A054799 -id:A054799 - cp's OEIS frontend

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

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

A098974 Primes p such that q-p = 24, where q is the next prime after p.

Original entry on oeis.org

1669, 2179, 4177, 4523, 4759, 5237, 6173, 6397, 6737, 7079, 7369, 7793, 8123, 8329, 9067, 11003, 11633, 11839, 12073, 12119, 13009, 13267, 16033, 16193, 16453, 16763, 16787, 17053, 17683, 17989, 18593, 18637, 19183, 19507, 20483, 22409, 22877, 23227

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 23 2004

Lower prime of a difference of 24 between consecutive primes.

23 successive numbers after prime number p are composite. - Artur Jasinski, Jan 15 2007

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A000040, A001223, A054541, A075526, A001359, A054799, A063091, A096292, A015913, A023200, A029710, A031934, A031936, A031938.

Cf. A000230, A023200, A031924, A031926, A031928, A031930, A031932, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Artur Jasinski, Jan 15 2007 *)

Entry revised by N. J. A. Sloane, Feb 13 2007

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

A050507 Solutions to sigma(x)+2=sigma(x+2) other than the smaller of twin primes.

Original entry on oeis.org

434, 8575, 8825

Offset: 1

Jud McCranie, Dec 27 1999

This sequence together with A001359 gives the solutions of sigma(x)+2 = sigma(x+2).
No others < 4.29*10^9.
No others < 5*10^10. - Charles R Greathouse IV, Oct 19 2010
They are also the solutions of A001065(x) = A001065(x+2), where A001065(n) is the sum of proper divisors of n. - Michel Marcus, Nov 14 2014
Makowski found these 3 solutions and verified that there are none others with x <= 9998. Haukkanen extended the bound to 2*10^8. - Amiram Eldar, Dec 28 2018
a(4) > 10^13, if it exists. - Giovanni Resta, Dec 12 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			sigma(434)+2=770=sigma(434+2), so 434 is in the sequence.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004, chapter B13, p. 104.
R. Sivaramakrishnan, Classical Theory of Arithmetical Functions, M. Dekker Inc., New York-Basel, 1989, p. 81, Problem 12.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62, Nos. 1-4 (1993), pp. 166-168.
Andrzej Makowski On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670.

Cf. A000203, A001359, A054799.

Select[Range[10000], CompositeQ[#] && DivisorSigma[1, #] + 2 == DivisorSigma[1, # + 2] &] (* Amiram Eldar, Dec 28 2018 *)

is(n)=sigma(n+2)==sigma(n)+2&&!isprime(n) \\ Charles R Greathouse IV, Feb 13 2013

A054902 Composite numbers n such that sigma(n)+12 = sigma(n+12).

Original entry on oeis.org

65, 170, 209, 1394, 3393, 4407, 4556, 11009, 13736, 27674, 38009, 38845, 47402, 76994, 157994, 162393, 184740, 186686, 209294, 680609, 825359, 954521, 1243574, 2205209, 3515609, 4347209, 5968502, 6539102, 6916241, 8165294, 10352294, 10595009, 10786814

Offset: 1

Labos Elemer, May 23 2000

nonn

			n = 65, sigma(65)+12 = 84+12 = 96 = sigma(65+12) = sigma(77).
n = 11009, sigma(11009)+12 = 11220+12 = 11232 = sigma(11009+12) = sigma(11021).

Donovan Johnson, Table of n, a(n) for n = 1..500

Complement of A046133 with respect to A015917.

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

isok(n) = !isprime(n) && ((sigma(n)+12) == sigma(n+12)); \\ Michel Marcus, Dec 18 2013

More terms from Jud McCranie, May 24 2000

Three more terms from Michel Marcus, Dec 18 2013

A054904 x = a(n) is the smallest composite number such that sigma(x+6n) = sigma(x)+6n, where sigma = A000203.

Original entry on oeis.org

104, 65, 20, 80, 44, 125, 45, 63, 40, 99, 56, 70, 296, 125, 88, 110, 104, 145, 212, 182, 80, 170, 333, 105, 369, 185, 184, 135, 180, 301, 356, 185, 1859, 329, 176, 195, 4916, 434, 612, 287, 140, 185, 776, 255, 524, 413, 344, 205, 272, 329, 567, 215, 320, 469

Offset: 1

Labos Elemer, May 23 2000

nonn

If sigma(x+d) = sigma(x)+d and d = 6k, then composite solutions seem to be more frequent and arise sooner.

a(725) > 3*10^11 (if it exists). - Donovan Johnson, Sep 23 2013

			n = 20, 6n = 120, a(20) = 182, sigma(182)+120 = 336+120 = 456 = sigma(182+120) = sigma(302).

Donovan Johnson, Table of n, a(n) for n = 1..724

Cf. A015914-A015917, A023200-A023203, A054799, A054903-A054905.

Table[x = 4; While[Nand[CompositeQ@ x, DivisorSigma[1, x + 6 n] == DivisorSigma[1, x] + 6 n], x++]; x, {n, 54}] (* Michael De Vlieger, Feb 18 2017 *)

/* finds first 696 terms */ mx=7695851; s=vector(mx); for(j=4, mx, if(isprime(j)==0, s[j]=sigma(j))); for(n=1, 696, n6=n*6; for(x=4, 7691753, if(s[x]>0, if(s[x+n6]==s[x]+n6, write("b054904.txt", n " " x); next(2))))) /* Donovan Johnson, Sep 23 2013 */

sigma(x+6n) = sigma(x)+6n, a(n) = min(x) and it is composite.

