A054268 -id:A054268 - cp's OEIS frontend

A054265 Sum of composite numbers between successive primes.

0, 4, 6, 27, 12, 45, 18, 63, 130, 30, 170, 117, 42, 135, 250, 280, 60, 320, 207, 72, 380, 243, 430, 651, 297, 102, 315, 108, 333, 1560, 387, 670, 138, 1296, 150, 770, 800, 495, 850, 880, 180, 1674, 192, 585, 198, 2255, 2387, 675, 228, 693, 1180, 240, 2214, 1270

Offset: 1

Patrick De Geest, Apr 15 2000

nonn

			Between 7 and 11 we have 8 + 9 + 10 which is a(4)=27.

James Spahlinger, Table of n, a(n) for n = 1..1000
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.

Cf. A000040, A046933, A054264, A054266, A054267, A054268.

a(n) = (prime(n+1) + prime(n))*(prime(n+1) - prime(n) - 1)/2; \\ Michel Marcus, Mar 24 2016

from sympy import nextprime, prime
def A054265(n): return ((p:=prime(n))+(q:=nextprime(p)))*(q-p-1)>>1 # Chai Wah Wu, Jun 01 2024

a(n) = (prime(n+1) + prime(n))*(prime(n+1) - prime(n) - 1)/2. - Zak Seidov, Sep 12 2002

A054264 Concatenation of composite numbers between the n-th prime and the following prime.

Original entry on oeis.org

4, 6, 8910, 12, 141516, 18, 202122, 2425262728, 30, 3233343536, 383940, 42, 444546, 4849505152, 5455565758, 60, 6263646566, 686970, 72, 7475767778, 808182, 8485868788, 90919293949596, 9899100, 102, 104105106, 108, 110111112

Offset: 2

Patrick De Geest, Apr 15 2000

Andrew Howroyd, Table of n, a(n) for n = 2..1000

Cf. A046933, A054265, A054266, A054267, A054268.

FromDigits[Flatten[IntegerDigits/@(Range[#[[1]]+1,#[[2]]-1])]]&/@Partition[ Prime[Range[2,30]],2,1] (* Harvey P. Dale, Mar 04 2014 *)

a(n)=my(r=prime(n)+1); fromdigits(concat(vector(nextprime(r)-r, i, digits(r+i-1)))) \\ Andrew Howroyd, Aug 14 2024

Offset changed and name edited by Andrew Howroyd, Aug 14 2024

A054266 Sum of composite numbers between prime p and nextprime(p) is palindromic.

Original entry on oeis.org

3, 5, 109, 193, 281, 509, 661, 827, 857, 1439, 2111, 3433, 3889, 3967, 4549, 6661, 7001, 8467, 10099, 17203, 18583, 21011, 21611, 23831, 24847, 25117, 26261, 26497, 26861, 28181, 29587, 30497, 31307, 47569, 47869, 49789, 53939, 54139, 66361

Offset: 1

Patrick De Geest, Apr 15 2000

			a(4)=193 since between 193 and next prime 197 we get the palindromic sum 194 + 195 + 196 = 585.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Zak Seidov)
P. De Geest, World!Of Numbers

Cf. A046933, A054264, A054265, A054267, A054268.

okQ[l_]:=Module[{x=IntegerDigits[Total[Range[First[l]+1, Last[l]-1]]]},x==Reverse[x]];Transpose[Select[Partition[ Prime[Range[2,6700]],2,1],okQ]][[1]] (* Harvey P. Dale, Mar 18 2011 *)

Corrected and extended b-file by Chai Wah Wu, Feb 25 2018

A104388 Lesser of two successive primes the average of which is a repdigit.

Original entry on oeis.org

3, 5, 7, 97, 109, 4441, 111111109, 555555555551, 9999999999999937, 111111111111111091, 444444444444444419, 555555555555555555491, 777777777777777777777767, 333333333333333333333333333333293, 444444444444444444444444444444444391

Offset: 1

Zak Seidov, Mar 04 2005

Chai Wah Wu, Table of n, a(n) for n = 1..31

Cf. A054268, A104386, A104387 (larger primes), A104389 (repdigits).

from itertools import count, islice
from sympy import isprime, prevprime
def agen():
    for d in count(1):
        ru = int("1"*d)
        for r in range(ru, 10*ru, ru):
            if r > 2:
                p = prevprime(r)
                if isprime(r + (r-p)) and prevprime(r+(r-p)) == p:
                    yield p
print(list(islice(agen(), 15))) # Michael S. Branicky, Jun 30 2022

a(n) = prime(A104386(n)).

More terms from Giovanni Resta, Feb 09 2006

A054267 Sum of composite numbers between prime p and nextprime(p) is palindromic with restriction 'p + 1 <> sum'.

Original entry on oeis.org

109, 193, 509, 661, 1439, 3433, 3889, 3967, 4549, 6661, 7001, 8467, 10099, 17203, 18583, 24847, 25117, 26497, 29587, 30497, 31307, 47569, 47869, 49789, 53939, 54139, 66361, 67061, 70901, 71011, 102199, 132229, 158269, 171179, 185699

Offset: 0

Patrick De Geest, Apr 15 2000

			a(4)=661 since between 661 and next prime 673 we get the palindromic sum 662 + 663 + 664 + 665 + 666 + 667 + 668 + 669 + 670 + 671 + 672 = 7337.

Cf. A046933, A054264, A054265, A054266, A054268.

A104387 Larger of two successive primes the average of which is a repdigit.

