A014689 -id:A014689 - cp's OEIS frontend

A331385 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = n + k.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2, 0, 2, 0, 0, 0, 0, 0, 2, 6, 6, 4, 6, 2, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 4, 8, 6, 6, 7, 2, 4, 2, 0, 1, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jan 17 2020

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 2 1
  0 0 1 3 1
  0 0 0 2 3 1 1
  0 0 0 1 4 3 1 2
  0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
  0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
Row n = 8 counts the following partitions (empty column not shown):
  (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
          (422)    (431)     (311111)   (62)        (5111)  (71)
          (3221)   (3311)    (2111111)  (521)
          (22211)  (4211)               (11111111)
                   (32111)
                   (221111)
Column k = 5 counts the following partitions:
  (11111)  (411)    (43)     (332)    (3222)   (22222)
           (3111)   (331)    (422)    (22221)
           (21111)  (421)    (3221)
                    (3211)   (22211)
                    (22111)

Row lengths are A331418.
Row sums are A000041.
Column sums are A331387.
Shifting row n to the right n times gives A331416.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415.

Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==m&]],{n,0,10},{m,n,Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]}]

A331387 Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 24, 34, 47, 64, 86, 113, 148, 191, 245, 310, 390, 486, 602, 740, 907, 1104, 1338, 1613, 1937, 2315, 2758, 3272, 3871, 4562, 5362, 6283, 7344, 8558, 9952, 11542, 13356, 15419, 17766, 20425, 23440, 26846, 30696, 35032, 39917, 45406

Offset: 0

Gus Wiseman, Jan 17 2020

nonn

Primes of parts means the prime counting function applied to the part sizes. Equivalently, a(n) is the number of integer partitions with part sizes in A014689(n) interpreted as a multiset. - Andrew Howroyd, Apr 17 2021

			The a(0) = 1 through a(5) = 16 partitions:
  ()  (1)  (3)   (4)    (33)    (43)
      (2)  (11)  (31)   (41)    (331)
           (21)  (32)   (42)    (332)
           (22)  (111)  (311)   (411)
                 (211)  (321)   (421)
                 (221)  (322)   (422)
                 (222)  (1111)  (3111)
                        (2111)  (3211)
                        (2211)  (3221)
                        (2221)  (3222)
                        (2222)  (11111)
                                (21111)
                                (22111)
                                (22211)
                                (22221)
                                (22222)
For example, the partition (3,2,2,1) is counted under n = 5 because it has sum of primes 5+3+3+2 = 13 and its sum of parts plus n is also 3+2+2+1+5 = 13.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column sums of A331385.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415, A331416, A331418.

Table[Sum[Length[Select[IntegerPartitions[k],Total[Prime/@#]==k+n&]],{k,0,2*n}],{n,0,10}]

seq(n)={my(m=1); while(prime(m)-m<=n, m++); Vec(1/prod(k=1, m, 1 - x^(prime(k)-k) + O(x*x^n)))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1/Product_{k>=1} 1 - x^(prime(k)-k). - Andrew Howroyd, Apr 16 2021

Terms a(31) and beyond from Andrew Howroyd, Apr 16 2021

A331418 If A331417(n) is the maximum sum of primes of the parts of an integer partition of n, then a(n) = A331417(n) - n + 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52, 53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121, 126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188, 189, 198, 203, 208, 213, 214, 219

Offset: 0

Gus Wiseman, Jan 17 2020

nonn

For n > 4, a(n) = A014692(n).

Converges to A014692.
Row lengths of A331385.
Sum of prime factors is A001414.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Cf. A000040, A014689, A056239, A330950, A330953, A330954, A331378, A331381, A331383, A331387, A331415, A331416.

Table[Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]-n+1,{n,0,30}]

a(n) = A331417(n) - n + 1.

A097935 Number of primes that are not less than prime(n)-n and not greater than prime(n)+n.

Original entry on oeis.org

2, 3, 4, 4, 3, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 8, 8, 9, 8, 9, 9, 10, 10, 12, 10, 10, 10, 11, 11, 12, 13, 13, 12, 13, 12, 12, 14, 16, 15, 15, 14, 15, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 18, 18, 20, 19, 19, 20, 21, 21, 21, 20, 21, 21, 23, 22, 23, 21, 22, 22, 22, 23, 24, 25

Offset: 1

Reinhard Zumkeller, Sep 05 2004

nonn

			a(10) = #{p prime: A000040(10)-10 <= p <= A000040(10)+10} = #{p prime: 19 <= p <= 39} = #{19,23,29,31,37} = 5.

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

Cf. A000720, A014688, A014689.

a[n_] := PrimePi[Prime[n] + n] - PrimePi[Prime[n] - n - 1];
Array[a, 100] (* Jean-François Alcover, Jun 11 2019 *)

a(n) = my(p=prime(n)); primepi(p+n) - primepi(p-n-1); \\ Michel Marcus, Jun 11 2019

[len([k for k in (nth_prime(n)-n..nth_prime(n)+n) if is_prime(k)]) for n in (1..75)]  # Peter Luschny, Sep 03 2013

a(n) = A000720(A014688(n)) - A000720(A014689(n)-1).

A098389 Prime(n) - floor(log_2(prime(n))).

