A234567 -id:A234567 - cp's OEIS frontend

A234475 Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

Conjecture: a(n) > 0 for all n > 5.

This implies that there are infinitely many primes p with p - 1 a term of A000009.

			a(6) = 1 since 6 = 3 + 3 with q(phi(3)*phi(3)/4) + 1 = q(1) + 1 = 2 prime.
a(76) = 1 since 76 = 18 + 58 with q(phi(18)*phi(58)/4) + 1 = q(42) + 1 = 1427 prime.
a(197) = 1 since 197 = 4 + 193 with q(phi(4)*phi(193)/4) + 1 = q(96) + 1 = 317789.
a(356) = 1 since 356 = 88 + 268 with q(phi(88)*phi(268)/4) + 1 = q(1320) + 1 = 35940172290335689735986241 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2525
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A232504, A233307, A233346, A233547, A233390, A233393, A234309, A234310, A234337, A234344, A234347, A234359, A234360, A234361, A234451, A234470, A234514, A234530, A234567, A234569

f[n_,k_]:=PartitionsQ[EulerPhi[k]*EulerPhi[n-k]/4]+1
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,3,n/2}]
Table[a[n],{n,1,100}]

A234530 Primes p with q(p) + 1 also prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 3, 11, 13, 29, 37, 47, 71, 79, 89, 103, 127, 131, 179, 181, 197, 233, 271, 331, 379, 499, 677, 691, 757, 887, 911, 1019, 1063, 1123, 1279, 1429, 1531, 1559, 1637, 2251, 2719, 3571, 4007, 4201, 4211, 4297, 4447, 4651, 4967, 5953, 6131, 7937, 8233, 8599, 8819, 9013, 11003, 11093, 11813, 12251, 12889, 12953, 13487, 13687, 15259

Offset: 1

Zhi-Wei Sun, Dec 27 2013

nonn

By the conjecture in A234514, this sequence should have infinitely many terms.
It seems that a(n+1) < a(n) + a(n-1) for all n > 4.
See A234366 for primes of the form q(p) + 1 with p prime.
See also A234644 for a similar sequence.

			a(1) = 2 since 2 and q(2) + 1 = 2 are both prime.
a(2) = 3 since 3 and q(3) + 1 = 3 are both prime.
a(3) = 11 since 11 and q(11) + 1 = 13 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Cf. A000009, A000040, A233346, A233393, A234366, A234470, A234475, A234514, A234567, A234569, A234572, A234615, A234644, A234647

n=0;Do[If[PrimeQ[PartitionsQ[Prime[k]]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,10^5}]
Select[Prime[Range[2000]],PrimeQ[PartitionsQ[#]+1]&] (* Harvey P. Dale, Apr 23 2017 *)

A234569 Primes p with P(p-1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

3, 5, 7, 37, 367, 499, 547, 659, 1087, 1297, 1579, 2137, 2503, 3169, 3343, 4457, 4663, 5003, 7459, 9293, 16249, 23203, 34667, 39971, 41381, 56383, 61751, 62987, 72661, 77213, 79697, 98893, 101771, 127081, 136193, 188843, 193811, 259627, 267187, 282913, 315467, 320563, 345923, 354833, 459029, 482837, 496477, 548039, 641419, 647189

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

By the conjecture in A234567, this sequence should have infinitely many terms. It seems that a(n+1) < a(n) + a(n-1) for all n > 5.
The b-file lists all terms not exceeding the 500000th prime 7368787. Note that P(a(113)-1) is a prime having 2999 decimal digits.
See also A234572 for primes of the form P(p-1) with p prime.

			a(1) = 3 since P(2-1) = 1 is not prime, but P(3-1) = 2 is prime.
a(2) = 5 since P(5-1) = 5 is prime.
a(3) = 7 since P(7-1) = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..113
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234572, A234615, A234644

n=0;Do[If[PrimeQ[PartitionsP[Prime[k]-1]],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]

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).

Original entry on oeis.org

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}]

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}]

A234572 Primes of the form P(p-1), where p is a prime and P(.) is the partition function (A000041).

Original entry on oeis.org

2, 5, 11, 17977, 790738119649411319, 2058791472042884901563, 27833079238879849385687, 8121368081058512888507057, 675004412390512738195023734124239, 1398703012615213588677365804960180341, 16193798232344933888778097136641377589301, 204931453786129197483756438132982529754356479553, 3019564607799532159016586951616642980389816614848623, 22757918197082858017617136646280039394687006502870793231847, 1078734573992480956821414895441907729656949308800686938161281

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Though the primes in this sequence are very rare, by the conjecture in A234567 there should be infinitely many such primes.

See A234569 for a list of known primes p with P(p-1) also prime.

			a(1) = 2 since 2 = P(3-1) with 2 and 3 both prime.
a(2) = 5 since 5 = P(5-1) with 5 prime.
a(3) = 11 since 11 = P(7-1) with 7 and 11 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50

Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234569

p[n_]:= A234569(n)
Table[PartitionsP[p[n]-1],{n,1,15}]

a(n) = A000041(A234569(n)-1).

