A316428 -id:A316428 - cp's OEIS frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

In other words, partitions whose average is an integer.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of partitions whose length divides their sum begins (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (1111), (7), (8), (42), (51), (9), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1327 terms from R. J. Mathar)

Cf. A056239, A067538, A074761, A143773, A237984, A289508, A289509, A290103, A296150, A298423, A316428, A316431.

isA326413 := proc(n)
    psigsu := A056239(n) ;
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326413(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019
# second Maple program:
q:= n-> (l-> nops(l)>0 and irem(add(i, i=l), nops(l))=0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..110])[];  # Alois P. Heinz, Nov 19 2021

Select[Range[2,100],Divisible[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]],PrimeOmega[#]]&]

A326568 Denominator of the average of the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Jul 13 2019

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 12 are {1,1,2}, with average 4/3, so a(12) = 3.

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences related to prime indices in the factorization of n.

a(n) is a divisor of Omega(n) = A001222(n).
Positions of 1's are A316413.
Cf. A001414, A056239, A067629, A112798, A123528/A123529, A289508, A289509, A290103, A316428, A326567, A326620.

Table[Denominator[Sum[q[[2]]*PrimePi[q[[1]]],{q,FactorInteger[n]}]/PrimeOmega[n]],{n,2,100}]

A326568(n) = { my(f=factor(n)); denominator(sum(i=1,#f~,f[i,2]*primepi(f[i,1]))/bigomega(n)); }; \\ Antti Karttunen, Jan 28 2025

Starting offset corrected from 0 to 2 and data section extended to a(108) by Antti Karttunen, Jan 28 2025

A078174 Numbers with an integer arithmetic mean of distinct prime factors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 111, 113, 114, 115

Offset: 1

Reinhard Zumkeller, Nov 20 2002

nonn

A008472(a(n)) == 0 modulo A001221(a(n)).

			42=2*3*7: (2+3+7)/3=4, therefore 42 is a term.

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

Union of A246655 and A070005.
Positions of 1's in A323172.
The version counting multiplicity is A078175.
The version for prime indices is A326621.
The average of the set of distinct prime factors is A323171/A323172.
The average of the multiset of prime factors is A123528/A123529.
Cf. A001221, A001414, A067629, A112798, A316413, A316428, A326567/A326568, A328966.

a078174 n = a078174_list !! (n-1)
a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
-- Reinhard Zumkeller, Jun 01 2013

Select[Range[2,200],IntegerQ[Mean[Transpose[FactorInteger[#]][[1]]]]&] (* Harvey P. Dale, Apr 18 2016 *)

is(n)=my(f=factor(n)[,1]);sum(i=1,#f,f[i])%#f==0 \\ Charles R Greathouse IV, May 30 2013

a(n) << n log n/(log log n)^k for any k. - Charles R Greathouse IV, May 30 2013

A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

Original entry on oeis.org

1, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 208, 243, 252, 264, 270, 280, 300, 544, 624, 729, 756, 784, 792, 810, 840, 880, 900, 1000, 1216, 1632, 1872, 2080, 2187, 2268, 2352, 2376, 2430, 2464, 2520, 2640, 2700, 2800, 2944, 3000, 3648, 4896, 5440, 5616

Offset: 1

Gus Wiseman, Jan 09 2021

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.

Also Heinz numbers of integer partitions whose sum is twice their length, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Like partitions in general (A000041), these are also counted by A000041.

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     27: {2,2,2}
     28: {1,1,4}
     30: {1,2,3}
     81: {2,2,2,2}
     84: {1,1,2,4}
     88: {1,1,1,5}
     90: {1,2,2,3}
    100: {1,1,3,3}
    208: {1,1,1,1,6}
    243: {2,2,2,2,2}
    252: {1,1,2,2,4}

Partitions of 2n into n parts are counted by A000041.
The number of prime indices alone is A001222.
The sum of prime indices alone is A056239.
Allowing sum to be any multiple of length gives A067538, ranked by A316413.
A000569 counts graphical partitions, ranked by A320922.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.
Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]

All terms satisfy A056239(a(n)) = 2*A001222(a(n)).

A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 52, 53, 56, 57, 58, 59, 61, 65, 67, 71, 73, 74, 75, 78, 79, 83, 84, 86, 87, 89, 91, 92, 95, 97, 101, 103, 106, 107, 109, 111, 113, 117, 122, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 27 2021

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 terms together with their prime indices begins:
     2: {1}        29: {10}       56: {1,1,1,4}
     3: {2}        30: {1,2,3}    57: {2,8}
     5: {3}        31: {11}       58: {1,10}
     6: {1,2}      35: {3,4}      59: {17}
     7: {4}        37: {12}       61: {18}
     9: {2,2}      38: {1,8}      65: {3,6}
    11: {5}        39: {2,6}      67: {19}
    13: {6}        41: {13}       71: {20}
    14: {1,4}      43: {14}       73: {21}
    17: {7}        45: {2,2,3}    74: {1,12}
    19: {8}        47: {15}       75: {2,3,3}
    20: {1,1,3}    49: {4,4}      78: {1,2,6}
    21: {2,4}      50: {1,3,3}    79: {22}
    23: {9}        52: {1,1,6}    83: {23}
    26: {1,6}      53: {16}       84: {1,1,2,4}

Robert Israel, Table of n, a(n) for n = 1..10000

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
The case where all parts are multiples, not just the maximum part, is A143773 (A316428), with strict case A340830, while the case of factorizations is A340853.
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340609.
The squarefree case is A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A244990/A244991, A326837, A326849 (A326848), A340653, A340691, A340693 (A340606), A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  g mod m = 0;
end proc:
select(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A001222(a(n)) divides A061395(a(n)).

