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.

A075565 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

5, 23, 58, 901, 1552, 1921, 4195, 6280, 10858, 19649, 20385, 32017, 63657, 65704, 83272, 84120, 86242, 105571, 145238, 181845, 271329, 271742, 316711, 322954, 331977, 345186, 379660, 381431, 409916, 424504, 490256, 524477, 542566, 550272, 561661, 565217, 566560
Offset: 1

Views

Author

Joseph L. Pe, Oct 18 2002

Keywords

Examples

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

Links

Crossrefs

Cf. A008472, A075784, A075846, A076525, A076527, A076531, A076532, A076533.

Programs

  • Magma
    [k:k in [5..560000]| &+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[ # - 1] + p[ # - 2] == p[ # ] &]
  • PARI
    sopf(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]);
isok(n) = sopf(n) == sopf(n-1) + sopf(n-2); \\ Michel Marcus, Feb 12 2020
  • Python
    from sympy import primefactors
def sopf(n): return sum(primefactors(n))
def afind(limit):
  sopfm2, sopfm1, sopf = 2, 3, 2
  for k in range(4, limit+1):
    if sopf == sopfm1 + sopfm2: print(k, end=", ")
    sopfm2, sopfm1, sopf = sopfm1, sopf, sum(primefactors(k+1))
afind(600000) # Michael S. Branicky, May 23 2021

Extensions

Edited and extended by Ray Chandler, Feb 13 2005

A081377 Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).

Original entry on oeis.org

1, 3, 14, 35, 42, 70, 105, 119, 209, 210, 238, 248, 297, 357, 418, 477, 594, 595, 616, 627, 714, 744, 954, 1045, 1178, 1190, 1240, 1254, 1463, 1485, 1672, 1674, 1736, 1785, 1848, 1863, 2079, 2090, 2376, 2385, 2540, 2728, 2926, 2945, 2970, 3080, 3135, 3302
Offset: 1

Views

Author

Labos Elemer, Mar 26 2003

Keywords

Comments

The multiplicities of the divisors are to be ignored.
Is it true that 1 is the only term in both this sequence and A055744? - Farideh Firoozbakht, Jul 01 2008. Answer from Luke Pebody, Jul 10 2008: No! In fact the numbers 103654150315463023813006470 and 6534150553412193640795377701190 are in both sequences.

Examples

			n=418=2*11*19: sigma(418)=720, phi[418]=180, common prime factor set ={2,3,5}
k = 477 = 3*3*53: sigma(477) = 702 = 2*3*3*3*13; phi(477) = 312 = 2*2*2*3*13; common factor set: {2,3,13}.
phi(89999)=66528=2^5*3^3*7*11 and sigma(89999)=118272=2^9*3*7*11 so 89999 is in the sequence.

Links

Crossrefs

Cf. A000010, A000203, A076533, A065642, A081378, A110751, A110819, A027598, A141718.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Do[s=ba[DivisorSigma[1, n]]; s1=ba[EulerPhi[n]]; If[Equal[s, s1], k=k+1; Print[n]], {n, 1, 10000}]
  • PARI
    is(n)=factor(eulerphi(n=factor(n)))[,1]==factor(sigma(n))[,1] \\ Charles R Greathouse IV, Nov 27 2013

Extensions

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Farideh Firoozbakht

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

A081378 Numbers k for which the sums of prime factors (ignoring multiplicity) of sigma(k) and phi(k) are equal but the sets of prime factors of sigma and phi are different.

Original entry on oeis.org

412, 1142, 1236, 1328, 1339, 1703, 2855, 2875, 2884, 3406, 3426, 3668, 3708, 3984, 4017, 5109, 5356, 5710, 5750, 5924, 6003, 6281, 6399, 6413, 6640, 6812, 7994, 8054, 8318, 8515, 8565, 8611, 8625, 8652, 8843, 8858, 9373, 9707, 9991
Offset: 1

Views

Author

Labos Elemer, Mar 26 2003

Keywords

Examples

			k = 412 = 2*2*103: sigma(412) = 728 = 2*2*2*7*13, phi(412) = 204 = 2*2*3*17, the sums of prime factors are identical (2 + 7 + 13 = 22 = 2 + 3 + 17) but the prime sets are different: {2,7,13} vs. {2,7,17}.

Links

Crossrefs

Cf. A000010, A000203, A065642, A076533, A081377.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; spf[x_] := Apply[Plus, ba[x]]; k=0; Do[s=ba[DivisorSigma[1, n]]; s1=ba[EulerPhi[n]]; s3=spf[DivisorSigma[1, n]]; s4=spf[EulerPhi[n]]; If[ !Equal[s, s1]&&Equal[s3, s4], k=k+1; Print[{n, s, s1, ba[n], s3}]], {n, 1, 10000}]
  • PARI
    is(n) = {my(f = factor(n), p1 = factor(sigma(f))[, 1], p2 = factor(eulerphi(f))[, 1]); p1 != p2 && vecsum(p1) == vecsum(p2) ;} \\ Amiram Eldar, Mar 25 2024
Showing 1-9 of 9 results.

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

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