A075526 -id:A075526 - cp's OEIS frontend

A125266 Number of repetitions in A007918.

3, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8

Offset: 1

Cino Hilliard, Jan 15 2007

Except for the first 2 terms, these numbers are always even. Conjecture: the number 2 occurs infinitely often in this sequence.
Essentially the same as A075526 and A054541. - R. J. Mathar, Jun 15 2008
3 together with A001223. - Omar E. Pol, Nov 01 2013

			A007918(0) = 2, A007918(1)=2, A007918(2) = 2. So 2 repeats 3 times, giving 3 as the first term in the table.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001223, A007918, A054541, A075526.

Join[{3},Differences[Prime[Range[100]]]] (* Paolo Xausa, Oct 25 2023 *)

nextprimerep(n) = { local(x,y,y1,c=0); y1=2; for(x=0,n, y=nextprime(x); if(y==y1,c++,y1=y;print1(c",");c=1); ) }

A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

These are the antidiagonal-sums of the absolute value of A175804.
First column of the same array is A281425.
For primes we have A376681 or A376684, signed A140119 or A376683.
For composites we have A377035, signed A377034.
For squarefree numbers we have A377040, signed A377039.
For nonsquarefree numbers we have A377048, signed A377049.
For prime powers we have A377053, signed A377052.
The signed version is A377056.
The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

nn=30;
q=Table[PartitionsP[n],{n,0,nn}];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A106588 Difference between n-th prime squared and n-th perfect square.

Original entry on oeis.org

3, 5, 16, 33, 96, 133, 240, 297, 448, 741, 840, 1225, 1512, 1653, 1984, 2553, 3192, 3397, 4128, 4641, 4888, 5757, 6360, 7345, 8784, 9525, 9880, 10665, 11040, 11869, 15168, 16137, 17680, 18165, 20976, 21505, 23280, 25125, 26368, 28329, 30360

Offset: 1

Alexandre Wajnberg, May 10 2005

			a(5) = 96 because 121 (fifth prime^2) - 25 (fifth square) = 96.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000290, A001248, A014688, A075526, A076368.

[NthPrime(n)^2 - n^2: n in [1..50]]; // G. C. Greubel, Sep 07 2021

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

a(n) = prime(n)^2 - n^2; \\ Michel Marcus, Sep 08 2021

[nth_prime(n)^2 - n^2 for n in (1..50)] # G. C. Greubel, Sep 07 2021

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

Extended by Ray Chandler, May 13 2005

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

Or, positive integers whose prime indices include at least one 1 or prime number.

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 terms together with their prime indices begin:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
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.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

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

A173400 n-th difference between consecutive primes=n-th difference between consecutive nonnegative nonprimes.

Original entry on oeis.org

1, 3, 7, 20, 26, 33, 43, 49, 52, 81, 116, 140, 176, 265, 288, 313, 320, 323, 373, 377, 395, 398, 405, 408, 486, 492, 530, 555, 567, 592, 671, 681, 772, 805, 849, 874, 884, 931, 936, 1016, 1030, 1149, 1204, 1324, 1347, 1406, 1464, 1550, 1621, 1639, 1707, 1712

Offset: 1

Juri-Stepan Gerasimov, Feb 17 2010

Numbers n such that A001223(n)=A054546(n).

Cf. A001223, A075526, A173390, A173401, A029707.

A001223(a(n))=A054546(a(n)).

Extended by Charles R Greathouse IV, Mar 25 2010

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).

cp's OEIS Frontend

A125266 Number of repetitions in A007918.

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A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A106588 Difference between n-th prime squared and n-th perfect square.

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A379313 Positive integers whose prime indices are not all composite.

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A173400 n-th difference between consecutive primes=n-th difference between consecutive nonnegative nonprimes.

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

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