A023203 -id:A023203 - cp's OEIS frontend

A079033 First differences of A023203.

4, 6, 6, 12, 6, 6, 18, 12, 6, 18, 6, 24, 12, 18, 6, 18, 42, 6, 12, 30, 12, 24, 30, 12, 24, 6, 30, 12, 12, 6, 18, 42, 48, 30, 30, 24, 12, 30, 18, 18, 24, 18, 36, 24, 18, 24, 24, 42, 18, 30, 42, 12, 18, 12, 36, 6, 60, 18, 42, 36, 30, 12, 6, 102, 24, 6, 42, 12, 6, 54, 6, 48, 12, 18, 30, 42

Offset: 1

N. J. A. Sloane, Jul 31 2003

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053323.

N:= 10^4: # to use primes <= N
Primes:= select(isprime, {2,seq(i,i=3..N,2)}):
A023203:= sort(convert(Primes intersect map(`+`,Primes,10),list)):
A023203[2..-1] - A023203[1..-2]; # Robert Israel, Jun 02 2016

Differences[Select[Prime[Range[500]],PrimeQ[#+10]&]] (* Harvey P. Dale, May 02 2011 *)

More terms from Labos Elemer, Aug 04 2003

A086138 Number of primes between p and p+10 if both p and (p+10) are prime, i.e., number of primes somewhere between 10+A023203(n) and A023203(n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 29 2003

nonn

			a(n)=0,1,2,3 correspond to {p,p+10} prime-pairs either consecutive ones or those with various d-patterns like as follows: a(n)=0 to cases like 139[10]149; a(n)=2 to 7[4,2,4]17 etc.; a(n)=3 to one case 3[2,2,4,2]13 and a(n)=2 to cases like 31[6,4]37 or 43[4,6]53.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A023303, A031928, A031929.

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d],k=k+1; Print[cp[s+1,s+d-1]]],{n,1,1000}]; k; d=10
Count[Range[#+1,#+9],?PrimeQ]&/@Select[Prime[Range[500]],PrimeQ[#+10]&] (* _Harvey P. Dale, Jan 17 2025 *)

forprime(p=2,1e5,if(isprime(p+10), print1(isprime(p+2)+isprime(p+4)+isprime(p+6)+isprime(p+8)", "))) \\ Charles R Greathouse IV, May 15 2013

Definition clarified by Harvey P. Dale, Jan 17 2025

A031928 Lower prime of a difference of 10 between consecutive primes.

Original entry on oeis.org

139, 181, 241, 283, 337, 409, 421, 547, 577, 631, 691, 709, 787, 811, 829, 919, 1021, 1039, 1051, 1153, 1171, 1249, 1399, 1471, 1627, 1699, 1723, 1801, 1879, 2017, 2029, 2053, 2089, 2143, 2521, 2647, 2719, 2731, 2767, 2887, 2917, 3001, 3109, 3361, 3517, 3547, 3571, 3583, 3709, 3769, 3823, 3853, 4201, 4219, 4231, 4243, 4261, 4273, 4327, 4339, 4363, 4483, 4663, 4861, 4909, 4957, 5011, 5179, 5323, 5581, 5659, 5701, 5791, 5869, 6079, 6091

Offset: 1

Lekraj Beedassy, Jul 23 2003

Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely, a(n)^(1/n) is a strictly decreasing function of n (see comments at A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A023203, A053323, A248855, A290450.

[p: p in PrimesUpTo(7000) | NextPrime(p)-p eq 10]; // Bruno Berselli, Apr 09 2013

Transpose[Select[Partition[Prime[Range[800]], 2, 1], #[[2]] - #[[1]] == 10&]] [[1]] (* Harvey P. Dale, Oct 02 2014 *)
p = Prime@Range@800; p[[Flatten@Position[Differences@p, 10]]] (* Hans Rudolf Widmer, Aug 28 2022 *)

forprime(p=o=1,1e4,10+o==(o=p)&&print1(p-10",")) \\ M. F. Hasler, Mar 10 2017

a(n) = prime(A320703(n)). - R. J. Mathar, Apr 30 2024

Edited by Labos Elemer, Jul 25 2003

A046136 Primes p such that p, p+4 and p+10 are primes.

