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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A050225 1/3-Smith numbers.

Original entry on oeis.org

6969, 19998, 36399, 39693, 66099, 69663, 69897, 89769, 99363, 99759, 109989, 118899, 181998, 191799, 199089, 297099, 306939, 333399, 336963, 339933, 363099, 396363, 397998, 399333, 399729, 588969, 606666, 606909, 639633, 660693, 666633
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Examples

			6969 is a 3^(-1) Smith number because the digit sum of 6969, i.e., S(6969) = 6 + 9 + 6 + 9 = 30, which is equal to 3 times the sum of the digits of its prime factors, i.e., 3*Sp(6969) = 3 * Sp(3 * 23 * 101) = 3 *( 3 + 2 + 3 + 1 + 0 + 1) = 30.

Links

Crossrefs

Cf. A006753, A050224.

Programs

  • Mathematica
    digSum[n_] := Plus @@ IntegerDigits[n]; thirdSmithQ[n_] := CompositeQ[n] && 3 * Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; Select[Range[666633], thirdSmithQ] (* Amiram Eldar, Aug 23 2020 *)

Extensions

More terms from Shyam Sunder Gupta, Mar 11 2005

A067173 Numbers n such that the sum of the prime factors of n equals the product of the digits of n.

Original entry on oeis.org

2, 3, 5, 7, 126, 154, 315, 329, 342, 418, 442, 1134, 1826, 2354, 3383, 4343, 5282, 5561, 6623, 7515, 7922, 9331, 9911, 12773, 13344, 14161, 15194, 17267, 18292, 21479, 22831, 26216, 26522, 29812, 32129, 33128, 33912, 57721, 81191, 81524
Offset: 1

Views

Author

Joseph L. Pe, Feb 18 2002

Keywords

Examples

			The prime factors of 315 are 3,5,7, which sum to 15, the product of the digits of 315, so 315 is a term of the sequence.

Links

Crossrefs

Cf. A065774, A006753.

Programs

  • Mathematica
    f[n_] := Module[{a, l, t, r}, a = FactorInteger[n]; l = Length[a]; t = Table[a[[i]][[1]], {i, 1, l}]; r = Sum[t[[i]], {i, 1, l}]]; g[n_] := Module[{b, m, s}, b = IntegerDigits[n]; m = Length[b]; s = Product[b[[i]], {i, 1, m}]]; Select[Range[10^5], f[ # ] == g[ # ] &]
Select[Range[2,100000],Total[FactorInteger[#][[All,1]]] == Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Feb 15 2017 *)

A103123 1/4-Smith numbers.

Original entry on oeis.org

19899699, 36969999, 36999699, 39699969, 39999399, 39999993, 66699699, 66798798, 67967799, 67987986, 69759897, 69889389, 69966699, 69996993, 76668999, 79488798, 79866798, 85994799, 86686886, 89769759, 89866568
Offset: 1

Views

Author

Shyam Sunder Gupta, Mar 16 2005

Keywords

Examples

			19899699 is a 4^(-1) Smith number because the digit sum of 19899699, i.e., S(19899699) = 1 + 9 + 8 + 9 + 9 + 6 + 9 + 9 = 60, which is equal to 4 times the sum of the digits of its prime factors, i.e., 4*Sp(19899699) = 4*Sp (3*2203*3011) = 4*(3 + 2 + 2 + 0 + 3 + 3 + 0 + 1 + 1) = 15.

Links

Crossrefs

Cf. A006753.

Programs

  • Mathematica
    digSum[n_] := Plus @@ IntegerDigits[n]; qSmithQ[n_] := CompositeQ[n] && 4 * Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) ==  digSum[n]; Select[Range[10^8], qSmithQ] (* Amiram Eldar, Aug 23 2020 *)

A103125 4-Smith numbers.

Original entry on oeis.org

2401, 5010, 7000, 10005, 10311, 10410, 10411, 11060, 11102, 11203, 12103, 13002, 13021, 13101, 14001, 14101, 14210, 20022, 20121, 20203, 20401, 21103, 21112, 21120, 21201, 22040, 22101, 22201, 23030, 30003, 30031, 30320, 31002, 31101
Offset: 1

Views

Author

Shyam Sunder Gupta, Mar 16 2005

Keywords

Examples

			2401 is a 4-Smith number because the sum of the digits of its prime factors, i.e., Sp(2401) = Sp(7*7*7*7) = 7 + 7 + 7 + 7 = 28, which is equal to 4 times the digit sum of 2401, i.e., 4*S(2401) = 4*(2 + 4 + 0 + 1) = 28.

