A071625 -id:A071625 - cp's OEIS frontend

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 2, 3, 0, 4, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 4, 3, 0, 1, 0, 4, 0, 2, 0, 0, 0, 1, 0, 0, 2, 5, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 3, 6, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 11 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.

			The prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and with distinct prime indices {1,2}, with sum 3, so a(96) = 7 - 3 = 4.

Positions of 0's are A005117, complement A013929.
For length instead of sum we have A046660.
Positions of 1's are A081770.
For factors instead of indices we have A280292, firsts A280286 (sorted A381075).
A multiplicative version is A290106.
Counting partitions by this statistic gives A364916.
Dominates A374248.
Positions of first appearances are A380956, sorted A380957.
For prime multiplicities instead of prime indices we have A380958.
For product instead of sum we have A380986.
A000040 lists the primes, differences A001223.
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, A071625, A075254, A075255, A116861, A124010, A136565, A178503, A325033, A325034, A366528, A366749, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Total[Union[prix[n]]],{n,100}]

a(n) = A056239(n) - A066328(n).

Additive: a(m*n) = a(m) + a(n) if gcd(m,n) = 1.

A113901 Product of omega(n) and bigomega(n) = A001221(n)*A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 6, 1, 4, 4, 4, 1, 6, 1, 6, 4, 4, 1, 8, 2, 4, 3, 6, 1, 9, 1, 5, 4, 4, 4, 8, 1, 4, 4, 8, 1, 9, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 8, 4, 8, 4, 4, 1, 12, 1, 4, 6, 6, 4, 9, 1, 6, 4, 9, 1, 10, 1, 4, 6, 6, 4, 9, 1, 10, 4, 4, 1, 12, 4, 4, 4, 8, 1, 12, 4, 6, 4, 4, 4, 12, 1, 6, 6, 8, 1, 9

Offset: 1

Cino Hilliard, Jan 29 2006

Positions of first appearances are A328964. - Gus Wiseman, Nov 05 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000

A307409(n) is (bigomega(n) - 1) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A066921, A066922, A068993, A070175, A071625, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965.

Table[PrimeNu[n]*PrimeOmega[n], {n,1,50}] (* G. C. Greubel, Apr 23 2017 *)

PARI
```
a(n) = omega(n)*bigomega(n);
```

a(n) = 1 iff n is prime.
A068993(a(n)) = 4. - Reinhard Zumkeller, Mar 13 2011
a(n) = A066921(n)*A066922(n). - Amiram Eldar, May 07 2025

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).

A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 7, 12, 16, 21, 33, 38, 50, 75, 87, 111, 150, 185, 244, 307, 373, 461, 585, 702, 856, 1043, 1255, 1498, 1822, 2143, 2565, 3064, 3607, 4251, 5064, 5920, 6953, 8174, 9503, 11064, 12927, 14921, 17320, 19986, 23067, 26485, 30499, 34894

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1, 2) and three distinct parts (1, 2, 3) so is counted under a(9).

The Heinz numbers of these partitions are given by A325259.

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)      (64)
              (41)  (51)    (52)    (62)     (63)      (73)
                    (2211)  (61)    (71)     (72)      (82)
                            (3211)  (3221)   (81)      (91)
                                    (3311)   (3321)    (3322)
                                    (4211)   (4221)    (4411)
                                    (32111)  (4311)    (5221)
                                             (5211)    (5311)
                                             (32211)   (6211)
                                             (42111)   (32221)
                                             (222111)  (33211)
                                             (321111)  (42211)
                                                       (43111)
                                                       (52111)
                                                       (421111)
                                                       (3211111)

Cf. A008284, A071625, A090858, A098859, A116608, A117571, A127002, A325242, A325259, A325270.

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==Length[Union[Length/@Split[#]]]+1&]],{n,0,30}]

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).

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 60, 8, 2310, 36, 420, 24, 30030, 180, 4620, 120, 510510, 1260, 72, 60060, 16, 900, 840, 9699690, 13860, 360, 1021020, 48, 6300, 9240, 223092870, 180180, 2520, 19399380, 240, 69300, 216, 120120, 6469693230, 1800, 3063060, 144, 44100, 27720, 446185740, 1680, 900900, 1080, 2042040, 200560490130, 12600, 58198140, 32, 720

Offset: 1

Matthew Vandermast, Dec 07 2010

A permutation of the members of A025487.

If integers m and n have conjugate prime signatures, then A001222(m) = A001222(n), A071625(m) = A071625(n), A085082(m) = A085082(n), and A181796(m) = A181796(n).

			A025487(5) = 8 = 2^3 has a prime signature of (3). The partition that is conjugate to (3) is (1,1,1), and the member of A025487 with that prime signature is 30 = 2*3*5 (or 2^1*3^1*5^1).  Therefore, a(5) = 30.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821.