A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n >= 60.
(ii) For any integer n > 1234, there is a positive integer k < n such that g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime, where g(n,k) = phi(k) + phi(n-k)/8.
Clearly, part (i) implies that there are infinitely many primes of the form q(m) + 1 with m - 1 and m + 1 also prime, and part (ii) implies that there are infinitely many primes of the form q(m) - 1 with m - 1 and m + 1 also prime. As log q(m) is asymptotically equivalent to  pi*sqrt(m/3), the conjecture is much stronger than the twin prime conjecture.
We have verified parts (i) and (ii) for n up to 100000 and 60000 respectively.

			a(50) = 1 since phi(34) + phi(16)/4 = 18 with 18 - 1, 18 + 1 and q(18) + 1 = 47 all prime.
a(215) = 1 since phi(87) + phi(128)/4 = 72 with 72 - 1, 72 + 1 and q(72) + 1 = 36353 all prime.
a(645) = 1 since phi(365) + phi(280)/4 = 312 with 312 - 1, 312 + 1 and q(312) + 1 = 207839472391 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235344, A235346, A235356, A235357, A235358.

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4
p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 3, 1, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Zhi-Wei Sun, Jan 07 2014

nonn

Conjecture: a(n) > 0 for all n > 1234.
See also part (ii) of the conjecture in A235343.
We have verified the conjecture for n up to 100000.

			a(50) = 1 since phi(10) + phi(40)/4 = 6 with 6 - 1, 6 + 1 and q(6) - 1 = 3 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235343, A235344, A235346, A235356, A235357.

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/8
p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]-1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A234647 Primes of the form q(p) - 1, where p is a prime and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 11, 17, 37, 53, 103, 1259, 1609, 5119, 9791, 70487, 570077, 20792119, 281138047, 23515017983, 35692320959, 48626519093, 3626048321047, 27077619952639, 1651411233432319, 10743948315198451, 13378670620050079, 39413984631175423, 58553713102334907283, 145464242180631569963, 25408177717067357968543, 1374387931601409538722802926765483199, 20557774525717988142856527912112710143, 326033386646595458662191828888146112979, 27403889354101748193301659902924397784656229

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

Though the primes in this sequence are very rare, by the conjecture in A234615 there should be infinitely many such primes.

See A234644 for a list of known primes p with q(p) - 1 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

Cf. A000009, A000040, A234366, A234470, A234475, A234514, A234530, A234567, A234569, A234572, A234615, A234644

a(1) = 2 since 2 = q(5) - 1 with 2 and 5 both prime.

p[n_]:=A234615(n)
Table[PartitionsQ[p[n]]-1,{n,1,30}]

a(n) = A000009(A234615(n)) - 1.

A234808 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*n - p are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 0, 3, 1, 2, 5, 2, 1, 5, 1, 2, 7, 2, 1, 4, 1, 2, 1, 4, 1, 4, 2, 4, 11, 4, 2, 3, 1, 5, 2, 3, 2, 6, 1, 5, 15, 4, 2, 9, 1, 6, 2, 5, 4, 6, 4, 4, 3, 8, 3, 6, 4, 7, 21, 2, 4, 7, 1, 7, 4, 6, 4, 6, 4, 8, 22, 7, 3, 13, 1, 10, 5, 3, 5, 7, 4, 9, 5, 10, 5, 8, 7, 7, 6, 8, 5, 6, 3, 8, 6, 7, 4, 8, 4

Offset: 1

Zhi-Wei Sun, Dec 30 2013

nonn

Conjecture: a(n) > 0 except for n = 1, 8.

Clearly, this implies Goldbach's conjecture.

			a(3) = 1 since 2 + phi(1) = 3 and 2*3 - 3 = 3 are both prime.
a(20) = 1 since 11 + phi(9) = 17 and 2*20 - 17 = 23 are both prime.
a(22) = 1 since 1 + phi(21) = 13 and 2*22 - 13 = 31 are both prime.
a(24) = 1 since 9 + phi(15) = 17 and 2*24 - 17 = 31 are both prime.
a(76) = 1 since 67 + phi(9) = 73 and 2*76 - 73 = 79 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A002372, A002375, A233547, A233918, A234200, A234246, A234360, A234470, A234475, A234514, A234567, A234615, A234694

f[n_,k_]:=k+EulerPhi[n-k]
p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[2n-f[n,k]]
a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A234475 Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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A234530 Primes p with q(p) + 1 also prime, where q(.) is the strict partition function (A000009).

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A234569 Primes p with P(p-1) also prime, where P(.) is the partition function (A000041).

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

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A234572 Primes of the form P(p-1), where p is a prime and P(.) is the partition function (A000041).

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A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

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A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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