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 1-10 of 14 results. Next

A076533 Numbers n such that sum of the distinct prime factors of phi(n) = sum of the distinct prime factors of sigma(n).

Original entry on oeis.org

1, 3, 14, 35, 42, 70, 105, 119, 209, 210, 238, 248, 297, 357, 412, 418, 477, 594, 595, 616, 627, 714, 744, 954, 1045, 1142, 1178, 1190, 1236, 1240, 1254, 1328, 1339, 1463, 1485, 1672, 1674, 1703, 1736, 1785, 1848, 1863, 2079, 2090, 2376, 2385, 2540, 2728
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			sopf(sigma(14)) = 5; sopf(phi(14)) = 5; hence 14 is a term of the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075784, A075846, A076525, A076527, A076531, A076532.

Programs

  • Mathematica
    p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[3, 10^4], p[DivisorSigma[1, # ]] == p[EulerPhi[ # ]] &]
Select[Range[3000],Total[FactorInteger[DivisorSigma[1,#]][[All,1]]] == Total[ FactorInteger[EulerPhi[#]][[All,1]]]&] (* Harvey P. Dale, Sep 20 2016 *)
  • PARI
    sopf(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i])
is(n)=sopf(sigma(n))==sopf(eulerphi(n)) \\ Charles R Greathouse IV, Mar 09 2014

Extensions

Edited by Ray Chandler, Feb 13 2005
a(1) inserted by Charles R Greathouse IV, Mar 09 2014

A075784 Numbers n such that sopf(n) = sopf(n-1) + sopf(n-2) + sopf(n-3), where sopf(x) = sum of the distinct prime factors of x.

Original entry on oeis.org

23156, 59785, 72521, 98426, 362231, 480223, 506123, 1049790, 1077252, 1133953, 1202068, 1277411, 1327229, 1627040, 2200058, 2317712, 2368026, 3610497, 4174012, 5668196, 6302128, 6324778, 6946075, 7179599, 7786163, 8053816
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			The sum of the distinct prime factors of 23156 is 2 + 7 + 827 = 836; the sum of the distinct prime factors of 23155 is 5 + 11 + 421 = 437; the sum of the distinct prime factors of 23154 is 2 + 3 + 17 + 227 = 249; the sum of the distinct prime factors of 23153 is 13 + 137 = 150; and 836 = 437 + 249 + 150. Hence 23156 belongs to the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075846, A076525, A076527, A076531, A076532, A076533.

Programs

  • Mathematica
    p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[5, 10^5], p[ # - 1] + p[ # - 2] + p[ # - 3] == p[ # ] &]
Flatten[Position[Partition[Table[Total[FactorInteger[n][[All,1]]],{n,8054000}],4,1],?(Total[Most[#]]==Last[#]&)]//Quiet]+3 (* _Harvey P. Dale, Feb 22 2020 *)

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A075846 Numbers k such that sopf(k) = (1/2)*(sopf(k+1) + sopf(k-1)), where sopf(x) = sum of the distinct prime factors of x.

Original entry on oeis.org

10, 21, 35, 82, 221, 296, 961, 2665, 12629, 13117, 30317, 54485, 99145, 125750, 132728, 142198, 156379, 185461, 225898, 241057, 265227, 265643, 280918, 281396, 284531, 326698, 379331, 393335, 400685, 437241, 437999, 548101, 584502, 641561
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			The sum of the distinct prime factors of 21 is 3 + 7 = 10; the sum of the distinct prime factors of 22 is 2 + 11 = 13; the sum of the distinct prime factors of 20 is 2 + 5 = 7; and 10 = (1/2)*(13 + 7). Hence 21 belongs to the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075784, A076525, A076527, A076531, A076532, A076533.

Programs

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A076525 Numbers n such that sopf(n) = sopf(n+1) - sopf(n-1), where sopf(x) = sum of the distinct prime factors of x.

Original entry on oeis.org

4, 22, 57, 900, 1551, 1920, 4194, 6279, 10857, 19648, 20384, 32016, 63656, 65703, 83271, 84119, 86241, 105570, 145237, 181844, 271328, 271741, 316710, 322953, 331976, 345185, 379659, 381430, 409915, 424503, 490255, 524476, 542565, 550271
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			The sum of the distinct prime factors of 22 is 2 + 11 = 13; the sum of the distinct prime factors of 23 is 23; the sum of the distinct prime factors of 21 is 3 + 7 = 10; and 13 = 23 - 10. Hence 22 belongs to the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075784, A075846, A076527, A076531, A076532, A076533.

Programs

  • Magma
    [k:k in [3..560000]| &+PrimeDivisors(k) eq &+PrimeDivisors(k+1)-&+PrimeDivisors(k-1)]; // Marius A. Burtea, Oct 10 2019
  • Mathematica
    p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[3, 10^5], p[ # ] == p[ # + 1] - p[ # - 1] &]

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A076527 Numbers n such that sopf(n) = sopf(n-1) - sopf(n-2), where sopf(x) = sum of the distinct prime factors of x.

Original entry on oeis.org

8, 66, 2883, 3264, 3552, 13872, 21386, 26896, 29698, 29768, 31980, 36567, 40517, 65305, 72012, 82719, 101639, 110848, 160230, 211646, 237136, 237568, 238303, 242606, 299186, 309960, 378014, 393208, 439105, 445795, 455182, 527078, 570021
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			The sum of the distinct prime factors of 66 is 2 + 3 + 11 = 16; the sum of the distinct prime factors of 65 is 5 + 13 = 18; the sum of the distinct prime factors of 64 is 2; and 16 = 18 - 2. Hence 66 belongs to the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075784, A075846, A076525, A076531, A076532, A076533.

