A073093 -id:A073093 - cp's OEIS frontend

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

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}]

A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

Gus Wiseman, May 26 2022

nonn

The run-sums transformation is described by Kimberling at A237685 and A237750.
The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.
Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.

			The terms together with their prime indices begin:
      1: {}            25: {3,3}           64: {1,1,1,1,1,1}
      2: {1}           27: {2,2,2}         67: {19}
      3: {2}           29: {10}            71: {20}
      4: {1,1}         31: {11}            73: {21}
      5: {3}           32: {1,1,1,1,1}     79: {22}
      7: {4}           37: {12}            81: {2,2,2,2}
      8: {1,1,1}       40: {1,1,1,3}       83: {23}
      9: {2,2}         41: {13}            84: {1,1,2,4}
     11: {5}           43: {14}            89: {24}
     12: {1,1,2}       47: {15}            97: {25}
     13: {6}           49: {4,4}          101: {26}
     16: {1,1,1,1}     53: {16}           103: {27}
     17: {7}           59: {17}           107: {28}
     19: {8}           61: {18}           109: {29}
     23: {9}           63: {2,2,4}        112: {1,1,1,1,4}
The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.

This sequence is a subset of A300273, counted by A275870.
The version for compositions is A353857, counted by A353847.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A325268 counts partitions by omicron, rank statistic A304465.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
A353853-A353859 pertain to composition run-sum trajectory.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.

ope[n_]:=Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k];
Select[Range[100],#==1||PrimeQ[NestWhile[ope,#,!SquareFreeQ[#]&]]&]

A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 1, 0, -1, 0, -1, 2, 0, -1, 0, -1, 2, -1, 0, -1, 1, 0, -1, 2, 0, -1, 0, -1, 4, -3, 0, -1, 0, -1, 2, 0, -1, 2, 0, -1, 3, -3, 1, 0, -1, 0, -1, 4, -3, 0, -1, 0, -1, 4, -3, 0, -1, 2, 0, -1, 2, 0, -1, 0, -1, 6, -9, 4, 0, -1, 1, 0, -1, 2, 0, -1, 2, -1, 0, -1, 4, -3, 0, -1, 0, -1, 6, -6, 0, -1, 0, -1, 4, -6, 4, -1, 0, -1, 2, 0, -1, 2, 0, -1

Offset: 1

Leroy Quet, Dec 15 2006

Row n has A001222(n)+1 terms. The polynomial P_n(y) = (sum{m=1 to A001222(n)+1} a(n,m)*y^m) is a generalization of the Mobius (Moebius) function, where P_n(1) = A008683(n).
From Gus Wiseman, Aug 24 2020: (Start)
Up to sign, also the number of strict length-k chains of divisors from n to 1, 1 <= k <= 1 + A001222(n). For example, row n = 36 counts the following chains (empty column indicated by dot):
  .  36/1  36/2/1   36/4/2/1   36/12/4/2/1
           36/3/1   36/6/2/1   36/12/6/2/1
           36/4/1   36/6/3/1   36/12/6/3/1
           36/6/1   36/9/3/1   36/18/6/2/1
           36/9/1   36/12/2/1  36/18/6/3/1
           36/12/1  36/12/3/1  36/18/9/3/1
           36/18/1  36/12/4/1
                    36/12/6/1
                    36/18/2/1
                    36/18/3/1
                    36/18/6/1
                    36/18/9/1
(End)