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 79, 97, 103, 127, 163, 223, 229, 307, 349, 379, 439, 457, 499, 643, 673, 853, 877, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1423, 1429, 1483, 1489, 1549, 1597, 1609, 1867, 1993, 2203, 2347, 2389, 2437, 2539, 2683, 2689, 2833, 2953

Offset: 1

Eric W. Weisstein

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A023200, A023203.

q:= p-> andmap(isprime, [p, p+4, p+10]):
select(q, [$2..3000])[];  # Alois P. Heinz, Feb 23 2020

Select[Range@ 2820, AllTrue[{#, # + 4, # + 10}, PrimeQ] &] (* Michael De Vlieger, Jul 24 2015, Version 10 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+4) && isprime(p+10), print1(p, ", "))); \\ Michel Marcus, Jul 24 2015

A023200 INTERSECT A023203. - R. J. Mathar, Jan 23 2009

A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

Original entry on oeis.org

434, 305635357, 104, 27, 195556, 65, 12, 39, 20, 56, 916, 80, 212282, 57, 44, 106645, 52, 125

Offset: 1

Labos Elemer May 23 2000

a(19) > 4293000000, if it exists. - Jud McCranie, May 25 2000

a(19) > 10^11, if it exists. - Charles R Greathouse IV, Oct 26 2022

			a(5) corresponds to n=3+2=5, d=2n=10 and the smallest composite integer is 195556. The next solution is 1152136225.

Mauro Fiorentini, Le soluzioni dell’equazione s(n + k) = s(n) + k, con n composto fino a 109 e k sino a 1000.

Cf. A015913, A015914, A015915, A015916, A015917, A054799.

Cf. A023200, A023201, A023202, A023203,

a(n)=forcomposite(x=3,10^66,if(sigma(x)+2*n==sigma(x+2*n),return(x)));
for(n=1,66,print1(a(n),", ")); \\ Joerg Arndt, Nov 15 2014

a19(lim,startAt=39)=startAt=ceil(startAt); my(v=vectorsmall(38),i=(startAt-1)%38); forfactored(n=startAt,lim\1+38, my(t=sigma(n)); if(i++>38,i=1); if(t==v[i]+38, return(n[1]-38)); v[i]=if(vecsum(n[2][,2])>1,t,0)) \\ Charles R Greathouse IV, Oct 25 2022

Description corrected by Jud McCranie, May 25 2000

A156104 Primes p such that p+36 is also prime.

Original entry on oeis.org

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607

Offset: 1

Vincenzo Librandi, Feb 08 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A156112.

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A015915 Numbers k such that sigma(k) + 8 = sigma(k+8).

Original entry on oeis.org

3, 5, 11, 23, 27, 29, 53, 59, 71, 89, 101, 131, 149, 173, 191, 233, 263, 269, 359, 389, 401, 431, 449, 479, 491, 563, 569, 593, 599, 653, 683, 701, 719, 743, 761, 821, 911, 929, 983, 1013, 1031, 1061, 1109, 1163, 1193, 1223, 1229, 1283, 1289

Offset: 1

Robert G. Wilson v

nonn

Different from A023202. Below 1000000 four composites were found [27, 1615, 1885, 218984] satisfying the "sigma(k) + 8 = sigma(k+8)" relation, together with more than 8000 primes. - Labos Elemer, May 23 2000

			sigma(27) + 8 = 48 = sigma(27+8), so 27 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000203, A015913-A015917, A023200-A023203, A046133, A001359, A054799.

Composite solutions are in A059118.

Select[Range[1300],DivisorSigma[1,#]+8==DivisorSigma[1,#+8]&] (* Harvey P. Dale, Jul 16 2011 *)

is(n)=sigma(n)+8==sigma(n+8) \\ Charles R Greathouse IV, Mar 11 2014

A015916 Numbers k such that sigma(k) + 10 = sigma(k+10).

