A377037 -id:A377037 - cp's OEIS frontend

A379300 Number of prime indices of n that are composite.

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

Offset: 1

Gus Wiseman, Dec 25 2024

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 prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Gus Wiseman, Dec 12 2024

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

A379303 Number of strict integer partitions of n with a unique composite part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 3, 6, 6, 8, 10, 10, 13, 15, 17, 20, 22, 24, 28, 31, 36, 40, 44, 50, 55, 62, 70, 75, 83, 89, 97, 108, 115, 128, 136, 146, 161, 172, 188, 203, 215, 233, 249, 269, 291, 309, 331, 353, 376, 405, 433, 459, 490, 518, 554, 592, 629, 670, 705

Offset: 0

Gus Wiseman, Dec 25 2024

nonn

			The a(4) = 1 through a(11) = 8 partitions:
  (4)  (4,1)  (6)    (4,3)    (8)      (9)      (10)       (6,5)
              (4,2)  (6,1)    (6,2)    (5,4)    (8,2)      (7,4)
                     (4,2,1)  (4,3,1)  (6,3)    (9,1)      (8,3)
                                       (8,1)    (5,4,1)    (9,2)
                                       (4,3,2)  (6,3,1)    (10,1)
                                       (6,2,1)  (4,3,2,1)  (5,4,2)
                                                           (6,3,2)
                                                           (8,2,1)

If no parts are composite we have A036497,  non-strict A034891 (ranks A302540).
If all parts are composite we have A204389, non-strict A023895 (ranks A320629).
The non-strict version is A379302, ranks A379301 (positions of 1 in A379300).
For a unique prime we have A379305, non-strict A379304 (ranks A331915).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Cf. A000070, A000586, A000607, A002095, A038348, A096258, A113646, A376680, A379308, A379309, A379314, A379315.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?CompositeQ]==1&]],{n,0,30}]

A377036 First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.

Original entry on oeis.org

4, 2, 0, -1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, -1, 78, -233, 687, -2363, 8160, -25670, 72352, -184451, 430937, -933087, 1888690, -3597221, 6479696, -11086920, 18096128, -28307626, 42644791, -62031001, 86466285, -110902034, 110907489, -52325, -483682930

Offset: 0

Gus Wiseman, Oct 18 2024

sign

The version for prime instead of composite is A007442.
For noncomposite numbers we have A030016.
This is the first column (n=1) of A377033.
For row-sums we have A377034, absolute version A377035.
First zero positions are A377037, cf. A376678, A376855, A377042, A377050, A377055.
For squarefree instead of composite we have A377041, nonsquarefree A377049.
For prime-power instead of composite we have A377054.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composite numbers, differences A073783, seconds A073445.
A008578 lists the noncomposites, differences A075526.
Cf: A018252, A065310, A065890, A140119, A173390, A333214, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]-1}]

The inverse zero-based binomial transform of a sequence (q(0), q(1), ..., q(m)) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset: 0

Gus Wiseman, Jan 12 2025

sign

Also the inverse zero-based binomial transform of the odd prime numbers.

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

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A379300 Number of prime indices of n that are composite.

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A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

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A379303 Number of strict integer partitions of n with a unique composite part.

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A377036 First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.

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A379542 Second term of the n-th differences of the prime numbers.

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