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 13, 15, 19, 25, 27, 32, 36, 38, 42, 48, 54, 56, 61, 65, 67, 73, 77, 83, 91, 95, 97, 101, 103, 107, 121, 124, 130, 132, 142, 144, 150, 156, 160, 166, 172, 174, 184, 186, 190, 192, 204, 216, 220, 222, 226, 232, 234, 244, 249, 255, 261, 263, 269, 273

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

a(n) = A000040(n) - A098388(n).

			a(10) = A000040(10) - A098388(10) = 29 - 4 = 25.

Cf. A000523, A098390, A014689, A098386, A098392.

#-Floor[Log[2,#]]&/@Prime[Range[60]] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=my(p=prime(n));p-log(p+.5)\log(2) \\ Charles R Greathouse IV, May 29 2013

A182174 a(n) = prime(n)^2 - n.

Original entry on oeis.org

3, 7, 22, 45, 116, 163, 282, 353, 520, 831, 950, 1357, 1668, 1835, 2194, 2793, 3464, 3703, 4470, 5021, 5308, 6219, 6866, 7897, 9384, 10175, 10582, 11421, 11852, 12739, 16098, 17129, 18736, 19287, 22166, 22765, 24612, 26531, 27850, 29889, 32000, 32719, 36438, 37205, 38764, 39555, 44474, 49681

Offset: 1

Alonso del Arte, Apr 16 2012

One way to find a run of n consecutive nonsquarefree numbers such that the first n primes appear in order as factors of numbers in the run is to use the Chinese remainder theorem (though this run is most likely not the earliest of length n).

The moduli are then of course the squares of the first n primes, while the remainders are then the first n terms of this sequence. (See A182433.)

			a(4) = 45 because the 4th prime is 7, and 7^2 - 4 = 49 - 4 = 45.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001248 squares of primes; A045882 and A078144 pertain to runs of consecutive nonsquarefree numbers.

Cf. A014689. [Bruno Berselli, Mar 19 2013]

[NthPrime(n)^2-n: n in [1..50]]; // Bruno Berselli, Apr 16 2012

Mathematica
```
Table[Prime[n]^2 - n, {n, 50}]
```

a(n) = A000040(n)^2 - n = A001248(n) - n. - Omar E. Pol, Apr 16 2012

a(36) inserted by Vincenzo Librandi, Mar 19 2013

A232862 Least positive integer m <= n^2/2 + 3 such that the set {prime(k) - k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

Original entry on oeis.org

1, 3, 4, 11, 9, 8, 10, 15, 29, 13, 23, 22, 23, 37, 32, 28, 48, 44, 53, 41, 45, 67, 76, 117, 119, 91, 121, 88, 89, 101, 72, 88, 100, 143, 144, 185, 145, 104, 176, 141, 144, 175, 187, 213, 121, 255, 128, 129, 189, 243, 122, 267, 275, 242, 209, 205, 130, 153, 263, 335

Offset: 1

Zhi-Wei Sun, Dec 01 2013

nonn

Conjecture: a(n) > 0 for all n > 0.

Note that a(4) = 11 = 4^2/2 + 3.

			a(3) = 4 since prime(1) - 1 = prime(2) - 2 = 1, prime(3) - 3 = 2, prime(4) - 4 = 3, and {1,2,3} is a complete system of residues modulo 3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Zhi-Wei Sun)
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A014689, A232616, A232463, A232548, A232861.

  L[m_,n_]:=Length[Union[Table[Mod[Prime[k]-k,n],{k,1,m}]]]
Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2/2+3}];
Print[n," ",0];Label[aa];Continue,{n,1,60}]

A331417 Maximum sum of primes of the parts of an integer partition of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 0

Gus Wiseman, Jan 17 2020

nonn

For n > 4, a(n) = prime(n).

Converges to A000040.
Row lengths of A331416.
The corresponding minimum is A007494.
Partitions whose sum of primes is divisible by their sum are A331379.
Cf. A000607, A001414, A014689, A056239, A179437, A330950, A330953, A330954, A331381, A331383, A331385, A331387, A331415, A331418.

Table[Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}],{n,0,10}]

a(n) = A331418(n) + n - 1.

A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

Original entry on oeis.org

5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.

Positions of 2's in A078147.
For prime numbers we have A029707.
For nonprime numbers we appear to have A014689.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.

Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

Complement of A375709 U A375711 U A375712.

A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

Original entry on oeis.org

3, 16, 23, 27, 31, 44, 46, 51, 55, 60, 68, 74, 79, 86, 95, 101, 105, 107, 112, 116, 121, 126, 129, 146, 147, 152, 159, 164, 167, 172, 177, 182, 185, 191, 195, 199, 204, 209, 220, 223, 229, 234, 237, 242, 244, 257, 262, 270, 275, 285, 286, 291, 299, 305, 312

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 3 after the third term.

Positions of 3's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A375927.

Join@@Position[Differences[Select[Range[1000],!SquareFreeQ[#]&]],3]

Complement of A375709 U A375710 U A375712.

cp's OEIS Frontend

A331385 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = n + k.

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A331387 Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.

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A331418 If A331417(n) is the maximum sum of primes of the parts of an integer partition of n, then a(n) = A331417(n) - n + 1.

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A097935 Number of primes that are not less than prime(n)-n and not greater than prime(n)+n.

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A098389 Prime(n) - floor(log_2(prime(n))).

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A182174 a(n) = prime(n)^2 - n.

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Magma

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A232862 Least positive integer m <= n^2/2 + 3 such that the set {prime(k) - k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

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A331417 Maximum sum of primes of the parts of an integer partition of n.

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