A181825 lists members of A025487 with self-conjugate prime signatures. See also A181823-A181824, A181826-A181827.

f[n_] := Block[{ww, dec}, dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 + Length@ NestWhileList[NextPrime@ # &, 1, Times @@ {##} <= n &, All] ]; {{{0}}}~Join~Map[Block[{w = #, k = 1}, Sort@ Apply[Join, {{ConstantArray[1, Length@ w]}, If[Length@ # == 0, #, #[[1]]] }] &@ Reap[Do[If[# <= n, Sow[w]; k = 1, If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]], {i, Infinity}] ][[-1]] ] &, ww]]; Sort[Map[{Times @@ MapIndexed[Prime[First@ #2]^#1 &, #], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Table[LengthWhile[#1, # >= j &], {j, #2}]] & @@ {#, Max[#]}} &, Join @@ f[2310]]][[All, -1]] (* Michael De Vlieger, Oct 16 2018 *)

partitionConj(v)=vector(v[1],i,sum(j=1,#v,v[j]>=i))
primeSignature(n)=vecsort(factor(n)[,2]~,,4)
f(n)=if(n==1, return(1)); my(e=partitionConj(primeSignature(n))~); factorback(concat(Mat(primes(#e)~),e))
A025487=[2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768];
concat(1, apply(f, A025487)) \\ Charles R Greathouse IV, Jun 02 2016

If A025487(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A025487(n))). a(n) also equals A108951(A181820(n)).

A238747 Row n of table gives prime metasignature of n: count total appearances of each distinct integer that appears in the prime signature of n, then arrange totals in nonincreasing order.

Original entry on oeis.org

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

Offset: 2

Matthew Vandermast, May 08 2014

A prime metasignature is analogous to the signature of a partition (cf. A115621); it is the signature of a prime signature.

Row n also gives prime signature of A181819(n).

			The prime signature of 72 (2^3*3^2) is {3,2}. The numbers 3 and 2 each appear once; therefore, the prime metasignature of 72 is {1,1}.
The prime signature of 120 (2^3*3*5) is {3,1,1}. 3 appears 1 time and 1 appears 2 times; therefore, the prime metasignature of 120 is {2,1}.

Length of row n equals A071625(n); sum of numbers in row n is A001221(n).

Cf. A115621, A181819, A238748.

Row n is identical to row A181819(n) of table A212171.

A307409 a(n) = (A001222(n) - 1)*A001221(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 3, 0, 4, 0, 4, 2, 2, 0, 6, 1, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 1, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 9, 0, 2, 4, 5, 2, 6, 0, 4, 2, 6, 0, 8, 0, 2, 4, 4, 2, 6, 0, 8, 3, 2, 0, 9, 2, 2, 2, 6, 0, 9, 2, 4, 2, 2, 2, 10, 0, 4, 4, 6, 0, 6, 0, 6

Offset: 1

Mats Granvik, Apr 07 2019

nonn

a(n) + 2 appears to differ from A000005 at n=1 and when n is a term of A320632. Verified up to n=3000.
If A320632 contains the numbers such that A001222(n) - A051903(n) > 1, then this sequence contains precisely the numbers p^k and p^k*q for distinct primes p and q. The comment follows, since d(p^k) = k+1 = (k-1)*1 + 2 and d(p^k*q) = 2k+2 = ((k+1)-1)*2 + 2. - Charlie Neder, May 14 2019
Positions of first appearances are A328965. - Gus Wiseman, Nov 05 2019
Regarding Neder's comment above, see also my comments in A322437. - Antti Karttunen, Feb 17 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Two less than A307408.
A113901(n) is bigomega(n) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A070175, A071625, A113901, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965, A322437.

a[n_] := (PrimeOmega[n] - 1)*PrimeNu[n];
aa = Table[a[n], {n, 1, 104}];

a(n) = (bigomega(n) - 1)*omega(n); \\ Michel Marcus, May 15 2019

a(n) = (A001222(n) - 1)*A001221(n).
a(n) = binomial(A001222(n) - 1, 1)*binomial(A001221(n), 1).
a(n) = A307408(n) - 2.

A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

Original entry on oeis.org

4, 9, 8, 25, 16, 49, 32, 81, 64, 121, 128, 169, 256, 625, 512, 289, 1024, 361, 2048, 1444, 1331, 529, 5324, 2116, 2197, 4232, 8788, 841, 17576, 961, 7569, 3844, 4913, 7688, 19652, 1369, 6859, 5476, 12321, 1681, 34225, 1849, 15129, 7396, 12167, 2209, 46225, 8836, 19881

Offset: 2

Michel Marcus, Dec 31 2016

nonn

Michel Marcus, Table of n, a(n) for n = 2..500

Cf. A001414 (sopfr), A008472 (sopf), A001248, A280163.
A multiplicative version is A064549 (sorted A001694), firsts of A003557.
For length instead of sum we have A151821.
These are the positions of first appearances in A280292 = A001414 - A008472.
For indices instead of factors we have A380956 (sorted A380957), firsts of A380955.
A multiplicative version for indices is A380987 (sorted A380988), firsts of A290106.
For prime exponents instead of factors we have A380989, firsts of A380958.
The sorted version is A381075.
For product instead of sum see A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A364916, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,2,mnrm[q/.(0->1)]}] (* Gus Wiseman, Feb 20 2025 *)

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]*f[j,2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]);
a(n) = {my(k = 2); while (sopfr(k) - sopf(k) != n, k++); k;}

For p prime, a(p) = p^2 (see A001248).

cp's OEIS Frontend

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

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A113901 Product of omega(n) and bigomega(n) = A001221(n)*A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.

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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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A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

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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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A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

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A238747 Row n of table gives prime metasignature of n: count total appearances of each distinct integer that appears in the prime signature of n, then arrange totals in nonincreasing order.

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