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.

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A105649 Smallest member of set of 4 consecutive numbers which are Smith numbers.

Original entry on oeis.org

4463535, 6356910, 8188933, 9425550, 11148564, 15966114, 15966115, 18542654, 21673542, 22821992, 23767287, 28605144, 36615667, 39227466, 47096634, 47395362, 48072396, 54054264, 55464835, 57484614, 57756450, 57761165, 58418508, 61843387, 62577157, 64572186, 65484066
Offset: 1

Views

Author

Shyam Sunder Gupta, May 03 2005

Keywords

Examples

			a(1) = 4463535 because 4463535 is the smallest member of a set of 4 consecutive numbers which are Smith numbers i.e. four consecutive numbers 4463535, 4463536, 4463537, 4463538 are all Smith numbers.

Links

Crossrefs

Cf. A006753, A050219, A059754, A105648, A105650.

Programs

  • Mathematica
    digSum[n_] := Plus @@ IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last @#* digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; sm = smithQ /@ Range[4]; seq = {}; Do[sm = Join[Rest[sm], {smithQ[k]}]; If[And @@ sm, AppendTo[seq, k - 3]], {k, 5, 10^7}]; seq (* Amiram Eldar, Aug 18 2020 *)

Extensions

a(7) inserted and more terms added by Amiram Eldar, Aug 18 2020

A105650 Smallest member of set of 5 consecutive numbers which are Smith numbers.

Original entry on oeis.org

15966114, 75457380, 162449165, 296049306, 296861735, 334792990, 429619207, 581097690, 581519244, 582548088, 683474015, 809079150, 971285861, 977218716, 1192453925, 1280731565, 1388562581, 1429560378, 1445388905, 1574950548, 1884167184, 1927281663, 2050918644
Offset: 1

Views

Author

Shyam Sunder Gupta, May 03 2005

Keywords

Examples

			a(1) = 15966114 because 15966114 is the smallest member of a set of 5 consecutive numbers which are Smith numbers i.e. five consecutive numbers 15966114, 15966115, 15966116, 15966117, 15966118 are all Smith numbers.

Links

Crossrefs

Cf. A006753.

Extensions

a(15)-a(23) from Donovan Johnson, Dec 31 2012

A170928 Least magic constant of magic squares using Smith numbers.

Original entry on oeis.org

822, 1195, 1636, 2472, 3720, 5856, 8737, 12202, 16335, 21333, 27612, 35185, 43968, 54013, 65464, 78281, 92422, 107932, 126404, 147816, 171556, 197041, 224506, 253587, 285314, 320620, 359151, 400064, 442886, 487920, 536844, 589129, 644797
Offset: 3

Views

Author

Stefano Tognon, Feb 04 2010

Keywords

Comments

a(n) >= (1/n)*Sum_{i=1..n^2} A006753(i).

Examples

			Magic square of order 3: see the book: M. Gardner. From the Penrose tilings to securely encrypted, 1993:
   94 382 346
  526 274  22
  202 166 454
.
The magic constant S = 822
Orders 4 to 6 are from participants of scientific forum dxdy.ru
The square of order 4:
   22 346 562 265
  778 274  85  58
    4 454 382 355
  391 121 166 517
.
S = 1195
The square of order 5:
  355 576   4 319 382
  454  85 391 648  58
   27 535 346 526 202
  706 166 378 121 265
   94 274 517  22 729
.
S = 1636
The square of order 6:
  729   4 636 762  22 319
   27 663 654 526  85 517
  391 645  58 378 438 562
  382 346 454 121 634 535
  355 648  94 483 627 265
  588 166 576 202 666 274

Links

Extensions

a(7), a(9) added by Natalia Makarova, Apr 02 2010
Edited by Max Alekseyev, May 26 2012

A213239 Numbers n such that sum of digits of n = sum of digits of anti-divisors of n.