			1/(zeta(x) - 1 + 1/y) = y - y^2/2^x - y^2/3^x + ( - y^2 + y^3)/4^x - y^2/5^x + ( - y^2 + 2y^3)/6^x - y^2/7^x + ...
From _Gus Wiseman_, Aug 24 2020: (Start)
The sequence of rows begins:
     1: 1              16: 0 -1 3 -3 1     31: 0 -1
     2: 0 -1           17: 0 -1            32: 0 -1 4 -6 4 -1
     3: 0 -1           18: 0 -1 4 -3       33: 0 -1 2
     4: 0 -1 1         19: 0 -1            34: 0 -1 2
     5: 0 -1           20: 0 -1 4 -3       35: 0 -1 2
     6: 0 -1 2         21: 0 -1 2          36: 0 -1 7 -12 6
     7: 0 -1           22: 0 -1 2          37: 0 -1
     8: 0 -1 2 -1      23: 0 -1            38: 0 -1 2
     9: 0 -1 1         24: 0 -1 6 -9 4     39: 0 -1 2
    10: 0 -1 2         25: 0 -1 1          40: 0 -1 6 -9 4
    11: 0 -1           26: 0 -1 2          41: 0 -1
    12: 0 -1 4 -3      27: 0 -1 2 -1       42: 0 -1 6 -6
    13: 0 -1           28: 0 -1 4 -3       43: 0 -1
    14: 0 -1 2         29: 0 -1            44: 0 -1 4 -3
    15: 0 -1 2         30: 0 -1 6 -6       45: 0 -1 4 -3
(End)

Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.

A008480 gives rows ends (up to sign).
A008683 gives row sums (the Moebius function).
A073093 gives row lengths.
A074206 gives unsigned row sums.
A097805 is the restriction to powers of 2 (up to sign).
A251683 is the unsigned version with zeros removed.
A334996 is the unsigned version (except with a(1) = 0).
A334997 is an unsigned non-strict version.
A337107 is the restriction to factorial numbers.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A337105 counts strict chains of divisors from n! to 1.
Cf. A000005, A001221, A002033, A124010, A167865.

f[l_List] := Block[{n = Length[l] + 1, c},c = Plus @@ Last /@ FactorInteger[n];Append[l, Prepend[ -Plus @@ Pick[PadRight[ #, c] & /@ l, Mod[n, Range[n - 1]], 0],0]]];Nest[f, {{1}}, 34] // Flatten(* Ray Chandler, Feb 13 2007 *)
chnsc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]];
Table[(-1)^k*Length[Select[chnsc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* Gus Wiseman, Aug 24 2020 *)

a(1,1)=1. a(n,1) = 0 for n>=2. a(n,m+1) = -sum{k|n,k < n} a(k,m), where, for the purpose of this sum, a(k,m) = 0 if m > A001222(k)+1.

Extended by Ray Chandler, Feb 13 2007

A336865 Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 06 2020

Row lengths are A073093(n) = A001222(n) + 1.

			The triangle begins as follows. The n-th row is shown to the right of "n:".
     1: (1)          16: (1,1,1,1,1)    31: (1,1)
     2: (1,1)        17: (1,1)          32: (1,1,1,1,1,1)
     3: (1,1)        18: (1,2,1,1)      33: (1,2,0)
     4: (1,1,1)      19: (1,1)          34: (1,2,0)
     5: (1,1)        20: (1,2,1,1)      35: (1,2,0)
     6: (1,2,0)      21: (1,2,0)        36: (1,2,2,2,0)
     7: (1,1)        22: (1,2,0)        37: (1,1)
     8: (1,1,1,1)    23: (1,1)          38: (1,2,0)
     9: (1,1,1)      24: (1,2,1,2,1)    39: (1,2,0)
    10: (1,2,0)      25: (1,1,1)        40: (1,2,1,2,1)
    11: (1,1)        26: (1,2,0)        41: (1,1)
    12: (1,2,1,1)    27: (1,1,1,1)      42: (1,3,0,0)
    13: (1,1)        28: (1,2,1,1)      43: (1,1)
    14: (1,2,0)      29: (1,1)          44: (1,2,1,1)
    15: (1,2,0)      30: (1,3,0,0)      45: (1,2,1,1)
Row n = 72 counts the following divisors:
  1  2  4   8  24  72
     3  9  12
           18
Row n = 1200 counts the following divisors:
  1  2   4   8  16   48  400  1200
     3  25  12  24   80  600
     5      20  40  200
            50
            75

A073093 gives row lengths.
A130092 gives positions of rows ending with 0.
A146291 is the version not requiring distinct prime multiplicities.
A181796 gives row sums.
A336499 is the restriction to factorial numbers.
A001222 counts prime factors, counting multiplicity.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336423 counts chains using A130091.
Cf. A000005, A001221, A098859, A118914, A124010, A336422, A336500, A336571.

