A014689 -id:A014689 - cp's OEIS frontend

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

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

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 9. Also, for any integer n > 51 there is a positive integer k < n such that p = k + prime(n-k) and prime(p) + p + 1 are both prime.
(ii) If n > 9 (or n > 21), then there is a positive integer k < n such that m - 1 and prime(m) + m (or prime(m) - m, resp.) are both prime, where m = k + prime(n-k).
(iii) If n > 483, then for some 0 < k < n both prime(m) + m and prime(m) - m are prime, where m = k + prime(n-k).
(iv) If n > 3, then there is a positive integer k < n such that prime(k + prime(n-k)) + 2 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with prime(p) - p + 1 (or prime(p) + p + 1) also prime.
See A234695 for primes p with prime(p) - p + 1 also prime.

			a(5) = 1 since 2 + prime(3) = 7 and prime(7) - 6 = 11 are both prime.
a(25) = 1 since 20 + prime(5) = 31 and prime(31) - 30 = 97 are both prime.
a(27) = 1 since 18 + prime(9) = 41 and prime(41) - 40 = 139 are both prime.
a(45) = 1 since 6 + prime(39) = 173 and prime(173) - 172 = 859 are both prime.
a(49) = 1 since 26 + prime(23) = 109 and prime(109) - 108 = 491 are both 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. A000040, A014688, A014689, A014692, A064269, A064270, A232861, A233150, A233183, A233206, A233296, A234695.

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

A378086 Number of nonsquarefree numbers < prime(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 5, 6, 7, 11, 11, 13, 14, 14, 16, 20, 22, 23, 25, 26, 27, 29, 31, 33, 36, 39, 39, 40, 41, 42, 49, 50, 53, 53, 57, 58, 61, 63, 64, 68, 70, 71, 74, 75, 76, 77, 81, 84, 86, 87, 88, 90, 91, 97, 99, 101, 103, 104, 107, 109, 109, 113, 119, 120, 121

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

			The nonsquarefree numbers counted under each term begin:
  n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9: n=10: n=11: n=12:
  --------------------------------------------------------------
   .    .    4    4    9    12   16   18   20   28    28    36
                       8    9    12   16   18   27    27    32
                       4    8    9    12   16   25    25    28
                            4    8    9    12   24    24    27
                                 4    8    9    20    20    25
                                      4    8    18    18    24
                                           4    16    16    20
                                                12    12    18
                                                9     9     16
                                                8     8     12
                                                4     4     9
                                                            8
                                                            4

For nonprime numbers we have A014689.
Restriction of A057627 to the primes.
First-differences are A061399 (zeros A068361), squarefree A061398 (zeros A068360).
For composite instead of squarefree we have A065890.
For squarefree we have A071403, differences A373198.
Greatest is A378032 (differences A378034), restriction of A378033 (differences A378036).
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A070321 gives the greatest squarefree number up to n.
A112925 gives the greatest squarefree number between primes, differences A378038.
A112926 gives the least squarefree number between primes, differences A378037.
A120327 gives the least nonsquarefree number >= n, first-differences A378039.
A377783 gives the least nonsquarefree > prime(n), differences A377784.
Cf. A013928, A046933, A053797, A053806, A072284, A076259, A224363, A337030, A377049, A378040, A378082, A378084.

Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}]

from math import isqrt
from sympy import prime, mobius
def A378086(n): return (p:=prime(n))-sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # Chai Wah Wu, Dec 05 2024

a(n) = A057627(prime(n)).

A331416 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) = k.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jan 17 2020

			Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 0 0 2 1
  0 0 0 0 0 0 1 3 1
  0 0 0 0 0 0 0 0 2 3 1 1
  0 0 0 0 0 0 0 0 0 1 4 3 1 2
  0 0 0 0 0 0 0 0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 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 = 19 counts the following partitions:
  (8)   (6111)   (532)        (443)       (33222)
  (71)  (51111)  (622)        (4331)      (42222)
                 (5221)       (4421)      (322221)
                 (4111111)    (33311)     (2222211)
                 (31111111)   (43211)
                 (211111111)  (332111)
                              (422111)
                              (3221111)
                              (22211111)

Row lengths are A331417.
Row sums are A000041.
Column sums are A000607.
Shifting row n to the left n times gives A331385.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions of 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, A330954, A331378, A331387, A331415, A331418.

maxm[n_]:=Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}];
Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==k&]],{n,0,10},{k,0,maxm[n]}]

A014692 a(n) = prime(n) - (n-1).

