cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Labos Elemer May 23 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A015913, A015914, A015915, A015916, A015917, A054799.
Cf. A023200, A023201, A023202, A023203,

Programs

  • PARI
    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
  • PARI
    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

Extensions

Description corrected by Jud McCranie, May 25 2000

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

Views

Author

Robert G. Wilson v

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

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

Views

Author

Labos Elemer, May 23 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    /* 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 */

Formula

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

Views

Author

Labos Elemer, May 23 2000

Keywords

Comments

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

Examples

			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.

References

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

Crossrefs

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

Programs

Extensions

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

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

Views

Author

Labos Elemer, May 29 2000

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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)

Extensions

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

Views

Author

Labos Elemer, May 29 2000

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[5000], CompositeQ[#] && Differences@ DivisorSigma[1, {#, # + 720}] == {720} &] (* Amiram Eldar, Mar 09 2025 *)
  • PARI
    isok(k) = !isprime(k) && sigma(k + 720) == sigma(k) + 720; \\ Amiram Eldar, Mar 09 2025

A054985 Composite numbers x such that sigma(x+120) = sigma(x)+120.

Original entry on oeis.org

182, 203, 287, 350, 407, 558, 611, 731, 779, 803, 963, 1424, 1643, 2627, 2747, 3431, 3806, 4187, 4223, 5063, 6767, 7946, 8927, 9047, 11904, 12707, 12878, 15794, 18923, 20567, 27263, 31175, 32111, 34427, 43139, 43811, 45854, 50165, 52592, 57479
Offset: 1

Views

Author

Labos Elemer, May 29 2000

Keywords

Comments

See also A015914, A054799, A033560.

Examples

			a(6)=558, sigma(558)+120=1248+120=1368=sigma(678)=sigma(558+120).

Links

Crossrefs

Cf. A015914-A015917, A054799.

Programs

  • Mathematica
    Select[Range[60000],CompositeQ[#]&&DivisorSigma[1,#]+120 == DivisorSigma[ 1,#+120]&] (* Harvey P. Dale, Nov 25 2022 *)
  • PARI
    isok(n) = !isprime(n) && (sigma(n+120) == (sigma(n) + 120)); \\ Michel Marcus, Dec 31 2013

A063680 Solutions to sigma(k) + 7 = sigma(k+7).

Original entry on oeis.org

74, 531434, 387420482, 2541865828322
Offset: 1

Views

Author

Jud McCranie, Jul 28 2001

Keywords

Comments

No other solutions < 4290000000. Sequence A063679 shows how to generate more solutions, but there may be solutions other than those produced by A063679.
No others < 10^17. - Seth A. Troisi, Oct 25 2022
k or k+7 must be a square or twice a square (A028982). See comment in A015886. - Seth A. Troisi, Oct 26 2022
From Jon E. Schoenfield, Oct 26 2022: (Start)
Each of the first 4 terms of the sequence is of the form k = 9^j - 7:
74 = 9^2 - 7,
531434 = 9^6 - 7,
387420482 = 9^9 - 7,
2541865828322 = 9^13 - 7.
The next terms of this form are 9^53 - 7 and 9^82 - 7.
Does the sequence contain any terms that are not of this form?
(End)
No other terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

Examples

			sigma(74) + 7 = 121 = sigma(74+7), so 74 is in the sequence.

Crossrefs

Cf. A000203, A063679.
Cf. A054799, A015913, A015914, A015915, A015916, A015917, A054905.

Programs

  • PARI
    isok(k) = sigma(k) + 7 == sigma(k+7); \\ Michel Marcus, Oct 25 2022

Extensions

a(4) from Seth A. Troisi, Oct 24 2022

A054983 Composite numbers n such that sigma(n+24) = sigma(n) + 24.

Original entry on oeis.org

80, 95, 119, 299, 527, 962, 1247, 1479, 1739, 2783, 4307, 4958, 5240, 6015, 7878, 8342, 10379, 11639, 16967, 20687, 21439, 29294, 34547, 36917, 49022, 51959, 54707, 59807, 76127, 97319, 153242, 181427, 203318, 203822, 213419, 363302, 423999, 494882, 582902
Offset: 1

Views

Author

Labos Elemer, May 29 2000

Keywords

Comments

See also A015914, A054799, A033560

Examples

			a(1) = 80, sigma(80)+24 = 186+24 = 210 = sigma(80+24) = sigma(104) = 104+52+26+13+8+4+2+1.

Links

Crossrefs

Cf. A015914-A015917, A054799.

Programs

  • Mathematica
    With[{nn=200000},Select[Complement[Range[nn],Prime[Range[ PrimePi[nn]]]], DivisorSigma[1,#+24] == DivisorSigma[1,#]+24&]] (* Harvey P. Dale, Jan 12 2013 *)
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.