A033831 -id:A033831 - cp's OEIS frontend

A033855 Numbers k such that j(k)*phi(k) = s(phi(k)), where j(k) = A033831(k), s(k) = sigma(k) - k.

1, 2, 7, 9, 29, 143, 155, 183, 731, 791, 1011, 1346, 35659, 60484, 65524, 525227, 525557, 525617, 526697, 529817, 531779, 567437, 1047554, 2541679, 33550337, 214486281, 1476844097, 1478227937, 1543409687, 14200144243, 14200244477, 14200257551, 14200349281, 14200779611, 14201040053, 14201501401

Offset: 1

Naohiro Nomoto

nonn

phi(a(n)) is a multiperfect number (A007691). - Max Alekseyev, Oct 09 2023

Max Alekseyev, Table of n, a(n) for n = 1..50000

Cf. A000010, A000203, A001065, A007691, A033831, A033856.

j[n_] := DivisorSum[n, 1&, # > 2 && n/# < #-1 &]; aQ[n_] := j[n] * (p = EulerPhi[ n]) == DivisorSigma[1, p] - p; Select[Range[10^4], aQ] (* Amiram Eldar, Jul 01 2019 *)

More terms from Amiram Eldar, Jul 01 2019

Terms a(27) onward from Max Alekseyev, Oct 09 2023

A033856 Numbers k such that j(k)*phi(k) = sigma(phi(k)), j(k) = A033831(k).

Original entry on oeis.org

14, 58, 175, 244, 833, 1017, 1348, 8653, 33263, 33497, 33611, 34099, 34151, 34529, 34771, 36281, 36449, 36743, 37367, 37373, 38311, 38695, 38735, 38915, 39035, 39263, 39295, 39431, 39995, 41015, 41635, 42119, 46209, 46617, 46731, 47067, 49911, 50007, 50871

Offset: 1

Naohiro Nomoto

Amiram Eldar, Table of n, a(n) for n = 1..406

Cf. A000010, A000203, A033831, A033855.

j[n_] := DivisorSum[n, 1&, # > 2 && n/# < #-1 &]; aQ[n_] := j[n]*(p = EulerPhi[ n]) == DivisorSigma[1, p]; Select[Range[10^4], aQ] (* Amiram Eldar, Jul 01 2019 *)

More terms from Amiram Eldar, Jul 01 2019

A033858 Numbers k such that j(k) + ud(k) = d(k), where j(k) = A033831, ud(k) = number of unitary divisors of k (A034444), d(k) = number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 4, 8, 9, 12, 20, 24, 25, 27, 40, 49, 54, 88, 104, 120, 121, 125, 135, 136, 152, 168, 169, 184, 189, 232, 248, 250, 264, 270, 280, 289, 296, 297, 312, 328, 343, 344, 351, 361, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 529, 536, 568, 584

Offset: 1

Naohiro Nomoto

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A033831, A034444.

j[n_] := DivisorSum[n, 1&, # > 2 && n/# < #-1 &]; Select[Range[10^3], j[#] + 2^PrimeNu[#] == DivisorSigma[0, #] &] (* Amiram Eldar, Jul 14 2019 *)

More terms from Amiram Eldar, Jul 14 2019

A033859 Numbers k such that A033831(k) = A034444(k) where A034444(k) = number of unitary divisors of k.

Original entry on oeis.org

8, 16, 24, 27, 36, 40, 54, 81, 88, 100, 104, 120, 125, 135, 136, 152, 168, 184, 189, 196, 225, 232, 248, 250, 264, 270, 272, 280, 296, 297, 312, 328, 343, 344, 351, 375, 376, 378, 408, 424, 440, 441, 456, 459, 472, 484, 488, 513, 520, 536, 568, 584, 594, 616

Offset: 1

Naohiro Nomoto

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033831, A034444.

with(numtheory): for n from 1 to 1200 do it := divisors(n): count := 0: for i from 1 to nops(it) do if it[i]>=3 and 1<=n/it[i] and n/it[i]<=(it[i]-2) then count := count+1 fi:od: if count=2^nops(ifactors(n)[2]) then printf(`%d,`,n) fi; od:

j[n_] := DivisorSum[n, 1&, # > 2 && n/# < #-1 &]; Select[Range[1000], j[#] == 2^PrimeNu[#] &] (* Amiram Eldar, Jun 11 2019 *)

More terms from James Sellers, Jun 20 2000

A033872 Numbers k such that A033831(k) is prime.

Original entry on oeis.org

8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 72, 74, 75, 76, 77, 80, 81, 82, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99

Offset: 1

Naohiro Nomoto

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033831.

f[n_] := DivisorSum[n, 1&, # > 2 && n/# < #-1 &]; Select[Range[1000], PrimeQ[f[#]] &] (* Amiram Eldar, Jun 11 2019 *)

isok(n) = isprime(sumdiv(n, d, (d>=3) && (q=n/d) && (q>=1) && (q<=d-2))); \\ Michel Marcus, Jun 11 2019

A033852 Integers k such that j(k)*phi(k) = sigma(k), where j(n) = A033831(n).