Table[Length[Select[Divisors[n],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,20},{k,0,PrimeOmega[n]}]

A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

Original entry on oeis.org

12, 24, 36, 40, 48, 60, 63, 72, 80, 84, 96, 108, 112, 120, 126, 132, 144, 156, 160, 168, 180, 189, 192, 200, 204, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 320, 324, 325, 336, 348, 351, 352, 360, 372, 378, 384, 396, 400, 408, 420, 432, 440

Offset: 1

Gus Wiseman, Jun 15 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 term rucksack is short for run-knapsack.

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.

Knapsack partitions are counted by A108917, ranked by A299702.
Non-knapsack partitions are ranked by A299729.
The non-partial version is A353839, complement A353838 (counted by A353837).
The complement is A353866, counted by A353864.
The complete complement is A353867, counted by A353865.
The complement for compositions is counted by A354580.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A333223 ranks knapsack compositions, counted by A325676.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353861 counts distinct partial run-sums of prime indices.
A354584 lists run-sums of prime indices, rows ranked by A353832.
Cf. A005811, A118914, A124010, A175413, A181819, A182857, A316413, A325862, A353834, A353835, A353836, A353931.

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

A075795 Number of k, 0

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 5, 4, 6, 7, 9, 8, 11, 11, 12, 11, 15, 14, 17, 16, 18, 19, 21, 19, 22, 23, 23, 24, 27, 26, 29, 26, 30, 31, 32, 31, 35, 35, 36, 35, 39, 38, 41, 40, 41, 43, 45, 42, 46, 46, 48, 48, 51, 49, 52, 51, 54, 55, 57, 55, 59, 59, 59, 57, 62, 62, 65, 64, 66, 66, 69, 66, 71

Offset: 1

Benoit Cloitre, Oct 13 2002

nonn

a(n) >= A000010(n)-1 since if 2<=kRobert Israel, Jul 24 2016

For n>1 a(n) = number of roots of the n-th polynomial in A275345, equal to 1. - Mats Granvik, Jul 24 2016

Robert Israel, Table of n, a(n) for n = 1..10000
T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.
T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.
Eric Weisstein's World of Mathematics, Cyclotomic polynomials.

Cf. A001222, A054372, A073093.

seq(n -numtheory:-bigomega(n)-1, n=1..1000); # Robert Israel, Jul 25 2016

Table[n - PrimeOmega@ n - 1, {n, 73}] (* Michael De Vlieger, Jul 26 2016 *)

a(n)=sum(k=1,n,if(1-polresultant(polcyclo(n),polcyclo(k)),0,1))

a(n) = n - A073093(n).

a(n) = n - A001222(n) - 1. - Michel Marcus, Jul 24 2016

a(30)=2 and a(31)=6 merged into a(30)=26 by Mats Granvik, Jul 24 2016

A144549 Triangular numbers p*(p+1)/2 with p prime such that 1+(number of prime factors of p+1) is prime.

Original entry on oeis.org

3, 6, 15, 91, 276, 703, 1431, 1770, 1891, 2701, 3486, 4005, 5356, 8646, 9730, 11175, 11476, 12403, 18721, 19503, 24976, 25878, 27261, 28680, 38503, 43071, 47278, 49141, 60378, 61075, 64620, 72010, 75855, 79003, 88831, 98346, 104653, 106491

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2008

nonn

Triangular numbers n(n+1)/2 such that n and A073093(n+1) are both prime.

			3 has one prime factor; 1+1 = 2 is prime, hence 2*3/2 = 3 is in the sequence.
14 = 2*7 has two prime factors; 1+2 = 3 is prime, hence 13*14/2 = 91 is in the sequence.
24 = 2*2*2*3 has four prime factors; 1+4 = 5 is prime, hence 23*24/2 = 276 is in the sequence.

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

Cf. A000217 (triangular numbers), A000040 (prime numbers), A001222 (number of prime divisors of n), A073093.