A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

Original entry on oeis.org

104, 147, 596, 1415, 4850, 5337, 370047, 1630622, 35020303, 120221396, 3954451796, 742514284703

Offset: 1

Labos Elemer, May 23 2000

Complement of A023201 with respect to A015914.
Intersection of A015914 and A018252.
Below 1000000 there are only 7 such composite numbers, compared with more than 16000 primes.
a(13) > 10^13. - Giovanni Resta, Jul 11 2013

			n=104, sigma(104)+6 = 210+6 = 216 = sigma(104+6) = sigma(110).
a(4) = 1415 = 5*283, 1415+6 = 1421 = 7*7*29:
sigma(1415) = 1+5+283+1415 = 1704,
sigma(1421) = 1+7+29+49+203+1421 = 1710 = sigma(1415)+6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 104, p. 37, Ellipses, Paris 2008.

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

forcomposite(n=9,1e7,if(sigma(n)+6==sigma(n+6),print1(n", "))) \\ Charles R Greathouse IV, Feb 14 2013

More terms from Jud McCranie, May 25 2000
New definition from Reinhard Zumkeller, Jan 27 2009
Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of R. J. Mathar.
a(12) from Giovanni Resta, Jul 11 2013

A054987 Smallest composite x such that sigma(x+2^n) = sigma(x) + 2^n.

Original entry on oeis.org

434, 305635357, 27, 39, 106645, 69, 2275, 63, 6475, 249, 7735, 3703, 10803, 16383, 58869, 51181, 87951, 1695, 9579, 105237, 98829, 1143369, 789609, 11625, 14038691, 178975, 48627929, 1881333, 402373721, 2667945, 82915599, 353195221, 70106601

Offset: 1

Labos Elemer, May 29 2000

nonn

The sequence is initiated by smallest of A015915. Special primes of A023202, A049488-A049491 also satisfy the Sigma[p+2^w]=Sigma[p]+2^w relation

			For the term 69: Sigma[69+2^6] = Sigma[133] = 1+7+19+133 = Sigma[69]+64 = (1+3+23+69)+64 = 160.

Cf. A049488-A049491, A001359, A054799, A015913, A015915, A023200, A023202, A054905.

Table[ Select[ Range[ 1, 110000 ], Equal[ EulerPhi[ #+2^k ]-EulerPhi[ # ]-2^k, 0 ] &&!PrimeQ[ # ]& ], {k, 1, 22} ]

a(n)=my(N=2^n,x=3); while(isprime(x++) || sigma(x+N) != sigma(x)+N,); x \\ Charles R Greathouse IV, Mar 11 2014

More terms from Labos Elemer, Aug 14 2003

a(21) corrected and a(27)-a(33) from Donovan Johnson, Nov 30 2008

A054982 a(n) = least composite number such that sigma(a(n)+n!) = sigma(a(n))+n! where sigma() = A000203.

Original entry on oeis.org

434, 104, 80, 182, 427, 1727, 4147, 7163, 42031, 165841, 569257, 2683909, 10040081, 39094849, 155533969, 717519401, 3041377519, 16076525809, 71749935913

Offset: 2

Labos Elemer, May 29 2000

a(21) <= 328823468719, a(22) <= 1542201899569, a(23) <= 9325753929619. - Donovan Johnson, Sep 22 2013

			a(7) = 1727 = 11*157, 4 divisors, sigma(1727)+5040 = 1896+5040 = 6936, sigma(1727+5040) = sigma(6767) = 1+67+101+6767 = 6936.
a(2) = A054799(24) = 434, a(3) = A015914(19) = 104, the first composites in that series.

Cf. A054904, A054905, A054799, A015914, A033560.

L = {}; Do[i = 1; While[ ! ((Plus @@ Divisors[i + j! ] == j! + Plus @@ Divisors[i]) && ! PrimeQ[i]), i++ ]; L = Append[L, i], {j, 2, 13}]; L (from Vit Planocka)

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 22 2003
a(14)-a(19) from Donovan Johnson, Nov 30 2008
a(20) from Donovan Johnson, Sep 19 2013

A054984 Composite numbers k such that sigma(k + 6!) = sigma(k + 720) = sigma(k) + 720.

Original entry on oeis.org

427, 553, 595, 623, 737, 871, 913, 923, 1199, 1207, 1241, 1507, 1582, 1817, 1848, 2193, 2226, 2337, 2398, 2407, 2553, 2561, 2728, 2758, 2929, 3016, 3115, 3248, 3346, 3502, 3503, 3598, 3705, 3762, 4171, 4293, 4343, 4462, 4587, 4633, 4841, 4867, 4984

Offset: 1

Labos Elemer, May 29 2000

nonn

			553 is a term because sigma(553) + 720 = 640 + 720 = 1360 = sigma(553 + 720) = sigma(1273) = 1 + 19 + 67 + 1273.

Amiram Eldar, Table of n, a(n) for n = 1..6000

Cf. A015914, A015915, A015916, A015917, A033560, A033561, A054799.

Select[Range[5000], CompositeQ[#] && Differences@ DivisorSigma[1, {#, # + 720}] == {720} &] (* Amiram Eldar, Mar 09 2025 *)

isok(k) = !isprime(k) && sigma(k + 720) == sigma(k) + 720; \\ Amiram Eldar, Mar 09 2025

cp's OEIS Frontend

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

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A098974 Primes p such that q-p = 24, where q is the next prime after p.

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

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A050507 Solutions to sigma(x)+2=sigma(x+2) other than the smaller of twin primes.

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A054902 Composite numbers n such that sigma(n)+12 = sigma(n+12).

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A054904 x = a(n) is the smallest composite number such that sigma(x+6n) = sigma(x)+6n, where sigma = A000203.

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A054903 Composite numbers n such that sigma(n)+6 = sigma(n+6), where sigma=A000203.

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A054987 Smallest composite x such that sigma(x+2^n) = sigma(x) + 2^n.