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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A236386 Numbers m such that phi(m) is an oblong number.

Original entry on oeis.org

3, 4, 6, 7, 9, 13, 14, 18, 21, 25, 26, 28, 31, 33, 36, 42, 43, 44, 49, 50, 62, 66, 73, 86, 87, 91, 95, 98, 111, 116, 117, 121, 135, 146, 148, 152, 157, 161, 169, 174, 182, 190, 201, 207, 211, 216, 222, 228, 234, 237, 241, 242, 252, 268, 270, 287, 289, 305
Offset: 1

Views

Author

Joseph L. Pe, Jan 24 2014

Keywords

Comments

An oblong number (A002378) is of the form k*(k+1) where k is a natural number.
From Bernard Schott, Feb 27 2023: (Start)
Subsequence of primes is A002383 because in this case phi(k^2+k+1) = k*(k+1).
Subsequence of oblong numbers is A359847 where k and phi(k) are both oblong numbers. (End)

Examples

			phi(13) = 12 = 3*4, an oblong number; so 13 is a term of the sequence.

Links

Crossrefs

Cf. A000010, A002378, A002383, A359847.
Similar, but where phi(m) is: A039770 (square), A039771 (cube), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A360944 (triangular).

Programs

  • Maple
    filter := m -> issqr(1 + 4*phi(m)) : select(filter, [$(1 .. 700)]); # Bernard Schott, Feb 26 2023
  • Mathematica
    Select[Range[500], IntegerQ@Sqrt[1 + 4*EulerPhi[#]] &] (* Giovanni Resta, Jan 24 2014 *)
  • PARI
    isok(m) = my(t=eulerphi(m)); !(t%2) && ispolygonal(t/2, 3); \\ Michel Marcus, Feb 27 2023
  • Python
    from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import totient
def A236386_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<2)+1), count(max(1,startvalue)))
A236386_list = list(islice(A236386_gen(),20)) # Chai Wah Wu, Feb 28 2023

Extensions

a(16)-a(58) from Giovanni Resta, Jan 24 2014

A096843 Primes of form repdigit - 1. Primes whose sum of divisors is a decimal repdigit.

Original entry on oeis.org

2, 3, 5, 7, 43, 443, 887, 2221, 8887, 444443, 888887, 444444443, 888888887, 444444444443, 888888888887, 222222222222222221, 444444444444444444444444444443, 44444444444444444444444444444443
Offset: 1

Views

Author

Labos Elemer, Jul 15 2004

Keywords

Comments

Union numbers 2, 5 and sequences A093171, A093163 and A091189.
Corresponding values of sigma(a(n)) are in A028987. - Jaroslav Krizek, Mar 19 2013

Examples

			n=43: sigma(43)=44;

Links

Crossrefs

Cf. A096503-A096508, A096841-A096846, A000203.

Extensions

Missing a(1)=2 and a(3)=5 added by Jaroslav Krizek, Mar 19 2013

A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 7, 9, 11, 14, 18, 22, 29, 37, 57, 58, 63, 67, 74, 76, 79, 108, 114, 126, 134, 137, 143, 155, 158, 175, 183, 191, 211, 225, 231, 244, 248, 274, 277, 286, 308, 310, 329, 341, 350, 366, 372, 379, 382, 396, 417, 422, 423, 450, 453, 462, 554, 556, 604, 623, 631, 658, 682
Offset: 1

Views

Author

Bernard Schott, Feb 26 2023

Keywords

Comments

Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.
Subsequence of triangular numbers is A287472.

Examples

			phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.

Links

Crossrefs

Cf. A000010, A000217, A055469, A287472.
Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).

Programs

  • Maple
    filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);
  • Mathematica
    Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* Amiram Eldar, Feb 27 2023 *)
  • PARI
    isok(m) = ispolygonal(eulerphi(m), 3); \\ Michel Marcus, Feb 27 2023
  • Python
    from itertools import islice, count
from sympy.ntheory.primetest import is_square
from sympy import totient
def A360944_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
A360944_list = list(islice(A360944_gen(),20)) # Chai Wah Wu, Feb 28 2023

A098217 Cototients of terms from A098216. By definitions these cototient values are decimal repdigits.

Original entry on oeis.org

0, 2, 4, 4, 3, 6, 8, 8, 7, 8, 9, 5, 9, 22, 11, 7, 44, 44, 66, 33, 88, 11, 33, 55, 88, 55, 99, 111, 77, 33, 99, 99, 111, 77, 77, 111, 55, 333, 55, 77, 444, 99, 333, 333, 77, 111, 77, 99, 77, 77, 777, 888, 888, 111, 99, 999, 777, 111, 99, 99, 555, 111, 999, 999, 999, 111, 1111
Offset: 1

Views

Author

Labos Elemer, Oct 22 2004

Keywords

Comments

It is believed that for every repdigit r>1, inverse[cototient[r]] has solution, usually more than one.

Crossrefs

Cf. A096503, A010785, A098216.

Programs

  • Mathematica
    ta={{0}};Do[s=Length[Union[IntegerDigits[n-EulerPhi[n]]]]; If[Equal[s, 1]&&!PrimeQ[n], Print[{n, n-EulerPhi[n]}];ta=Append[ta, n]], {n, 1, 100000}];ta=Delete[ta, 1];ta-EulerPhi[ta]

A098218 Nonprime numbers whose cototient is a decimal repunits >1 from A002275.

Original entry on oeis.org

35, 121, 231, 327, 535, 1111, 2047, 2407, 2911, 3127, 3327, 20767, 45967, 64111, 75847, 81607, 103927, 177367, 202207, 210767, 224295, 234607, 275647, 277807, 290911, 295447, 305887, 308911, 321407, 333327, 453911, 475967, 586127, 1199327
Offset: 1

Views

Author

Labos Elemer, Oct 22 2004

Keywords

Comments

It is believed that for every repdigit r>1, inverse(cototient(r)) has a solution, usually more than one. For r=1, primes are the solutions.

Examples

			n=1111 and cototient(1111)=111. By accident, both n and its cototient are decimal repunits.

Links

Crossrefs

Cf. A096503, A010785, A002275, A098216, A098217.

Programs

  • Mathematica
    ta={{0}};Do[s=Length[u=Union[IntegerDigits[n-EulerPhi[n]]]]; If[Equal[s, 1]&&!PrimeQ[n]&&Equal[u, {1}], Print[{n, n-EulerPhi[n]}]; ta=Append[ta, n]], {n, 1, 100000}];ta=Delete[ta, 1];ta-EulerPhi[ta]

A096842 Sigma applied to A096841 produces these repdigits: a[n]=A000203[A096841(n)].

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 44, 222, 444, 888, 444, 888, 888, 2222, 6666, 8888, 8888, 222222, 88888, 222222, 444444, 444444, 888888, 444444, 444444, 666666, 888888, 888888, 888888, 888888, 888888, 444444, 444444, 888888, 888888, 888888, 888888, 888888
Offset: 1

Views

Author

Labos Elemer, Jul 15 2004

Keywords

Examples

			n=43:sigma[43]=44;

Crossrefs

Cf. A096503-A096508, A096841-A096846, A000203.

Programs

  • Mathematica
    rd[x_] := Length[Union[IntegerDigits[x]]] Do[s = rd[DivisorSigma[1, n]]; s1 = DivisorSigma[1, n]; If[Equal[s, 1], Print[{n, s1}]; ta[[u]] = n; u = u + 1], {n, 1, 1000000}];ta;DivisorSigma[1, ta]
Previous Showing 11-16 of 16 results.

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