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

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 terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A344415 Numbers whose greatest prime index is half their sum of prime indices.

Original entry on oeis.org

4, 9, 12, 25, 30, 40, 49, 63, 70, 84, 112, 121, 154, 165, 169, 198, 220, 264, 273, 286, 289, 325, 351, 352, 361, 364, 390, 442, 468, 520, 529, 561, 595, 624, 646, 714, 741, 748, 765, 832, 841, 850, 874, 918, 931, 952, 961, 988, 1020, 1045, 1173, 1197, 1224

Offset: 1

Gus Wiseman, May 19 2021

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 terms together with their prime indices begins:
       4: {1,1}           198: {1,2,2,5}
       9: {2,2}           220: {1,1,3,5}
      12: {1,1,2}         264: {1,1,1,2,5}
      25: {3,3}           273: {2,4,6}
      30: {1,2,3}         286: {1,5,6}
      40: {1,1,1,3}       289: {7,7}
      49: {4,4}           325: {3,3,6}
      63: {2,2,4}         351: {2,2,2,6}
      70: {1,3,4}         352: {1,1,1,1,1,5}
      84: {1,1,2,4}       361: {8,8}
     112: {1,1,1,1,4}     364: {1,1,4,6}
     121: {5,5}           390: {1,2,3,6}
     154: {1,4,5}         442: {1,6,7}
     165: {2,3,5}         468: {1,1,2,2,6}
     169: {6,6}           520: {1,1,1,3,6}

The partitions with these Heinz numbers are counted by A035363.
The conjugate version is A340387.
This sequence is the case of equality in A344414 and A344416.
A001222 counts prime factors with multiplicity.
A025065 counts palindromic partitions, ranked by A265640.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A322109 ranks partitions of n with no part > n/2, counted by A110618.
A334201 adds up all prime indices except the greatest.
A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.
A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.
Cf. A000070, A001414, A209816, A301988, A316413, A316428, A320924, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[primeMS[#]]==Total[primeMS[#]]/2&]

A061395(a(n)) = A056239(a(n))/2.

A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, May 19 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     2: {1}        20: {1,1,3}    39: {2,6}
     3: {2}        21: {2,4}      40: {1,1,1,3}
     4: {1,1}      22: {1,5}      41: {13}
     5: {3}        23: {9}        42: {1,2,4}
     6: {1,2}      25: {3,3}      43: {14}
     7: {4}        26: {1,6}      44: {1,1,5}
     9: {2,2}      28: {1,1,4}    46: {1,9}
    10: {1,3}      29: {10}       47: {15}
    11: {5}        30: {1,2,3}    49: {4,4}
    12: {1,1,2}    31: {11}       51: {2,7}
    13: {6}        33: {2,5}      52: {1,1,6}
    14: {1,4}      34: {1,7}      53: {16}
    15: {2,3}      35: {3,4}      55: {3,5}
    17: {7}        37: {12}       56: {1,1,1,4}
    19: {8}        38: {1,8}      57: {2,8}
For example, 56 has prime indices {1,1,1,4} and 7 <= 2*4, so 56 is in the sequence. On the other hand, 224 has prime indices {1,1,1,1,1,4} and 9 > 2*4, so 224 is not in the sequence.

These partitions are counted by A025065 but are different from palindromic partitions, which have Heinz numbers A265640.
The opposite even-weight version appears to be A320924, counted by A209816.
The opposite version appears to be A322109, counted by A110618.
The case of equality in the conjugate version is A340387.
The conjugate opposite version is A344291, counted by A110618.
The conjugate opposite 5-smooth case is A344293, counted by A266755.
The conjugate version is A344296, also counted by A025065.
The case of equality is A344415.
The even-weight case is A344416.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A330950, A344294, A344297.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[primeMS[#]]>=Total[primeMS[#]]/2&]

A056239(a(n)) <= 2*A061395(a(n)).

A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227, 229, 233, 239, 241, 251

Offset: 1

Gus Wiseman, May 15 2022

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   36: {1,1,2,2}

Robert Israel, Table of n, a(n) for n = 1..10000

The first term that is not a prime power A000961 is 36.
The first term that is not a prime or a perfect power A001597 is 1260. - Corrected by Robert Israel, Mar 10 2025
The non-recursive version is A325755, counted by A325702.
Removing all primes gives A353389.
These partitions are counted by A353426.
The version for compositions is A353431.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with all distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

pshadow:= proc(n) local F,i;
  F:= ifactors(n)[2];
  mul(ithprime(i),i=F[..,2])
end proc:
filter:= proc(n) local s;
  if isprime(n) then return true fi;
  s:= pshadow(n);
  n mod s = 0 and member(s,R)
end proc:
R:= {}:
for i from 2 to 2000 do if filter(i) then R:= R union {i} fi od:
sort(convert(R,list)); # Robert Israel, Mar 10 2025

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,200],suQ[#]&]

Equals A353389 U A000040.

A353394 Product of prime shadows of prime indices of n (with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

Offset: 1

Gus Wiseman, May 17 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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.

Positions of first appearances are A353397.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, with an inverse A181821.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Times@@red/@primeMS[n],{n,100}]

a(n) = Product_i A181819(A112798(n,i)).
Positions where a(n) = A003963(n) are A003586.
Positions where a(n) = A005361(n) are A353399, counted by A353398.
Positions where a(n) = A181819(n) are A353395, counted by A353396.

cp's OEIS Frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

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Maple

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A326568 Denominator of the average of the multiset of prime indices of n.

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A078174 Numbers with an integer arithmetic mean of distinct prime factors.

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A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

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A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

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Maple

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A348551 Heinz numbers of integer partitions whose mean is not an integer.

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Maple

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A344415 Numbers whose greatest prime index is half their sum of prime indices.

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A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.