A175508 -id:A175508 - cp's OEIS frontend

A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

1, 15, 21, 25, 27, 33, 35, 39, 42, 45, 49, 50, 51, 54, 55, 57, 63, 65, 66, 69, 70, 75, 77, 78, 81, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 110, 111, 114, 115, 117, 119, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is greater than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A114324.

			The sequence of terms together with their prime indices begins:
   1: {}
  15: {2,3}
  21: {2,4}
  25: {3,3}
  27: {2,2,2}
  33: {2,5}
  35: {3,4}
  39: {2,6}
  42: {1,2,4}
  45: {2,2,3}
  49: {4,4}
  50: {1,3,3}
  51: {2,7}
  54: {1,2,2,2}
  55: {3,5}
  57: {2,8}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  69: {2,9}
  70: {1,3,4}
  75: {2,3,3}
  77: {4,5}
  78: {1,2,6}
  81: {2,2,2,2}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325038, A325041, A325042, A325044.

q:= n-> (l-> mul(i, i=l)>add(i, i=l))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..200])[];  # Alois P. Heinz, Mar 27 2019

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

A003963(a(n)) > A056239(a(n)).

A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 88, 89, 92, 94, 96, 97, 101

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than or equal to their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  24: {1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042.

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

A003963(a(n)) <= A056239(a(n)).
a(n) = A325038(n)/2.
Union of A301987 and A325038.

A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276 shifted once to the right.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  46: {1,9}
  48: {1,1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325041, A325042, A325044.

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

A003963(a(n)) < A056239(a(n)).

a(n) = 2 * A325044(n).

A379721 Numbers whose prime indices have sum <= product.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103

Offset: 1

Gus Wiseman, Jan 05 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.
Partitions of this type are counted by A319005.
The complement is A325038.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}

The case of equality is A301987, inequality A325037.
Nonpositive positions in A325036.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A379681 gives sum plus product of prime indices, firsts A379682.
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721 (this)
- different: A379736, ranks A379722, see A111133
Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.

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

Number k such that A056239(k) <= A003963(k).

A325036 Difference between product and sum of prime indices of n.

Original entry on oeis.org

1, 0, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -1, 1, -3, 0, -1, 0, -2, 2, -1, 0, -3, 3, -1, 2, -2, 0, 0, 0, -4, 3, -1, 5, -2, 0, -1, 4, -3, 0, 1, 0, -2, 5, -1, 0, -4, 8, 2, 5, -2, 0, 1, 7, -3, 6, -1, 0, -1, 0, -1, 8, -5, 9, 2, 0, -2, 7, 4, 0, -3, 0, -1, 10, -2, 11, 3, 0, -4, 8, -1, 0, 0, 11, -1, 8, -3, 0, 4, 14, -2, 9

Offset: 1

Gus Wiseman, Mar 25 2019

sign

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 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.
Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.
Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@primeMS[n]-Total[primeMS[n]],{n,100}]
dps[n_]:=Module[{pi=Flatten[Table[PrimePi[#[[1]]],#[[2]]]&/@FactorInteger[n]]},Times@@pi-Total[pi]]; Join[{1},Array[dps,100,2]] (* Harvey P. Dale, May 26 2023 *)

A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };
A325036(n) = (A003963(n) - A056239(n)); \\ Antti Karttunen, May 08 2022

a(n) = A003963(n) - A056239(n).

For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - Antti Karttunen, May 08 2022

Data section extended up to a(93) by Antti Karttunen, May 08 2022

A075254 a(n) = n + (sum of prime factors of n taken with repetition).

Original entry on oeis.org

1, 4, 6, 8, 10, 11, 14, 14, 15, 17, 22, 19, 26, 23, 23, 24, 34, 26, 38, 29, 31, 35, 46, 33, 35, 41, 36, 39, 58, 40, 62, 42, 47, 53, 47, 46, 74, 59, 55, 51, 82, 54, 86, 59, 56, 71, 94, 59, 63, 62, 71, 69, 106, 65, 71, 69, 79, 89, 118, 72, 122, 95, 76, 76, 83, 82, 134, 89, 95, 84, 142

Offset: 1

Zak Seidov, Sep 10 2002

a(n) = n + A001414(n).

Product of prime factors plus sum of prime factors of n. For minus instead of plus we have A075255, zeros A175787. - Gus Wiseman, Jan 26 2025

			a(6)=11 because 6=2*3, sopfr(6)=2+3=5 and 6+5=11.