Links

Crossrefs

Cf. A006753.

Programs

  • Mathematica
    sn4Q[n_]:=Module[{a=Total[Flatten[IntegerDigits/@(Table[First[#],{Last[ #]}]&/@FactorInteger[n])]],b=4Total[IntegerDigits[n]]},a==b] (* Harvey P. Dale, Oct 03 2011 *)

A103126 5-Smith numbers.

Original entry on oeis.org

2030, 10203, 12110, 20210, 20310, 21004, 21010, 24000, 24010, 31010, 41001, 50010, 70000, 100004, 100012, 100210, 100310, 100320, 101020, 101041, 102022, 103200, 104010, 104101, 104110, 105020, 106001, 110020, 110202, 110212, 110400, 111013
Offset: 1

Views

Author

Shyam Sunder Gupta, Mar 16 2005

Keywords

Examples

			2030 is a 5-Smith number because the sum of the digits of its prime factors, i.e., Sp(2030) = Sp(2*5*7*29) = 2 + 5 + 7 + 2 + 9 = 25, which is equal to 5 times the digit sum of 2030, i.e., 5*S(2030) = 5*(2 + 0 + 3 + 0) = 25.

Links

Crossrefs

Cf. A006753.

Programs

  • Mathematica
    digSum[n_] := Plus @@ IntegerDigits[n]; fiveSmithQ[n_] := CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == 5 *digSum[n]; Select[Range[10^5], fiveSmithQ] (* Amiram Eldar, Aug 23 2020 *)

A230354 Even numbers n such that digit sum of n = digit sum of largest odd divisor of n.

Original entry on oeis.org

12, 18, 36, 54, 60, 72, 90, 108, 126, 132, 144, 156, 162, 180, 198, 204, 216, 228, 234, 240, 252, 270, 276, 306, 320, 324, 342, 348, 360, 372, 378, 396, 414, 420, 432, 450, 504, 516, 522, 540, 558, 594, 612, 624, 630, 636, 660, 702, 708, 720, 732, 738, 756, 774, 780, 792, 810, 900
Offset: 1

Views

Author

Antonio Roldán, Oct 16 2013

Keywords

Examples

			Largest odd divisor of 162 is 81. Digit_sum(162)=9, digit_sum(81)=9

Crossrefs

Cf. A006753, A219340, A230355, A230356, A230357.

Programs

  • PARI
    mdi(n)= n / 2^valuation(n, 2)
digsum(n)={local (d, p); d=0; p=n; while(p, d+=p%10; p=floor(p/10)); return(d)}
{for (n=2, 10^3,m=mdi(n);if(digsum(n)==digsum(mdi(n))&&m<>n,print(n)));}

A230355 Nonsquarefree numbers n such that digit sum of n = digit sum of squarefree part of n.

Original entry on oeis.org

12, 24, 60, 100, 120, 132, 150, 156, 200, 204, 228, 240, 264, 276, 300, 320, 348, 372, 420, 500, 516, 552, 600, 624, 636, 660, 700, 708, 732, 744, 780, 912, 1000, 1014, 1050, 1056, 1068, 1092, 1100, 1128, 1164, 1200, 1212, 1216, 1236, 1248, 1272, 1300, 1308, 1320, 1356, 1380, 1392, 1400
Offset: 1

Views

Author

Antonio Roldán, Oct 16 2013

Keywords

Examples

			Squarefree part of 624=2^4*3*13 is 39. Digit_sum(624)=12, digit_sum(39)=12

Crossrefs

Cf. A006753, A230354, A230356, A230357.

Programs

  • PARI
    digsum(n)={local (d, p); d=0; p=n; while(p, d+=p%10; p=floor(p/10)); return(d)}
{for (n=4, 10^3,m=core(n);if(digsum(n)==digsum(m)&&m<>n,print(n)));}

A230356 Nonsquare numbers n such that digit sum of n = digit sum of square part of n.