Original entry on oeis.org

2, 2, 3, 4, 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, 222, 223

Offset: 1

Mohammad K. Azarian

Also, number of primes between prime(n) and prime(prime(n)) inclusive. For example, prime(1) = 2, prime(prime(1)) = prime(2) = 3 and there are two primes between 2 and 3 inclusive. - Zak Seidov, Sep 22 2003; N. J. A. Sloane, Mar 07 2007
Since a(n+1) - a(n) = prime(n+1) - prime(n) - 1 >= 1 for n > 1, the sequence is monotonic for n > 1. - N. J. A. Sloane, Mar 07 2007
a(n) = number of terms < prime(n) in A141468. - David James Sycamore, Oct 14 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A141468.

Equals A014689 + 1.

A014692:=n->ithprime(n)-(n-1): seq(A014692(n), n=1..100); # Wesley Ivan Hurt, Oct 15 2017

Table[Prime[n] - n + 1, {n, 61}]  (* Geoffrey Critzer, May 02 2013 *)

first(n) = {my(t, res = vector(n)); forprime(p=2, , t++; res[t] = p - t + 1; if(t>=n, return(res)))} \\ David A. Corneth, Oct 04 2017

a(n) = prime(n)-n+1; \\ Altug Alkan, Oct 05 2017

from sympy import prime
def A014692(n): return prime(n)-n+1 # Chai Wah Wu, Oct 11 2024

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

Original entry on oeis.org

0, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 26

Offset: 0

Labos Elemer, Jul 31 2002

nonn

a(n-1) = A018252(n) - n. a(n-1) = inverse (frequency distribution) sequence of A014689(n), i.e. number of terms of sequence A014689(n) less than n. a(n) = A073169(n+1) - 1, for n >= 1. For n >= 1: a(n) + 1 = A073169(n) = the number of set {1, primes}, i.e. (A008578) less than (n)-th composite numbers (A002828(n)). a(n-1) = The number of primes (A000040(n)) less than n-th nonprime (A018252(n)). - Jaroslav Krizek, Jun 27 2009

			n=100: composite[100]=133,Pi[133]=32=a(100)

Cf. A065890, A073426, A000720, A002808, A000040, A018252, A158611, A073169.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[PrimePi[c[w]], {w, 1, 128}]
With[{nn=150},PrimePi/@Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, Jun 26 2013 *)

from sympy import composite
def A073425(n): return composite(n)-n-1 if n else 0 # Chai Wah Wu, Oct 11 2024

a(n) = A000720(A002808(n)).
a(n) ~ n. - Charles R Greathouse IV, Sep 02 2015
a(n) = A002808(n)-n-1 for n > 0. - Chai Wah Wu, Oct 11 2024

Edited by N. J. A. Sloane, Jul 04 2009 at the suggestion of R. J. Mathar

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A331415 Sum of prime factors minus sum of prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 2, 3, 6, 3, 7, 4, 3, 4, 10, 3, 11, 4, 4, 7, 14, 4, 4, 8, 3, 5, 19, 4, 20, 5, 7, 11, 5, 4, 25, 12, 8, 5, 28, 5, 29, 8, 4, 15, 32, 5, 6, 5, 11, 9, 37, 4, 8, 6, 12, 20, 42, 5, 43, 21, 5, 6, 9, 8, 48, 12, 15, 6, 51, 5, 52, 26, 5, 13, 9, 9

Offset: 1

Gus Wiseman, Jan 17 2020

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 factors of 12 are {2,2,3}, while the prime indices are {1,1,2}, so a(12) = 7 - 4 = 3.

The number of k's is A331387(k) = sum of k-th column of A331385.
The sum of prime factors of n is A001414(n).
The sum of prime indices of n is A056239(n).
Numbers divisible by the sum of their prime factors are A036844.
Sum of prime factors is divisible by sum of prime indices: A331380
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A000720, A001222, A014689, A056239, A112798, A301987, A318995, A325036, A330953, A331378, A331379, A331416, A331418.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k*(p-PrimePi[p])]],{n,30}]

Totally additive with a(prime(k)) = prime(k) - k = A014689(k).

a(n) = A001414(n) - A056239(n).

A064269 Numbers k such that prime(k) - k is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 14, 16, 18, 28, 30, 42, 44, 50, 54, 66, 68, 76, 84, 90, 94, 110, 144, 148, 154, 168, 174, 178, 192, 196, 214, 220, 242, 264, 266, 268, 278, 280, 282, 294, 304, 308, 336, 346, 348, 354, 358, 360, 370, 376, 380, 382, 384, 390, 400, 402, 408, 414

Offset: 1

Jason Earls, Sep 23 2001

			n=54: prime(54) - 54 = 197, a prime.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A014689, A064270.