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

A000027 gives product of prime factors, indices A003963.
A000040 lists the primes, differences A001223.
A001414 gives sum of prime factors, indices A056239.
A027746 lists prime factors, indices A112798, count A001222.
A075255 gives product of prime factors minus sum of prime factors.
Cf. A000720, A001055, A001222, A008472, A090340, A096461 (iteration), A175508, A319000, A325036, A325037, A325038, A379681, A379682.

a075254 n = n + a001414 n  -- Reinhard Zumkeller, Feb 27 2012

[n eq 1 select 1 else (&+[p[1]*p[2]: p in Factorization(n)]) + n: n in [1..80]]; // G. C. Greubel, Jan 10 2019

A075254 := proc(n)
    n+A001414(n) ;
end proc: # R. J. Mathar, Jul 27 2015

Table[If[n==1,1, n +Plus@@Times@@@FactorInteger@n], {n, 80}] (* G. C. Greubel, Jan 10 2019 *)

a(n) = my(f = factor(n)); n + sum(k=1, #f~, f[k,1]*f[k,2]); \\ Michel Marcus, Feb 22 2017

[n + sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0, len(factor(n)))) for n in range(1, 80)] # G. C. Greubel, Jan 10 2019

From Gus Wiseman, Jan 26 2025: (Start)
First differences are 1 - A090340(n).
a(n) = 2*n - A075255(n).
a(n) = 2*A001414(n) + A075255(n).
(End)

A325041 Heinz numbers of integer partitions whose product of parts is one greater than their sum.

Original entry on oeis.org

1, 15, 42, 54, 100, 132, 312, 560, 720, 816, 1824, 3520, 4416, 6272, 8064, 10368, 11136, 16640, 23808, 38400, 56832, 78848, 87040, 101376, 125952, 264192, 389120, 577536, 745472, 958464, 1302528, 1720320, 1884160, 1982464, 2211840, 2899968, 5996544

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one more than their sum of prime indices (A056239).

			The sequence of terms together with their prime indices begins:
      1: {}
     15: {2,3}
     42: {1,2,4}
     54: {1,2,2,2}
    100: {1,1,3,3}
    132: {1,1,2,5}
    312: {1,1,1,2,6}
    560: {1,1,1,1,3,4}
    720: {1,1,1,1,2,2,3}
    816: {1,1,1,1,2,7}
   1824: {1,1,1,1,1,2,8}
   3520: {1,1,1,1,1,1,3,5}
   4416: {1,1,1,1,1,1,2,9}
   6272: {1,1,1,1,1,1,1,4,4}
   8064: {1,1,1,1,1,1,1,2,2,4}
  10368: {1,1,1,1,1,1,1,2,2,2,2}
  11136: {1,1,1,1,1,1,1,2,10}
  16640: {1,1,1,1,1,1,1,1,3,6}
  23808: {1,1,1,1,1,1,1,1,2,11}
  38400: {1,1,1,1,1,1,1,1,1,2,3,3}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325042, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],Times@@primeMS[#]==Total[primeMS[#]]+1&]

A003963(a(n)) = A056239(a(n)) + 1.

A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.

Original entry on oeis.org

4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 168, 178, 194, 202, 206, 214, 216, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 400, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
  106: {1,16}
  118: {1,17}
  122: {1,18}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325044.

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

A003963(a(n)) = A056239(a(n)) - 1.

a(n) = 2 * A301987(n).

A352486 Heinz numbers of non-self-conjugate integer partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Mar 20 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 sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.

			The terms together with their prime indices begin:
   3: (2)
   4: (1,1)
   5: (3)
   7: (4)
   8: (1,1,1)
  10: (3,1)
  11: (5)
  12: (2,1,1)
  13: (6)
  14: (4,1)
  15: (3,2)
  16: (1,1,1,1)
  17: (7)
  18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.

The complement is A088902, counted by A000700.
These partitions are counted by A330644.
These are the positions of nonzero terms in A352491.
A000041 counts integer partitions, strict A000009.
A098825 counts permutations by unfixed points.
A238349 counts compositions by fixed points, rank statistic A352512.
A325039 counts partitions w/ same product as conjugate, ranked by A325040.
A352523 counts compositions by unfixed points, rank statistic A352513.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition
- A122111 = rank of conjugate partition
- A296150 = parts of partition, reverse A112798, conjugate A321649
- A352487 = less than conjugate, counted by A000701
- A352488 = greater than or equal to conjugate, counted by A046682
- A352489 = less than or equal to conjugate, counted by A046682
- A352490 = greater than conjugate, counted by A000701
Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524,  A324846, A350841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]

a(n) != A122111(a(n)).