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967, 1009

Offset: 1

Robert G. Wilson v

nonn

Different from A023203. Below 1000000 the only composite number here is 195556: sigma(195556) + 10 = 342230 + 10 = sigma(195566). - Labos Elemer, May 23 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A023203, A015913, A015914, A015915, A015917.

Select[Range[2000], DivisorSigma[1, #] + 10==DivisorSigma[1, # + 10] &] (* Vincenzo Librandi, Mar 10 2014 *)
Select[Partition[DivisorSigma[1,Range[1100]],11,1],#[[1]]+10==#[[-1]]&][[All,1]]-1 (* Harvey P. Dale, May 20 2021 *)

A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13

Offset: 1

Labos Elemer, May 23 2000

nonn

Least (prime) solutions for phi(x+2n)=phi(x)+2n seems to be identical to this sequence, while prime solutions are indeed identical to this sequence.
2nd definition = smallest number x such that phi(x+2n)=phi(x)+2n.
3rd definition = smallest primes p such that p+2n=q prime (A020483).
The 3 definitions are identical or conjectured to be identical.
The definitions are not identical if we do not take the smallest numbers. These smallest solutions are believed to be always prime numbers.
Duplicate of A020483, assuming that the 3rd definition is also correct. - R. J. Mathar, Apr 26 2015
If it can be proved that all these definitions are identical, then this entry should be merged with A020483. - N. J. A. Sloane, Feb 06 2017

			n-th primes 2,3,5,7,11,13, are solutions to sigma(x+2n)=2n+sigma(x) at 2n=2,6,22,116,88.

Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions. Marcel Dekker,Inc., New York.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A023200-A023203, A015913-A015917, A000203, A000010, A020483.

A054906 := proc(n)
    local x;
    for x from 0 do
        if numtheory[sigma](x+2*n) = numtheory[sigma](x)+2*n then
            return x;
        end if;
    end do:
end proc:
seq(A054906(n),n=1..40); # R. J. Mathar, Sep 23 2016

Table[x = 1; While[DivisorSigma[1, x + 2 n] != DivisorSigma[1, x] + 2 n, x++]; x, {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

a(n) = my(x = 1); while(sigma(x+2*n) != sigma(x)+2*n, x++); x; \\ Michel Marcus, Dec 17 2013

Minimal solutions to A000203(x+2n)=A000203(x)+2n or to A000010(x+2n)=A000010(x)+2n or to p+2n=q; p, q primes, a(n)=p.

a(n) <= A054905(n). - R. J. Mathar, Apr 28 2015

A092146 Primes of the form p + 10 where p is a prime.

Original entry on oeis.org

13, 17, 23, 29, 41, 47, 53, 71, 83, 89, 107, 113, 137, 149, 167, 173, 191, 233, 239, 251, 281, 293, 317, 347, 359, 383, 389, 419, 431, 443, 449, 467, 509, 557, 587, 617, 641, 653, 683, 701, 719, 743, 761, 797, 821, 839, 863, 887, 929, 947, 977, 1019, 1031

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Mar 31 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A006512, A023203, A046132, A046117, A092402.

lst={};Do[p=Prime[n];If[PrimeQ[p=p+10],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Prime[Range[200]]+10,PrimeQ] (* Harvey P. Dale, Aug 03 2018 *)

a(n) = 10 + A023203(n). - Alois P. Heinz, Feb 27 2020

cp's OEIS Frontend

A079033 First differences of A023203.

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A086138 Number of primes between p and p+10 if both p and (p+10) are prime, i.e., number of primes somewhere between 10+A023203(n) and A023203(n).

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A031928 Lower prime of a difference of 10 between consecutive primes.

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A046136 Primes p such that p, p+4 and p+10 are primes.

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A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

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A156104 Primes p such that p+36 is also prime.

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Magma

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A015915 Numbers k such that sigma(k) + 8 = sigma(k+8).

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A015916 Numbers k such that sigma(k) + 10 = sigma(k+10).

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