
             134       1346        13468  13689  1289A  148AB  1248AC
                       1456        14568  15689  1348A  1689B  12689C
                                   14678  16789  1389A  189AB  1289AC
                                                 1458A         13468C
                                                 1478A         1348AC
                                                 1589A         13689C
                                                 1789A         1389AC
                                                               14568C
                                                               1458AC
                                                               14678C
                                                               1468BC
                                                               1478AC
                                                               148ABC
                                                               15689C
                                                               1589AC
                                                               16789C
                                                               1689BC
                                                               1789AC
                                                               189ABC

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@Length/@Split[Sort[Array[prisig,n]]]/Count[Array[prisig,n],prisig[n]],{n,30}]

a(n) = A326439(n)/A064839(n).

A085612 Triangle in which the n-th row contains n numbers (beginning with the smallest unused value) that have the same prime signature as the first entry on that row.

Original entry on oeis.org

1, 2, 3, 4, 9, 25, 5, 7, 11, 13, 6, 10, 14, 15, 21, 8, 27, 125, 343, 1331, 2197, 12, 18, 20, 28, 44, 45, 50, 16, 81, 625, 2401, 14641, 28561, 83521, 130321, 17, 19, 23, 29, 31, 37, 41, 43, 47, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 24, 40, 54, 56, 88, 104, 135, 136, 152

Offset: 1

Alford Arnold, Jul 04 2003

Permutation of the natural numbers when the triangle is seen flattened, inverse = A226387, see also A224401. - Reinhard Zumkeller, Jun 05 2013

			a(7) through a(10) = 5, 7, 11, 13 because 5 has not been used.
1;
2,3;
4,9,25;
5,7,11,13;
6,10,14,15,21;
8,27,125,343,1331,2197;
12,18,20,28,44,45,50;
16,81,625,2401,14641,28561,83521,130321;

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, PARI program for A085612
Index entries for sequences that are permutations of the natural numbers

Cf. A025487, A064839, A077558.
Cf. A085836.
Cf. A046523.

import Data.List ((\\))
a085612 n k = a085612_row n !! (k-1)
a085612_row n = a085612_tabf !! (n-1)
a085612_tabf = f 0 $ zip [1..] a046523_list where
   f x zs'@(z:zs) = (map fst ys) : f (x + 1) (zs' \\ ys) where
     ys = z : take x (filter ((== snd z) . snd) zs)
a085612_list = concat a085612_tabf
-- Reinhard Zumkeller, Jun 05 2013

PARI
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See Links section.
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More terms from Ray Chandler, Aug 21 2003

A326439 Number of maximal subsets of {1..n} such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 8, 16, 20, 20, 24, 36, 48, 48, 56, 112, 128, 192, 240, 288, 324, 324, 486, 567, 1134, 1512, 1680, 1680, 1848, 1848, 2112, 2376, 2640, 2640, 2880, 3168, 3456, 6912, 7488, 14976, 16128, 20160, 24192, 26208, 28080, 28080, 37440, 43680

Offset: 0

Gus Wiseman, Jul 06 2019

nonn

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

			The a(0) = 1 through a(9) = 8 subsets:
  {}  {1}  {12}  {12}  {124}  {124}  {1246}  {1246}  {12468}  {12468}
                 {13}  {134}  {134}  {1346}  {1346}  {13468}  {12689}
                              {145}  {1456}  {1456}  {14568}  {13468}
                                             {1467}  {14678}  {13689}
                                                              {14568}
                                                              {14678}
                                                              {15689}
                                                              {16789}

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

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(Length/@Split[Sort[Array[prisig,n]]]),{n,0,30}]

a(n)={if(n==0, 1, my(M=Map()); for(i=1, n, my(f=factor(i)[,2], s=sum(k=1, #f, x^f[k]), z); mapput(M, s, if(mapisdefined(M, s, &z), z + 1, 1))); vecprod(Mat(M)[,2]))} \\ Andrew Howroyd, Aug 30 2019

A325263 Number of subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 2, 2, 6, 4, 16, 8, 40, 40, 60, 36, 216, 72, 168, 168, 840, 240, 960, 360, 1080, 864, 864, 672, 6720, 4480, 2560, 10240, 7680, 3840, 42240, 7680, 92160, 23040, 23040, 23040, 253440, 42240, 49920, 49920, 324480, 74880, 524160, 112320, 336960, 336960, 181440

Offset: 1

Gus Wiseman, Jul 06 2019

nonn

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

			The a(1) = 1 through a(5) = 16 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}        {7}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}      {1,7}
                     {2,4}    {4,5}    {2,6}      {4,7}
                     {3,4}    {1,4,5}  {3,6}      {6,7}
                     {1,2,4}           {4,6}      {1,4,7}
                     {1,3,4}           {5,6}      {1,6,7}
                                       {1,2,6}    {4,6,7}
                                       {1,3,6}    {1,4,6,7}
                                       {1,4,6}
                                       {1,5,6}
                                       {2,4,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {1,2,4,6}
                                       {1,3,4,6}
                                       {1,4,5,6}

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A325365, A326438, A326439, A326441.

prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(1+Length/@Split[Sort[Array[prisig,n]]])/(1+Count[Array[prisig,n],prisig[n]]),{n,30}]

a(n) = A326438(n)/(1 + A064839(n)).

cp's OEIS Frontend

A372133 Prime shadow (A181819) with shifted e_i to generate odd numbers, multiplied by 2^(A064839 - 1).

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A372134 Prime shadows (A181819) linearly transformed to generate odd numbers, multiplied by 2^(A064839 - 1).

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A343133 Numbers k such that A064839(k) = A064839(k+1).

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A101296 n has the a(n)-th distinct prime signature.

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A065036 Product of the cube of a prime (A030078) and a different prime.

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A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

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A325365 Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

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A085612 Triangle in which the n-th row contains n numbers (beginning with the smallest unused value) that have the same prime signature as the first entry on that row.

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