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 1-7 of 7 results.

A244438 Terms of the sequence A145749 which are not of the form 2*p where p is prime.

Original entry on oeis.org

8, 188, 285, 902, 2013, 8493, 37406, 40977, 61918, 68210, 90094, 303853, 352941, 360446, 375565, 467654, 501693, 724934, 889285, 940093, 1079662, 1473565, 1488957, 1517206, 1573045, 1692302, 1864285, 2048973, 2077405, 2346226, 2584106, 2693517, 3393934, 3509997, 3802029, 4083526, 4194406
Offset: 1

Views

Author

Farideh Firoozbakht, Aug 14 2014

Keywords

Links

Crossrefs

Cf. A000010, A000203, A005384, A145749.

Programs

  • Mathematica
    Select[Range[5000000], !PrimeQ[#/2] && DivisorSigma[1, #] + EulerPhi[#] == DivisorSigma[1, # + 1] + EulerPhi[# + 1] &]
  • PARI
    for(n=1,10^7,s=eulerphi(n)+sigma(n);if(s==eulerphi(n+1)+sigma(n+1) && ((n%2==0 && !isprime(n/2)) || n%2),print1(n,", "))) \\ Derek Orr, Aug 14 2014

A066198 Numbers n where phi changes as fast as sigma, i.e., abs(phi(n+1) - phi(n)) = abs(sigma(n+1) - sigma(n)).

Original entry on oeis.org

2, 6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 854, 862, 886, 902, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2013, 2026, 2038, 2062, 2098, 2206, 2446, 2458
Offset: 1

Views

Author

Joseph L. Pe, Dec 16 2001

Keywords

Comments

This sequence is the union of two sequences A145748 and A145749. See comment lines of A145749. [Farideh Firoozbakht, Nov 01 2008]

Examples

			|phi(7) - phi(6)| = |6 - 2| = |8 - 12| = |sigma(7) - sigma(6)|.
|phi(9) - phi(8)| = |6 - 4| = 2 = |13 - 15| = |sigma(9) - sigma(8)|.

Links

Crossrefs

Cf. A065387, A145748, A145749.

Programs

  • Mathematica
    Select[ Range[ 1, 10^4 ], Abs[ DivisorSigma[ 1, # + 1 ] - DivisorSigma[ 1, # ] ] == Abs[ EulerPhi[ # + 1 ] - EulerPhi[ # ] ] & ]
  • PARI
    { n=0; for (m=1, 10^9, if (abs(eulerphi(m + 1) - eulerphi(m)) == abs(sigma(m + 1) - sigma(m)), write("b066198.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 05 2010

Extensions

More terms from Jason Earls, Jun 05 2002

A145748 Numbers n such that phi(n+1)-phi(n)=sigma(n+1)-sigma(n).

Original entry on oeis.org

2, 854, 751358, 1421637, 8775206, 8892195, 16485944, 31845344, 95494035, 277653495, 380438505, 744048855, 1091725394, 1615353002, 2284844925, 2491028745, 6345217034, 8490513014, 12784909335, 14177454885, 15669084375, 17694356295, 17836667354, 24180347115
Offset: 1

Views

Author

Farideh Firoozbakht, Nov 01 2008

Keywords

Comments

This sequence is a subsequence of A066198.

Crossrefs

Cf. A066198, A145749.

Programs

  • Mathematica
    de[n_]:=DivisorSigma[1,n]-EulerPhi[n];Do[If[de[n]==de[n+1],Print[n]],{n,50000000}] (* Firoozbakht *)
Select[Range[10^6], (EulerPhi[# + 1] - EulerPhi[#]) == (DivisorSigma[1, # + 1] - DivisorSigma[1, #]) &] (* Alonso del Arte, Feb 08 2012 *)

Extensions

a(9)-a(16) from Donovan Johnson, Dec 14 2009
a(17)-a(24) from Donovan Johnson, Feb 08 2012

A244439 Numbers n such that phi(n)*sigma(n) = phi(n+1)*sigma(n+1).

Original entry on oeis.org

5, 55, 56, 123, 135, 147, 175, 304, 351, 644, 1015, 2464, 19304, 61131, 162524, 476671, 567644, 712724, 801944, 2435488, 3346399, 3885056, 4555999, 8085560, 8369360, 12516692, 22702119, 29628800, 83884031, 83994624, 84789247, 354812535, 860616295, 1091535704
Offset: 1

Views

Author

Farideh Firoozbakht, Aug 14 2014

Keywords

Comments

Since both numbers 55 and 56 are in the sequence we have sigma(55)*phi(55) = sigma(56)*phi(56) = sigma(57)*phi(57). It seems that 56 is the only number n which has the nice property sigma(n-1)*phi(n-1) = sigma(n)*phi(n) = sigma(n+1)*phi(n+1).
Up to n < 6*10^11 the similar equation phi(n)*sigma(n+1) = phi(n+1)*sigma(n) is satisfied only by n = 696003. - Giovanni Resta, Jun 08 2020

Examples

			5 is in the sequence because sigma(5)*phi(5) = sigma(6)*phi(6) = 24.
55 is in the sequence because sigma(55)*phi(55) = sigma(56)*phi(56) = 2880.

Links

Crossrefs

Cf. A000010, A000203, A145749.

Programs

  • Maple
    with(numtheory): A244439:=n->`if`(phi(n)*sigma(n) = phi(n+1)*sigma(n+1), n, NULL): seq(A244439(n), n=1..10^4); # Wesley Ivan Hurt, Aug 16 2014
  • Mathematica
    Select[Range[10^5], Equal @@ (EulerPhi[{#, # + 1}] DivisorSigma[1, {#, # + 1}]) &] (* Giovanni Resta, Jun 08 2020 *)
  • PARI
    for(n=1,10^6,s=eulerphi(n)*sigma(n);if(s==eulerphi(n+1)*sigma(n+1),print1(n,", "))) \\ Derek Orr, Aug 14 2014

Extensions

More terms from Jens Kruse Andersen, Aug 16 2014

A259495 Numbers k such that sigma(k) + phi(k) + d(k) = sigma(k+1) + phi(k+1) + d(k+1), where sigma(k) is the sum of the divisors of k, phi(k) the Euler totient function of k and d(k) the number of divisors of k.