A280292 a(n) = sopfr(n) - sopf(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 2, 0, 0, 0, 6, 0, 3, 0, 2, 0, 0, 0, 4, 5, 0, 6, 2, 0, 0, 0, 8, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 3, 0, 0, 6, 7, 5, 0, 2, 0, 6, 0, 4, 0, 0, 0, 2, 0, 0, 3, 10, 0, 0, 0, 2, 0, 0, 0, 7, 0, 0, 5, 2, 0, 0, 0, 6, 9, 0, 0, 2, 0, 0, 0, 4, 0, 3, 0, 2, 0, 0, 0, 8, 0, 7, 3, 7, 0, 0, 0, 4, 0

Offset: 1

Michel Marcus, Dec 31 2016

nonn

Alladi and Erdős (1977) proved that for all numbers m>=0, m!=1, the sequence of numbers k such that a(k) = m has a positive asymptotic density which is equal to a rational multiple of 1/zeta(2) = 6/Pi^2 (A059956). For example, when m=0, the sequence is the squarefree numbers (A005117), whose density is 6/Pi^2, and when m=2 the sequence is A081770, whose density is 1/Pi^2. - Amiram Eldar, Nov 02 2020

Sum of prime factors minus sum of distinct prime factors. Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916. - Gus Wiseman, Feb 21 2025

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See pp. 164-166.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 165.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Jean-Marie De Koninck, Paul Erdős and Aleksandar Ivić, Reciprocals of certain large additive functions, Canadian Mathematical Bulletin, Vol. 24, No. 2 (1981), pp. 225-231.
Aleksandar Ivić, On certain large additive functions, arXiv:math/0311505 [math.NT], 2003.

Cf. A001414 (sopfr), A008472 (sopf), A057521, A059956, A081770, A280163, A338559, A338560.
A multiplicative version is A003557, firsts A064549 (sorted A001694).
For length instead of sum we have A046660.
For product instead of sum we have A066503, firsts A381076.
Positions of first appearances are A280286 (sorted A381075).
For indices instead of factors we have A380955, firsts A380956 (sorted A380957).
For exponents instead of factors we have A380958, firsts A380989.
A000040 lists the primes, differences A001223.
A001222 counts prime factors (distinct A001221).
A003963 gives product of prime indices, distinct A156061, excess A380986.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A027746 lists prime factors, distinct A027748.
A112798 lists prime indices (sum A056239), distinct A304038 (sum A066328).
Cf. A071625, A075255, A116861, A136565, A175508, A290106, A364916, A381077.

Array[Total@ # - Total@ Union@ # &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 105] (* Michael De Vlieger, Feb 25 2019 *)

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) = sopfr(n) - sopf(n);

a(n) = A001414(n) - A008472(n).
a(A005117(n)) = 0.
a(n) = A001414(A003557(n)). - Antti Karttunen, Oct 07 2017
Additive with a(1) = 0 and a(p^e) = p*(e-1) for prime p and e > 0. - Werner Schulte, Feb 24 2019
From Amiram Eldar, Nov 02 2020: (Start)
a(n) = a(A057521(n)).
Sum_{n<=x} a(n) ~ x*log(log(x)) + O(x) (Alladi and Erdős, 1977).
Sum_{n<=x, n nonsquarefree} 1/a(n) ~ c*x + O(sqrt(x)*log(x)), where c = Integral_{t=0..1} (F(t)-6/Pi^2)/t dt, and F(t) = Product_{p prime} (1-1/p)*(1-1/(t^p - p)) (De Koninck et al., 1981; Finch, 2018), or, equivalently c = Sum_{k>=2} d(k)/k = 0.1039..., where d(k) = (6/Pi^2)*A338559(k)/A338560(k) is the asymptotic density of the numbers m with a(m) = k (Alladi and Erdős, 1977; Ivić, 2003). (End)

More terms from Antti Karttunen, Oct 07 2017

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A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

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A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

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A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

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A379721 Numbers whose prime indices have sum <= product.

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A325036 Difference between product and sum of prime indices of n.

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A075254 a(n) = n + (sum of prime factors of n taken with repetition).

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A325041 Heinz numbers of integer partitions whose product of parts is one greater than their sum.

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