A325036 -id:A325036 - cp's OEIS frontend

A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

2, 1, 21, 25, 39, 35, 57, 55, 49, 65, 75, 77, 129, 95, 91, 105, 183, 119, 125, 143, 133, 185, 147, 161, 169, 195, 175, 209, 339, 217, 255, 253, 259, 305, 247, 285, 273, 245, 301, 299, 345, 323, 325, 357, 371, 435, 669, 391, 361, 403, 399, 473, 343, 469, 481

Offset: 0

Gus Wiseman, Jan 21 2025

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 sum and product of prime indices are A056239 and A003963 respectively.

			The least number whose prime indices satisfy (product) - (sum) = 3 is 25 (prime indices {3,3}), so a(3) = 25.

Position of first appearance of n in A325036.
For sum instead of difference we have A379682, firsts of A379681.
A000040 lists the primes, differences A001223.
A003963 multiplies together prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
The subtraction A325036 takes the following values:
- zero: A301987, counted by A001055 (strict A045778).
- negative: A325037, counted by A114324, see A318029
- positive: A325038, counted by A096276 shifted right
- negative one: A325041, counted by A028422
- one: A325042, counted by A001055 shifted right
- nonnegative: A325044, counted by A096276
- nonpositive: A379721, counted by A319005
Cf. A000720, A178503, A175508, A318950, A319000, A325034, A326151, A326153/A326154, A379319, A379722.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pp=Table[Total[prix[n]]-Times@@prix[n],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[pp,-i][[1,1]],{i,0,mnrm[-DeleteCases[pp,0|_?Positive]]}]

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

A014689 a(n) = prime(n)-n, the number of nonprimes less than prime(n).

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 10, 11, 14, 19, 20, 25, 28, 29, 32, 37, 42, 43, 48, 51, 52, 57, 60, 65, 72, 75, 76, 79, 80, 83, 96, 99, 104, 105, 114, 115, 120, 125, 128, 133, 138, 139, 148, 149, 152, 153, 164, 175, 178, 179, 182, 187, 188, 197, 202, 207, 212, 213, 218, 221, 222

Offset: 1

Mohammad K. Azarian

a(n) = A048864(A000040(n)) = number of nonprimes in RRS of n-th prime. - Labos Elemer, Oct 10 2002
A000040 - A014689 = A000027; in other words, the sequence of natural numbers subtracted from the prime sequence produces A014689. - Enoch Haga, May 25 2009
a(n) = A000040(n) - n. a(n) = inverse (frequency distribution) sequence of A073425(n), i.e., number of terms of sequence A073425(n) less than n. a(n) = A065890(n) + 1, for n >= 1. a(n) - 1 = A065890(n) = the number of composite numbers, i.e., (A002808) less than n-th primes, (i.e., < A000040(n)). - Jaroslav Krizek, Jun 27 2009
a(n) = A162177(n+1) + 1, for n >= 1. a(n) - 1 = A162177(n+1) = the number of composite numbers, i.e., (A002808) less than (n+1)-th number of set {1, primes}, (i.e., < A008578(n+1)). - Jaroslav Krizek, Jun 28 2009
Conjecture: Each residue class contains infinitely many terms of this sequence. Similarly, for any integers m > 0 and r, we have prime(n) + n == r (mod m) for infinitely many positive integers n. - Zhi-Wei Sun, Nov 25 2013
First differences are A046933 = differences minus one between successive primes. - Gus Wiseman, Jan 18 2020

T. D. Noe, Table of n, a(n) for n=1..1000

Equals A014692 - 1.
Cf. A000040, A033286, A158611, A002808, A065890.
Cf. A232463, A232443.
The sum of prime factors of n is A001414(n).
The sum of prime indices of n is A056239(n).
Their difference is A331415(n).
Cf. A000720, A046933, A056239, A318995, A325036, A331380, A331416, A331418.

a014689 n = a000040 n - fromIntegral n
-- Reinhard Zumkeller, Apr 09 2012

[NthPrime(n)-n: n in [1..70]]; // Vincenzo Librandi, Mar 20 2013

Table[Prime[n] - n, {n, 61}] (* Alonso del Arte *)

a(n) = prime(n)-n \\ Charles R Greathouse IV, Sep 05 2011

from sympy import prime
def A014689(n): return prime(n)-n # Chai Wah Wu, Oct 11 2024

G.f: b(x) - x/((1-x)^2), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016

More terms from Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

Correction for Aug 2009 change of offset in A158611 and A008578 by Jaroslav Krizek, Jan 27 2010

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

Original entry on oeis.org

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).