Original entry on oeis.org

4, 285, 902, 2013, 8493, 37406, 61918, 90094, 120001, 184484, 250550, 303853, 352941, 360446, 375565, 501693, 724934, 889285, 940093, 995630, 1079662, 1473565, 1488957, 1517206, 1573045, 1581806, 1692302, 1864285, 2048973, 2693517, 3393934, 3509997, 4083526, 4194406
Offset: 1

Views

Author

Paolo P. Lava, Jun 29 2015

Keywords

Examples

			sigma(4) + phi(4) + d(4) = 7 + 2 + 3 = 12 and sigma(5) + phi(5) + d(5) = 6 + 4 + 2 = 12.
sigma(285) + phi(285) + d(285) = 480 + 144 + 8 = 632 and sigma(286) + phi(286) + d(286) = 504 + 120 + 8 = 632.

Links

Crossrefs

Cf. A000005, A000010, A000203, A233541.
Cf. A054004, A145749, A259496.

Programs

  • Maple
    with(numtheory): P:=proc(q) local n; for n from 1 to q do
if sigma(n)+phi(n)+tau(n)=sigma(n+1)+phi(n+1)+tau(n+1)
then print(n); fi; od; end: P(10^9);
  • Mathematica
    f[n_] := Module[{fct = FactorInteger[n]}, p = fct[[All, 1]]; e = fct[[All, 2]]; Times @@ (e + 1) + Times @@ ((p^(e + 1) - 1)/(p - 1)) + Times @@ ((p - 1)*p^(e - 1))]; f1 = 0; s = {}; Do[f2 = f[n]; If[f2 == f1, AppendTo[s, n - 1]];  f1 = f2, {n, 2, 10^5}]; s (* Amiram Eldar, Jul 12 2019 *)

A259496 Numbers n such that phi(n) + d(n) = phi(n+1) + d(n+1), where phi(n) is the Euler totient function of n and d(n) the number of divisors of n.

Original entry on oeis.org

5, 7, 104, 105, 1754, 3255, 16215, 22935, 67431, 93074, 983775, 1025504, 2200694, 2619705, 3365438, 4163355, 4447064, 4695704, 6372794, 7838265, 9718904, 11903775, 23992215, 26879684, 29357475, 37239735, 40588485, 41207144, 48615735, 56424555, 76466985, 81591194, 83864055
Offset: 1

Views

Author

Paolo P. Lava, Jun 29 2015

Keywords

Comments

So far, less than 10^9, except for 7, 67431 & 3365438, all terms have been congruent to 5 or 4 (mod 10). - Robert G. Wilson v, Jul 06 2015

Examples

			phi(5) + d(5) = 4 + 2 = 6 and phi(6) + d(6) = 2 + 4 = 6.
phi(7) + d(7) = 6 + 2 = 8 and phi(8) + d(8) = 4 + 4 = 8.

Links

Crossrefs

Cf. A000005, A000010, A061468, A054004, A145749, A259495.

Programs

  • Magma
    [n: n in [1..6*10^6] | EulerPhi(n) + NumberOfDivisors(n) eq EulerPhi(n+1) + NumberOfDivisors(n+1)]; // Vincenzo Librandi, Jun 30 2015
  • Maple
    with(numtheory): P:=proc(q) local n; for n from 1 to q do
if phi(n)+tau(n)=phi(n+1)+tau(n+1) then print(n); fi;
od; end: P(10^9);
  • Mathematica
    a = k = 2; lst = {}; While[k < 100000001, b = EulerPhi[k] + DivisorSigma[0, k]; If[a == b, AppendTo[lst, k - 1]]; k++; a = b]; lst

Extensions

a(23)-a(33) from Robert G. Wilson v, Jul 05 2015

A297366 Numbers k such that uphi(k) + usigma(k) = uphi(k+1) + usigma(k+1), where uphi is the unitary totient function (A047994) and usigma the sum of unitary divisors (A034448).

Original entry on oeis.org

6, 10, 12, 15, 18, 22, 24, 26, 28, 36, 40, 46, 48, 52, 58, 63, 72, 80, 82, 88, 96, 100, 106, 108, 112, 124, 136, 148, 162, 166, 172, 178, 192, 196, 226, 232, 242, 250, 262, 268, 285, 288, 292, 316, 346, 352, 358, 382, 388, 400, 432, 448, 466, 478, 486, 502
Offset: 1

Views

Author

Amiram Eldar, Dec 29 2017

Keywords

Comments

The unitary version of A145749.

Examples

			6 is in the sequence since uphi(6) + usigma(6) = 2 + 12 = uphi(7) + usigma(7) = 6 + 8 = 14.

Links

Crossrefs

Cf. A034448, A047994, A145749.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
uphi[n_] := (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]; u[n_] := uphi[n]+usigma[n]; aQ[n_] := u[n] == u[n + 1]; Select[Range[10^3], aQ]
  • PARI
    u(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,1]^f[i,2]-1) + prod(i = 1, #f~, f[i,1]^f[i,2]+1);}
list(kmax) = {my(u1 = u(1), u2); for(k = 2, kmax, u2 = u(k); if(u1 == u2, print1(k-1, ", ")); u1 = u2);} \\ Amiram Eldar, Jun 30 2025
Showing 1-7 of 7 results.

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