A000009 -id:A000009 - cp's OEIS frontend

A234615 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) - 1 are both prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 7.
(ii) Any integer n > 7 not equal to 15 can be written as k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) + 1 are both prime.
(iii) Any integer n > 83 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a square. Also, each integer n > 45 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a triangular number.
Clearly, part (i) of this conjecture implies that there are infinitely many primes p with q(p) - 1 also prime (cf. A234644).

			a(6) = 1 since 6 = 2 + 4 with prime(2) + phi(4) = 5 and q(5) - 1 = 2 both prime.
a(58) = 1 since 58 = 12 + 46 with prime(12) + phi(46) = 59 and q(59) - 1 = 9791 both prime.
a(526) = 1 since 526 = 389 + 137 with prime(389) + phi(137) = 2819 and q(2819) - 1 = 326033386646595458662191828888146112979 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A234475, A234514, A234530, A234567, A234569, A234572, A234644

f[n_,k_]:=Prime[k]+EulerPhi[n-k]
q[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[PartitionsQ[f[n,k]]-1]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset: 0

Gus Wiseman, Dec 13 2024

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
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, A225485 or A325280.

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A232504 Number of ways to write n = k + m (k, m > 0) with p(k) + q(m) prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 1, 2, 2, 1, 1, 4, 1, 5, 4, 5, 4, 4, 3, 5, 5, 6, 2, 4, 8, 4, 3, 6, 5, 3, 5, 5, 8, 5, 6, 4, 7, 5, 5, 2, 6, 9, 8, 3, 10, 7, 9, 7, 4, 7, 8, 8, 5, 6, 8, 5, 4, 8, 5, 5, 7, 11, 7, 7, 9, 8, 7, 9, 11, 8, 10, 4, 7, 8, 7, 9, 13, 7, 8, 4, 6, 11, 8, 13, 3, 8, 10, 5, 7, 11, 11, 6, 9, 6, 5, 10, 6, 9, 5, 10, 11, 9, 8, 11, 8

Offset: 1

Zhi-Wei Sun, Nov 25 2013

nonn

Conjecture: a(n) > 0 for all n > 1.

			a(5) = 1 since 5 = 1 + 4 with p(1) + q(4) = 1 + 2 = 3 prime.
a(8) = 1 since 8 = 4 + 4 with p(4) + q(4) = 5 + 2 = 7 prime.

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

Cf. A000009, A000040, A000041, A202650, A231201.

a[n_]:=Sum[If[PrimeQ[PartitionsP[k]+PartitionsQ[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

Original entry on oeis.org

1, 1, 3, 19, 121, 1041, 10651, 121843, 1575729, 22970881, 366805171, 6365365491, 120044573353, 2430782532049, 52677233993931, 1217023986185491, 29799465317716321, 771272544315151233, 21044341084622337379, 603173026772647474771

Offset: 0

Seiichi Manyama, Oct 17 2017

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 9, 1, 1, 1, 3, 9, 1, ...], a purely periodic sequence with exact period 5.
3 divides a(3*n+2); 9 divides a(9*n+8); 11 divides a(11*n+4); 19 divides a(19*n+3). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A293839, A293841.

Cf. A058892.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k)*a(n-k)/(n-k)! for n > 0.

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Differences[Table[PartitionsQ[n],{n,0,100}],2]

A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

The non-strict version is A357977.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsQ],100]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f9(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

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

A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, -4, 18, -20, 47, -56, 110, -153, 309, -532, 1045, -1768, 2855, -3620, 2928, 2927, -20371, 62261, -148774, 314112, -613835, 1155936, -2175658, 4244218, -8753316, 19006746, -42471491, 95234915, -210395017, 453414314, -949507878, 1931940045

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = -4.

For primes we have A140119 or A376683, absolute value A376681 or A376684.
For composites we have A377034, absolute value A377035.
For squarefree numbers we have A377039, absolute value A377040.
For nonsquarefree numbers we have A377047, absolute value A377048.
For prime powers we have A377052, absolute value A377053.
For partition numbers we have A377056, absolute value A378621.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The unsigned version is A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A234644 Primes p with q(p) - 1 also prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 41, 43, 53, 59, 79, 103, 151, 191, 269, 277, 283, 373, 419, 521, 571, 577, 607, 829, 859, 1039, 2503, 2657, 2819, 3533, 3671, 4079, 4153, 4243, 4517, 4951, 4987, 5689, 5737, 5783, 7723, 8101, 9137, 9173, 9241, 9539, 11467, 12323, 12697, 15017, 15277, 15427, 15803, 16057, 17959, 18661

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

By the conjecture in A234615, this sequence should have infinitely many terms.
See A234647 for primes of the form q(p) - 1 with p prime.
See also A234530 for a similar sequence.

			a(1) = 5 since neither q(2) - 1 = 0 nor q(3) - 1 = 1 is prime, but q(5) - 1 = 2 is prime.
a(2) = 11 since q(7) - 1 = 4 is composite, but q(11) - 1 = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..140
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000040, A234470, A234475, A234514, A234530, A234567, A234569, A234572, A234615, A234647.

q[k_]:=q[k]=PrimeQ[PartitionsQ[Prime[k]]-1]
n=0;Do[If[q[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^5}]

A050393 Reversion of partitions into distinct parts A000009.

Original entry on oeis.org

1, -1, 0, 3, -7, 3, 31, -105, 101, 419, -1971, 2923, 5800, -40388, 81147, 64075, -854408, 2204543, -56096, -18070916, 58866158, -38939227, -371701743, 1544696638, -1870286829, -7166094999, 39743193694, -68677654555

Offset: 1

Christian G. Bower, Nov 15 1999

sign

Vincenzo Librandi, Table of n, a(n) for n = 1..200
N. J. A. Sloane, Transforms
Index entries for reversions of series

Cf. A000009, A007312, A050394.

InverseSeries[QPochhammer[-1, x]/2 + O[x]^20][[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)

G.f. A(x) satisfies: A(x) = -1 + (1 + x) * Product_{k>=2} 1/(1 + A(x)^k). - Ilya Gutkovskiy, Apr 23 2020

cp's OEIS Frontend

A234615 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) - 1 are both prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

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A232504 Number of ways to write n = k + m (k, m > 0) with p(k) + q(m) prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

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A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

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Formula

A378972 Second differences of the strict partition numbers A000009.

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A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

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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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A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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A234644 Primes p with q(p) - 1 also prime, where q(.) is the strict partition function (A000009).

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