Original entry on oeis.org

10, 18, 27, 40, 45, 54, 63, 72, 90, 108, 117, 126, 135, 153, 160, 162, 171, 180, 207, 216, 220, 234, 243, 250, 252, 261, 270, 304, 306, 315, 333, 342, 351, 360, 405, 414, 423, 432, 450, 490, 504, 513, 522, 531, 540, 603, 612, 621, 630, 640, 702, 711, 720, 801, 810, 931
Offset: 1

Views

Author

Antonio Roldán, Oct 16 2013

Keywords

Examples

			135 = 2^3*5. Square part of 135 is 9. Digit_sum(135) =9, digit_sum(9) = 9.

Crossrefs

Cf. A006753, A230354, A230355, A230357.

Programs

  • PARI
    digsum(n)={local (d, p); d=0; p=n; while(p, d+=p%10; p=floor(p/10)); return(d)}
{for (n=2, 10^3,m=n/core(n);if(digsum(n)==digsum(m)&&m<>n,print(n)));}

A230357 Numbers n such that digit sum of n equals digit sum of sopf(n) (sum of the distinct prime factors of n).

Original entry on oeis.org

22, 94, 105, 114, 136, 140, 160, 166, 202, 222, 234, 250, 265, 274, 346, 355, 361, 382, 424, 438, 445, 454, 516, 517, 526, 532, 562, 634, 702, 706, 712, 732, 812, 913, 915, 922, 1036, 1071, 1086, 1111, 1116, 1122, 1138, 1165, 1185, 1204, 1206, 1219, 1221, 1230, 1239, 1255, 1282, 1312, 1316, 1318, 1345, 1363, 1400, 1404, 1432, 1507, 1520, 1530, 1550
Offset: 1

Views

Author

Antonio Roldán, Oct 16 2013

Keywords

Examples

			166=2*83. Sopf(166)=85. Digit_sum(166)=13, digit_sum(85)=13.

Crossrefs

Cf. A006753, A036920, A230354, A230355, A230356.

Programs

  • PARI
    sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
digsum(n)={local (d, p); d=0; p=n; while(p, d+=p%10; p=floor(p/10)); return(d)}
{for (n=4, 2*10^3,m=sopf(n);if(digsum(n)==digsum(m)&&m<>n,print(n)))}

A329936 Binary hoax numbers: composite numbers k such that sum of bits of k equals the sum of bits of the distinct prime divisors of k.

Original entry on oeis.org

4, 8, 9, 15, 16, 32, 45, 49, 50, 51, 55, 64, 75, 85, 100, 117, 126, 128, 135, 153, 159, 162, 171, 185, 190, 200, 205, 207, 215, 222, 225, 238, 246, 249, 252, 253, 256, 287, 303, 319, 324, 333, 338, 350, 369, 374, 378, 380, 400, 407, 438, 442, 444, 469, 471
Offset: 1

Views

Author

Amiram Eldar, Nov 24 2019

Keywords

Comments

Analogous to A278909 (binary Smith numbers) as A019506 (hoax numbers) is analogous to A006753 (Smith numbers).
Includes all the powers of 2 except for 1 and 2.

Examples

			4 = 2^2 is in the sequence since the binary representation of 4 is 100 and 1 + 0 + 0 = 1, and the binary representation of 2 is 10 and 1 + 0 = 1.

Links

Crossrefs

Cf. A006753, A019506, A278909.

Programs

  • Maple
    filter:= proc(n)
if isprime(n) then return false fi;
convert(convert(n,base,2),`+`) = add(convert(convert(t,base,2),`+`),t=numtheory:-factorset(n))
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 28 2019
  • Mathematica
    binWt[n_] := Total @ IntegerDigits[n, 2]; binHoaxQ[n_] := CompositeQ[n] && Total[binWt /@ FactorInteger[n][[;; , 1]]] == binWt[n]; Select[Range[500], binHoaxQ]
  • PARI
    is(n)= my(f=factor(n)[,1]); sum(i=1,#f, hammingweight(f[i]))==hammingweight(n) && !isprime(n) \\ Charles R Greathouse IV, Nov 28 2019
Previous Showing 31-40 of 90 results. Next

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