Original entry on oeis.org

5, 7, 11, 101, 113, 4447, 111111113, 555555555559, 10000000000000061, 111111111111111131, 444444444444444469, 555555555555555555619, 777777777777777777777787, 333333333333333333333333333333373, 444444444444444444444444444444444497

Offset: 1

Zak Seidov, Mar 04 2005

Lesser primes A104388, repdigits A104389. What is the next term?

The next term is 444444444444444444444444444444444497. The first term with more than 100 digits is a(22) which has 109 digits. - Harvey P. Dale, Jun 28 2011

Michael S. Branicky, Table of n, a(n) for n = 1..31

Cf. A054268, A104386, A104388, A104389.

Union[Flatten[Table[NextPrime/@Select[FromDigits/@Table[PadLeft[{i},n,i], {n,45}], Mean[{NextPrime[#],NextPrime[#,-1]}]==#&], {i,9}]]] (* Harvey P. Dale, Jun 28 2011 *)

from itertools import count, islice
from sympy import isprime, prevprime
def agen():
    for d in count(1):
        ru = int("1"*d)
        for r in range(ru, 10*ru, ru):
            if r > 2:
                p = prevprime(r)
                if isprime(r + (r-p)) and prevprime(r+(r-p)) == p:
                    yield 2*r - p
print(list(islice(agen(), 15))) # Michael S. Branicky, Jun 30 2022

a(n) = prime(A104386(n)+1).

More terms from Giovanni Resta, Apr 05 2006

A104389 Repdigits which are the average of two successive primes.

Original entry on oeis.org

4, 6, 9, 99, 111, 4444, 111111111, 555555555555, 9999999999999999, 111111111111111111, 444444444444444444, 555555555555555555555, 777777777777777777777777, 333333333333333333333333333333333, 444444444444444444444444444444444444

Offset: 1

Zak Seidov, Mar 04 2005

Michael S. Branicky, Table of n, a(n) for n = 1..31

Corresponding primes: A104387, A104388.

Cf. A054268, A104386, A104387, A104388.

from itertools import count, islice
from sympy import isprime, prevprime
def agen():
    for d in count(1):
        ru = int("1"*d)
        for r in range(ru, 10*ru, ru):
            if r > 2:
                p = prevprime(r)
                if isprime(r + (r-p)) and prevprime(r+(r-p)) == p:
                    yield r
print(list(islice(agen(), 15))) # Michael S. Branicky, Jun 30 2022

a(n) = (A104387(n)+A104388(n))/2.

More terms from Giovanni Resta, Apr 05 2006

A104386 Numbers k such that the average of the k-th and (k+1)-th primes is a repdigit.

Original entry on oeis.org

2, 3, 4, 25, 29, 603, 6363181, 21366409911, 279238341033925, 2907021742443974, 11220808305309952, 11885037375341198280

Offset: 1

Zak Seidov, Mar 04 2005

Cf. A054268.

Corresponding primes A104387, A104388, repdigits A104389.

from itertools import count, islice
from sympy import isprime, prevprime, primepi
def agen():
    for d in count(1):
        ru = int("1"*d)
        for r in range(ru, 10*ru, ru):
            if r > 2:
                p = prevprime(r)
                if isprime(r + (r-p)) and prevprime(r+(r-p)) == p:
                    yield primepi(p)
print(list(islice(agen(), 7))) # Michael S. Branicky, Jun 30 2022

(prime(k) + prime(k+1))/2 = repdigit.

a(8) from Giovanni Resta, Apr 05 2006
a(9) from Michael S. Branicky, Jul 02 2022
a(10)-a(12) from Chai Wah Wu, Jun 01 2024

A114370 Primes p such that the sum of numbers from prime p to nextprime(p)-1 is a repdigit.

Original entry on oeis.org

2, 3, 5, 53, 55555553, 55555555555555555555555553, 2777777777777777777777777777777777777

Offset: 1

Giovanni Resta, Feb 09 2006

The sequence is built under the (reasonable) assumption that 100+2*log(p)^2 is an upper bound to the largest gap between a prime p and nextprime(p). Under this assumption there are no other terms with less than 100 digits.

			nextprime(55555555555555555555555553) is 55555555555555555555555559 and the sum
from 55555555555555555555555553 to 55555555555555555555555558 gives the repdigit 333333333333333333333333333.

Cf. A010785, A054268.

from itertools import count, islice
from sympy import isprime, nextprime
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A114370_gen(): # generator of terms
    for l in count(1):
        c = []
        for m in range(1,10):
            k = m*(10**l-1)//9<<1
            for a, b in diop_quadratic((x-y)*(x+y-1)-k):
                if isprime(b) and a == nextprime(b):
                    c.append(b)
        yield from sorted(c)
A114370_list = list(islice(A114370_gen(),6)) # Chai Wah Wu, Jun 02 2024

cp's OEIS Frontend

A054265 Sum of composite numbers between successive primes.

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A054264 Concatenation of composite numbers between the n-th prime and the following prime.

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A054266 Sum of composite numbers between prime p and nextprime(p) is palindromic.

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A104388 Lesser of two successive primes the average of which is a repdigit.

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A054267 Sum of composite numbers between prime p and nextprime(p) is palindromic with restriction 'p + 1 <> sum'.

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A104387 Larger of two successive primes the average of which is a repdigit.

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Mathematica

Python

Formula

Extensions

A104389 Repdigits which are the average of two successive primes.

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Python

Formula

Extensions

A104386 Numbers k such that the average of the k-th and (k+1)-th primes is a repdigit.

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Programs

Python

Formula

Extensions

A114370 Primes p such that the sum of numbers from prime p to nextprime(p)-1 is a repdigit.