Original entry on oeis.org

5, 8, 64, 691, 1779, 2851, 6361, 9066, 9606, 9771, 10789, 10996, 18996, 21481, 22569, 27529, 27691, 31516, 36709, 36776, 42649, 48651, 53296, 56586, 58749, 60369, 64794, 72889, 76754, 78766, 79374, 79896, 80989, 86596, 90606, 90879, 92766, 96171, 98979, 108529
Offset: 1

Views

Author

Paolo P. Lava, Jun 07 2012

Keywords

Examples

			Sum of digits of 1779 is 1+7+7+9=24.
Anti-divisors of 1779 are 2, 6, 1186 and their digits’ sum is 2+6+1+1+8+6=24.

Crossrefs

Cf. A006753, A066417, A213240.

Programs

  • Maple
    with(numtheory);
A213239:=proc(q)
local a,b,c,d,k,n;
for n from 1 to q do
  a:=0; b:=0;
  for k from 2 to n-1 do
    if abs((n mod k)-k/2)<1 then
       c:=k; while c>0 do b:=b+(c mod 10); c:=trunc(c/10); od; fi; od;
  c:=n; d:=0; while c>0 do d:=d+(c mod 10); c:=trunc(c/10); od;
  if b=d then print(n); fi; od; end:
A213239(100000);
  • Python
    [n for n in range(1,10**5) if sum([sum([int(x) for x in str(d)]) for d in range(2,n) if n % d and 2*n % d in [d-1,0,1]]) == sum([int(x) for x in str(n)])] # Chai Wah Wu, Aug 08 2014

A213240 Numbers n such that sum of digits of n = sum of digits of n’, where n’ is the arithmetic derivative of n.

Original entry on oeis.org

4, 22, 27, 54, 56, 81, 94, 108, 121, 135, 166, 202, 216, 243, 245, 260, 265, 266, 272, 274, 308, 344, 346, 355, 382, 405, 445, 454, 459, 513, 517, 518, 526, 562, 567, 594, 634, 648, 675, 702, 706, 729, 784, 806, 832, 837, 841, 891, 913, 920, 922, 945, 950, 972
Offset: 1

Views

Author

Paolo P. Lava, Jun 07 2012

Keywords

Examples

			n=344 and 3+4+4=11; n’=524 and 5+2+4=11.

Links

Crossrefs

Cf. A003415, A006753, A213239

Programs

  • Maple
    with(numtheory);
A213240:=proc(q)
local a,b,d,n,p,pfs;
for n from 1 to q do
  pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs); b:=0;
  while a>0 do b:=b+(a mod 10); a:=trunc(a/10); od;
  a:=n; d:=0; while a>0 do d:=d+(a mod 10); a:=trunc(a/10); od;
  if b=d then print(n); fi;
od; end:
A213240(100000000);

A329935 Numbers k such that k and k+1 are both hoax numbers (A019506).

Original entry on oeis.org

84, 516, 644, 860, 2325, 3344, 4188, 4980, 5268, 5484, 6259, 6603, 6692, 6980, 7051, 7195, 8076, 8420, 9716, 10704, 11774, 12795, 12955, 12956, 13747, 14475, 14715, 14724, 16473, 17148, 17149, 17225, 17661, 19175, 21828, 22143, 22347, 24259, 24272, 24980
Offset: 1

Views

Author

Amiram Eldar, Nov 24 2019

Keywords

Comments

Analogous to A050219 (smaller of Smith brothers) as A019506 (hoax numbers) is analogous to A006753 (Smith numbers).

Examples

			84 is in the sequence since 84 is a hoax number: 84 = 2^2 * 3 * 7 and 8 + 4 = 2 + 3 + 7 = 12, and 85 = 84 + 1 is also a hoax number: 85 = 5 * 17 and 8 + 5 = 5 + 1 + 7 = 13.

Links

Crossrefs

Cf. A006753, A019506, A050219, A235766.