Select[ Range[ 415 ], PrimeQ[ Prime[ # ] - # ] & ]

j=[]; for(n=1,500, if(isprime(prime(n)-n), j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (isprime(prime(m) - m), write("b064269.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 10 2009

A375929 Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 76, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

Positions of 1's in A073783 (see also A054546, A065310).

			The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after positions 3, 4, 7, 8, ...

Positions in A002808 of each element of A068780.
The complement is A065890 shifted.
First differences are A373403 (except first).
The version for non-prime-powers is A375713, differences A373672.
The version for prime-powers is A375734, differences A373671.
The version for non-perfect-powers is A375740.
The version for nonprime numbers is A375926.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprime numbers, differences A065310.
A046933 counts composite numbers between primes.
Cf. A014689, A054546, A176246, A246655, A251092.

Join@@Position[Differences[Select[Range[100],CompositeQ]],1]

from sympy import primepi
def A375929(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+bisection(lambda y:primepi(x+2+y))-2
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

# faster for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    pic, prevc = 0, -1
    for i in count(4):
        if not isprime(i):
            if i == prevc + 1:
                yield pic
            pic, prevc = pic+1, i
print(list(islice(agen(), 10000))) # Michael S. Branicky, Sep 17 2024

a(n) = A375926(n) - 1.

A375926 Numbers k such that A018252(k+1) = A018252(k) + 1. In other words, the k-th nonprime number is 1 less than the next.

Original entry on oeis.org

4, 5, 8, 9, 12, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 27, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Sep 11 2024

nonn

			The nonprime numbers are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after term 4, term 5, term 8, etc.

The complement appears to be A014689, except the first term.
Positions of 1's in A065310 (see also A054546, A073783).
First differences are A373403 (except first).
The version for non-prime-powers is A375713, differences A373672.
The version for prime-powers is A375734, differences A373671.
The version for non-perfect-powers is A375740.
The version for composite numbers is A375929.
A000040 lists the prime numbers, differences A001223.
A018252 lists the nonprimes, exclusive A002808.
A046933 counts composite numbers between primes.
Cf. A000961, A006549, A057820, A176246, A246655, A251092, A375708.

Join@@Position[Differences[Select[Range[100],!PrimeQ[#]&]],1]

from sympy import primepi
def A375926(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+bisection(lambda y:primepi(x+1+y))-1
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, kprime(k) - 1, kprime(k) + 1 all prime.

Original entry on oeis.org

22110, 23742, 128238, 275592, 346560, 1061910, 1281522, 1339002, 1378188, 1461600, 1850130, 2064150, 2354952, 2478270, 2523708, 2689260, 2694300, 3916638, 4422618, 4933530, 6179082, 6541080, 6641562, 6740478, 6759030, 7315812, 8484798, 8711010, 9133308, 9687720

Offset: 1

Zhi-Wei Sun, Dec 01 2013

nonn

Obviously, each term of the sequence is a multiple of 6.
Conjecture: (i) This sequence contains infinitely many terms.
(ii) Let P(x) be a non-constant integer-valued polynomial with positive leading coefficient. Then, there are infinitely many positive integers k with prime(k) - k in the range P(Z) = {P(m): m is an integer}, if and only if the degree of P(x) is at most 3. We may also replace prime(k) - k by prime(k) + k.

			a(1) = 22110 with the six numbers 22110 - 1 = 22109, 22110 + 1 = 22111, prime(22110) - 22110 = 228841, prime(22110) + 22110 = 273061, 22110*prime(22110) - 1 = 5548526609, 22110*prime(22110) + 1 = 5548526611 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017.

Cf. A000040, A014574, A014689, A064269, A064270, A064370, A105645, A232443, A232463, A232548.

n=0
Do[If[PrimeQ[k-1]&&PrimeQ[k+1]&&PrimeQ[Prime[k]-k]&& PrimeQ[Prime[k]+k]&& PrimeQ[k*Prime[k]-1]&& PrimeQ[k*Prime[k]+1],n=n+1;Print[n," ",k]],{k,1,9700000}]

cp's OEIS Frontend

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

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A378086 Number of nonsquarefree numbers < prime(n).

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A331416 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) = k.

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A014692 a(n) = prime(n) - (n-1).

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A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

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A331415 Sum of prime factors minus sum of prime indices of n.

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A064269 Numbers k such that prime(k) - k is prime.

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A375929 Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.

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A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, kprime(k) - 1, kprime(k) + 1 all prime.