A379722 Numbers whose prime indices do not have the same sum as product.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Gus Wiseman, Jan 08 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 A379736.
The complement is A301987, counted by A001055.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

Nonzeros of 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.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
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 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
- different: A379736, ranks A379722 (this)
Cf. A000720, A003963, A025147, A075254, A111133, A178503, A319000, A325041.

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

A379736 Number of integer partitions of n whose product of parts is not n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 9, 14, 19, 28, 40, 55, 73, 100, 133, 174, 226, 296, 381, 489, 623, 790, 1000, 1254, 1568, 1956, 2434, 3007, 3714, 4564, 5599, 6841, 8342, 10141, 12308, 14881, 17968, 21636, 26013, 31183, 37331, 44582, 53169, 63260, 75171, 89130, 105556

Offset: 0

Gus Wiseman, Jan 07 2025

nonn

These partitions are ranked by A379722, complement A301987.

			The a(2) = 1 through a(7) = 14 partitions:
  (11)  (21)   (31)    (32)     (33)      (43)
        (111)  (211)   (41)     (42)      (52)
               (1111)  (221)    (51)      (61)
                       (311)    (222)     (322)
                       (2111)   (411)     (331)
                       (11111)  (2211)    (421)
                                (3111)    (511)
                                (21111)   (2221)
                                (111111)  (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

The complement is counted by A001055.
The strict case is A111133 (except first term).
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1, see A379734, strict A379735.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736 (this), ranks A379722
Cf. A028422, A069016, A319000, A319916, A325036, A325041, A326152, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],Times@@#!=n&]],{n,0,30}]

a(n) = A000041(n) - A001055(n).

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)

A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite multisets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  1   1   0   0   0   0   0   0   0   0   0   0
   n=3:  1   1   1   0   0   0   0   0   0   0   0   0
   n=4:  1   1   1   2   0   0   0   0   0   0   0   0
   n=5:  1   1   1   2   1   1   0   0   0   0   0   0
   n=6:  1   1   1   2   1   2   0   2   1   0   0   0
   n=7:  1   1   1   2   1   2   1   2   1   1   0   2
   n=8:  1   1   1   2   1   2   1   3   1   1   0   3
   n=9:  1   1   1   2   1   2   1   3   2   1   0   3
  n=10:  1   1   1   2   1   2   1   3   2   2   0   3
  n=11:  1   1   1   2   1   2   1   3   2   2   1   3
  n=12:  1   1   1   2   1   2   1   3   2   2   1   4
For example, the A(9,12) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 1
   n+k=4: 0 0 1 1
   n+k=5: 0 0 0 1 1
   n+k=6: 0 0 0 1 1 1
   n+k=7: 0 0 0 0 1 1 1
   n+k=8: 0 0 0 0 2 1 1 1
   n+k=9: 0 0 0 0 0 2 1 1 1
  n+k=10: 0 0 0 0 0 1 2 1 1 1
  n+k=11: 0 0 0 0 0 1 1 2 1 1 1
  n+k=12: 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=13: 0 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=14: 0 0 0 0 0 0 2 1 2 1 2 1 1 1
  n+k=15: 0 0 0 0 0 0 1 2 1 2 1 2 1 1 1
  n+k=16: 0 0 0 0 0 0 0 1 3 1 2 1 2 1 1 1
For example, antidiagonal n+k=10 counts the following partitions:
  n=5: (5)
  n=6: (411), (2211)
  n=7: (31111)
  n=8: (2111111)
  n=9: (111111111)
so the 10th antidiagonal is: (0,0,0,0,0,1,2,1,1,1).

Row sums are A000041 = partitions of n, strict A000009, no ones A002865.
Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
Antidiagonal sums are A379667.
The case without ones is A379668, antidiagonal sums A379669 (zeros A379670).
The strict case is A379671, antidiagonal sums A379672.
The strict case without ones is A379678, antidiagonal sums A379679 (zeros A379680).
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A096765, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

cp's OEIS Frontend

A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

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A014689 a(n) = prime(n)-n, the number of nonprimes less than prime(n).

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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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A379722 Numbers whose prime indices do not have the same sum as product.

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A379736 Number of integer partitions of n whose product of parts is not n.

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