Programs

  • Mathematica
    digitSum[n_]  := Total @ IntegerDigits[n]; hoaxQ[n_] := CompositeQ[n] && Total[ digitSum /@ FactorInteger[n][[;; , 1]] ] == digitSum[n]; seq = {}; isHoax1 = hoaxQ[1]; Do[isHoax2 = hoaxQ[n]; If[isHoax1 && isHoax2, AppendTo[seq, n-1]]; isHoax1 = isHoax2, {n, 2, 25000}]; seq

A334528 Palindromic numbers that are also Niven numbers and Smith numbers.

Original entry on oeis.org

4, 666, 28182, 45054, 51315, 82628, 239932, 454454, 864468, 2594952, 2976792, 3189813, 3355533, 4172714, 4890984, 5319135, 5367635, 5777775, 7149417, 7247427, 8068608, 8079708, 8100018, 8280828, 8627268, 9227229, 9423249, 21699612, 22544522, 24166142, 27677672
Offset: 1

Views

Author

Amiram Eldar, May 05 2020

Keywords

Comments

Witno (2014) proved that this sequence is infinite.

Examples

			666 is a term since it is palindromic, a Niven number (6 + 6 + 6 = 18 is a divisor of 666) and a Smith number (666 = 2 * 3 * 3 * 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7).

Links

Crossrefs

Intersection of A002113, A005349 and A006753.
Intersection of any two of the sequences A082232, A098834 and A334527.

Programs

  • Mathematica
    digSum[n_] := Plus @@ IntegerDigits[n]; palNivenSmithQ[n_] := PalindromeQ[n] && Divisible[n, (ds = digSum[n])] && CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == ds; Select[Range[10^5], palNivenSmithQ]

A033662 Possible digital sums of Smith numbers (conjectural).

Original entry on oeis.org

4, 9, 13, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Numbers not in sequence are not digital sums of Smith numbers. Numbers in sequence are almost certainly so.

Crossrefs

Cf. A006753.

Formula

a(n) in {4, 9, 13, 15} or a(n) >= 17

A063844 Smith numbers which are also base-2 pseudoprimes.

Original entry on oeis.org

645, 4369, 13747, 15709, 88357, 157641, 642001, 656601, 1507963, 2269093, 2313697, 4101637, 7428421, 7429117, 8388607, 22669501, 26296401, 27218269, 27336673, 28011001, 32701297, 34487601, 36507801, 37167361, 47903701, 54215161, 71804161, 72498253, 74411131, 82279741, 86438857, 86530621, 93614521, 96135601, 97863529
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 25 2001

Keywords

Examples

			645 is a member since 645=3*5*43, sum of digits of 645 is 6+4+5=15, sum of digits of prime factors = 3+5+4+3=15 and 2^644 (mod 645) == 1.

Links

Crossrefs

Intersection of A001567 and A006753.

Programs

  • Mathematica
    fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[#[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; id != fid && Plus @@ id == Plus @@ fid]; k = 1; lst = {}; While[k < 10^8, If[!PrimeQ@ k && PowerMod[2, k-1, k] == 1, AppendTo[lst, k]]; k++]; Select[ lst, fQ]

A098835 Smith abundant numbers.

Original entry on oeis.org

378, 438, 576, 588, 636, 648, 654, 666, 690, 728, 762, 852, 1086, 1284, 1376, 1626, 1736, 1776, 1842, 1872, 1908, 1952, 1962, 2286, 2484, 2556, 2576, 2688, 2934, 2944, 2958, 2964, 2970, 3138, 3168, 3174, 3246, 3258, 3294, 3366, 3390, 3564, 3690, 3852
Offset: 1

Views

Author

Shyam Sunder Gupta, Oct 10 2004

Keywords

Examples

			a(1) = 378 because 378 is a Smith number as well as an abundant number.

Links

Crossrefs

Intersection of A005101 and A006753.

Programs

  • Mathematica
    digSum[n_] := Plus @@ IntegerDigits[n]; abSmithQ[n_] := DivisorSigma[1, n] > 2*n && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; Select[Range[4000], abSmithQ] (* Amiram Eldar, Aug 23 2020 *)
Previous Showing 51-60 of 90 results. Next

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

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