Subsequence of A034953. - R. J. Mathar, Jan 03 2009

[ p*(p+1)/2: p in PrimesUpTo(490) | IsPrime(1 + &+[ f[2]: f in Factorization(p+1) ]) ];

aQ[n_] := PrimeQ[n] && PrimeQ[PrimeOmega[n + 1] + 1]; p = Select[Range[470], aQ]; p*(p + 1)/2 (* Amiram Eldar, Aug 31 2019 *)

Edited, corrected (3 inserted) and extended beyond a(16) by Klaus Brockhaus, Jan 05 2009

3 inserted and extended by R. J. Mathar, Jan 03 2009

A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

Original entry on oeis.org

0, 1, 2, 5, 8, 3, 16, 37, 66, 9, 128, 7, 256, 17, 10, 549, 1024, 67, 2048, 13, 18, 129, 4096, 39, 8200, 257, 16450, 21, 32768, 11, 65536, 131621, 130, 1025, 24, 71, 262144, 2049, 258, 45, 524288, 19, 1048576, 133, 74, 4097, 2097152, 551, 4194320, 8201

Offset: 1

Franklin T. Adams-Watters, Mar 06 2015

nonn

			The prime power divisors of 12 are 2, 3, and 4. These are indices 1, 2, and 3 in the list of prime powers, so a(12) = 2^(1-1) + 2^(2-1) + 2^(3-1) = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A246655, A065515, A034729, A000961.

Cf. A095874, A210208, A073093, A000079.

a248906 = sum . map ((2 ^) . subtract 2 . a095874) . tail . a210208_row
-- Reinhard Zumkeller, Mar 07 2015

al(n) = my(r=vector(n),pps=[p| p <- [1..n], isprimepower(p)],p2); for(k=1,#pps,p2=2^(k-1);forstep(j=pps[k],n,pps[k],r[j]+=p2));r

Additive with a(p^k) = Sum_{j=1..k} 2^(A065515(p^j)-1).
a(A051451(k)) = 2^k - 1.
a(n) = Sum_{k=1..A073093(n)} 2^(A095874(A210208(n,k))-2). - Reinhard Zumkeller, Mar 07 2015

A378309 a(n) = A010051(n) + A001222(n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 22 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001222, A010051, A073093.

First differences of A113903.

Array[Boole[PrimeQ[#]] + PrimeOmega[#] &, 100]

a(n) = isprime(n) + bigomega(n); \\ Michel Marcus, Nov 23 2024

a(n) = A113903(n) - A113903(n-1).
a(n) = 2 for prime n.
a(2*k) = A073093(k), for k >= 2.

A191278 Count of Mosaic numbers that equal n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 6, 1, 3, 3, 1, 1, 6, 1, 6, 3, 3, 1, 10, 1, 3, 1, 6, 1, 16, 1, 1, 3, 3, 3, 20, 1, 3, 3, 10, 1, 16, 1, 6, 6, 3, 1, 15, 1, 6, 3, 6, 1, 10, 3, 10, 3, 3, 1, 50, 1, 3, 6, 1, 3, 16, 1, 6, 3, 16, 1, 50, 1, 3, 6, 6, 3, 16, 1, 15, 1, 3, 1, 50, 3, 3, 3, 10, 1, 50

Offset: 1

R. J. Mathar, May 29 2011

The number of solutions x to A000026(x)=n.

R. J. Mathar, Table of n, a(n) for n = 1..1000

A191278 := proc(n)
    local f, beta, a, j ;
    f := ifactors(n)[2] ;
    beta := A073093(n) ;
    a := 1/beta ;
    for j in ifactors(n)[2] do
        a := a*binomial(beta, op(2, j) ) ;
    end do:
    a ;
end proc:

Let n=product_j p_j^e(j) be the prime factorization of n and beta=A073093(n). Then a(n)*beta = product_j binomial(beta,e(j)). [Gordon-Robertson in A000026, Theorem 1]

cp's OEIS Frontend

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

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A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

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A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

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A336865 Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

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A075795 Number of k, 0

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A144549 Triangular numbers p*(p+1)/2 with p prime such that 1+(number of prime factors of p+1) is prime.

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A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

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