Programs

  • Magma
    [k:k in [4..571000]| &+PrimeDivisors(k-1) - &+PrimeDivisors(k-2) eq (&+PrimeDivisors(k))]; // Marius A. Burtea, Feb 12 2020
  • Mathematica
    p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[4, 10^5], p[ # ] == p[ # - 1] - p[ # - 2] &]

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A076531 Numbers n such that sopf(phi(n)) = phi(sopf(n)), where sopf(x) = sum of the distinct prime factors of x.

Original entry on oeis.org

3, 203, 322, 377, 644, 851, 931, 1166, 1211, 1288, 1421, 1666, 1815, 1862, 2332, 2576, 3332, 3724, 4664, 4830, 5152, 6401, 6517, 6664, 7042, 7241, 7448, 9075, 9328, 9555, 9660, 9845, 9922, 9947, 10304, 10465, 11662, 11814, 11830, 12558, 12903, 13034
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			sopf(phi(203)) = sopf(168) = 12; phi(sopf(203)) = phi(36) = 12 hence 203 is a term of the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075784, A075846, A076525, A076527, A076532, A076533.

Programs

  • Mathematica
    p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[3, 10^4], p[EulerPhi[ # ]] == EulerPhi[ p[ # ]] &]
  • PARI
    sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]); \\ A008472
isok(n) = eulerphi(sopf(n)) == sopf(eulerphi(n)); \\ Michel Marcus, Oct 04 2019

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A076532 Numbers n such that sopf(sigma(n)) = sigma(sopf(n)), where sopf(x) = sum of the distinct prime factors of x.

Original entry on oeis.org

2, 90, 425, 490, 605, 630, 726, 735, 750, 816, 2250, 2695, 3185, 3234, 3420, 3822, 4176, 5096, 5250, 6591, 7644, 8470, 9100, 9425, 10296, 10780, 11616, 11638, 12321, 15750, 16940, 18096, 22736, 23276, 25578, 27360, 27783, 28500, 31900, 36400
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

			sopf(sigma(90)) = sopf(234) = 18; sigma(sopf(90)) = sigma(10) = 18, hence 90 is a term of the sequence.

Links

Crossrefs

Cf. A008472, A075565, A075784, A075846, A076525, A076527, A076531, A076533.

Programs

  • Mathematica
    p[n_] := Apply[Plus, Transpose[FactorInteger[n]][[1]]]; Select[Range[2, 10^4], p[DivisorSigma[1, # ]] == DivisorSigma[1, p[ # ]] &]
  • PARI
    isok(n) = (n>1) && sigma(sopf(n)) == sopf(sigma(n)); \\ Michel Marcus, Oct 04 2019

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A294995 Numbers n such that sopfr(n) = sopfr(n-1) + sopfr(n-2), where sopfr is the sum of prime factors of n with multiplicity (A001414).

Original entry on oeis.org

23, 610, 1162, 1243, 1651, 7385, 13066, 37129, 38123, 41194, 41361, 48511, 59452, 72179, 83151, 87375, 98877, 103528, 126497, 138190, 141037, 148657, 157994, 162410, 175077, 262788, 296482, 299398, 351226, 354321, 418134, 425099, 452130, 465254, 470494
Offset: 1

Views

Author

Amiram Eldar, Nov 12 2017

Keywords

Examples

			610 is in the sequence since sopfr(608) = 29, sopfr(609) = 39 and sopfr(610) = 68 = 39 + 29.

Links

Crossrefs

Cf. A001414, A075565.

Programs

  • Mathematica
    f[n_]:=Plus @@ Times @@@ FactorInteger@ n; Select[Range[10^5], f[#]==f[#-1]+f[#-2] &]
  • PARI
    sopfr(n,f=factor(n))=f[,1]~*f[,2]
list(lim)=my(v=List(),a=0,b=2,c); forfactored(k=3,lim\1, c=sopfr(k[2]); if(c==a+b, listput(v,k[1])); a=b; b=c); Vec(v) \\ Charles R Greathouse IV, Nov 12 2017

A332973 Solutions k of the equation usigma(k) = usigma(k-1) + usigma(k-2) where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

3, 42, 188970, 998670, 51670374, 91397154, 236280786, 259172826, 792554574, 1106710914, 1468869930, 1957827498, 2467823442, 2496238590, 3324585210, 4055970282, 4183629690, 4384566870, 13479861630, 20681058270, 29343074178, 43449285210, 68705958690, 71418085926
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			42 is a term since s(42) = 96 and s(40) + s(41) = 54 + 42 = 96.

Links

Crossrefs

The unitary version of A065900.
Cf. A034448, A065557, A075565, A076136, A145469, A291126, A291176, A292033, A294995, A332974.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3, 10^8], usigma[#] == usigma[# - 1] + usigma[# - 2] &]
  • PARI
    usigma(k) = sumdivmult(k, d, if(gcd(d, k/d)==1, d)); \\ A034448
isok(k) = usigma(k) == usigma(k-1) + usigma(k-2); \\ Jinyuan Wang, Mar 08 2020

Extensions

Terms a(22) and beyond from Giovanni Resta, Mar 10 2020

A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

3, 24, 360, 5016, 28440, 42066, 50568, 60456, 187176, 998670, 1454706, 12055512, 14365608, 25726728, 27896424, 51670374, 91702962, 141084774, 236280786, 249854952, 386668344, 439362504, 792554574, 1115866152, 1931976696, 2467823442, 2496238590, 2655297558, 2715505440
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			24 is a term since isigma(24) = 60 and isigma(22) + isigma(23) = 36 + 24 = 60.

Links

Crossrefs

The infinitary version of A065900.
Cf. A049417, A065557, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995, A332976.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[3, 10^5], isigma[#] == isigma[# - 1] + isigma[# - 2] &]
Showing 1-10 of 14 results. Next

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

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