Original entry on oeis.org

35, 105, 248, 418, 594, 744, 812, 1254

Offset: 1

Naohiro Nomoto

a(9) > 10^7. - Michel Marcus, Nov 05 2014

a(9) > 10^8. - Manfred Scheucher, May 30 2015

Cf. A000010 (phi), A000203 (sigma), A033831.

isok(n) = sumdiv(n, d, (d>=3) && (q=n/d) && (q>=1) && (q<=d-2))*eulerphi(n) == sigma(n); \\ Michel Marcus, Nov 05 2014

A033853 Integers k such that j(k)*d(k)=phi(k), where j = A033831.

Original entry on oeis.org

3, 15, 56, 102, 228, 234, 384, 408, 510, 864, 936, 1092, 1140, 1170, 1920, 2040, 4320, 4680, 5460

Offset: 1

Naohiro Nomoto

If it exists, a(20) > 10^9. - Sean A. Irvine, Jul 19 2020

Cf. A000005 (d), A000010 (phi), A033831.

a(17)-a(19) from Sean A. Irvine, Jul 19 2020

A033854 Numbers k such that j(k)*ud(k)=phi(k), j(n) = A033831, ud(n) = number of unitary divisors of n (A034444).

Original entry on oeis.org

3, 4, 8, 15, 20, 28, 40, 102, 132, 140, 252, 360, 510, 780

Offset: 1

Naohiro Nomoto

If it exists, a(15) > 10^9. - Sean A. Irvine, Jul 20 2020

Cf. A000010 (phi), A033831, A034444.

A033857 Numbers n such that j(n)*phi(n) = usigma(n), where j(n) = A033831, usigma(n) = sum of unitary divisors of n (A034448).

Original entry on oeis.org

35, 44, 56, 105, 198, 418, 1254

Offset: 1

Naohiro Nomoto

If it exists, a(8) > 10^9. - Sean A. Irvine, Jul 20 2020

Cf. A000010 (phi), A034448 (usigma), A033831.

A100563 Number of bases less than sqrt(n) in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 2, 0, 1, 0, 1, 2, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 2, 0, 0, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 3, 0, 2, 1, 0, 0, 1, 1, 2, 1, 0, 0, 0, 2, 0, 2, 1, 2, 1, 0, 3, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 1, 2, 1, 3, 1, 0, 0, 2, 2

Offset: 1

Gordon Hamilton, Nov 29 2004

Is there a number m such that a(n) > 0 for all n > m? I call the set of numbers for which a(n)=0 "unkempt" for refusing to use a mirror in any base. Is there an infinite number of unkempt numbers? a(n) can be arbitrarily large.
The sequence A123586 gives the values of n where a(n)=0. - Robert G. Wilson v, Nov 01 2014
Is there a closed-form formula for this function? - Robert G. Wilson v, Nov 01 2014
From Robert G. Wilson v, Nov 26 2014: (Start)
The first occurrence, beginning at 0, of n is: 1, 5, 17, 65, 121, 562, 1432, 1477, 4369, 36582, 35101, 86677, 83161, 360361, 291721, 720721, 887041, 1496881, 1670761, 3931201, 3341521, 5654881, 7207201, 7761601,...
Positions where a(n)=k:
k = 0:  A123586;
k = 1:  5, 7, 9, 10, 13, 15, 16, 20, 23, 25, 27, 28, 29, 33, 34, 36, 37, 38, 40, ...;
k = 2:  17, 21, 26, 31, 46, 51, 52, 55, 57, 63, 67, 73, 78, 80, 82, 91, 92, 93, 98, ...;
k = 3:  65, 85, 100, 130, 154, 164, 170, 178, 191, 195, 203, 209, 242, 282, 292, ...;
k = 4:  121, 235, 255, 257, 273, 300, 325, 341, 343, 373, 400, 495, 601, 610, 626, 666, ...;
k = 5:  562, 676, 771, 819, 1009, 1111, 1220, 1333, 1365, 1441, 1543, 1978, 1981, 2000, ...;
k = 6:  1432, 2380, 2666, 2925, 3280, 4035, 4095, 4161, 4225, 4401, 4525, 4561, 4681, ...;
k = 7:  1477, 4097, 4591, 7141, 7993, 8191, 9640, 10081, 10297, 10626, 10858, 11761, ...; etc.
(End)

			100 is a palindrome in bases 3, 7 and 9, so a(100) = 3.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Wikipedia, Base 1

Cf. A060873, A060874, A060875, A060876, A060877, A060878, A060879, A060947, A060948, A060949, A123586.

Cf. A000042.

f[n_] := Module[{p}, Table[ p = IntegerDigits[n, b]; If[p == Reverse@ p, {b, p}, Sequence @@ {}], {b, 2, Sqrt@ n}]]; Array[ Length@ f@# &, 105] (* Robert G. Wilson v, Nov 01 2014 *)

a(n) = {my(nb = 0); for (b=2, sqrt(n), d = digits(n, b); nb+= (Vecrev(d) == d);); nb;} \\ Michel Marcus, Nov 05 2014

a(n) = A135551(n) - A033831(n). - Robert G. Wilson v, Nov 01 2014

a(58) from Robert G. Wilson v, Nov 05 2014

cp's OEIS Frontend

A033855 Numbers k such that j(k)*phi(k) = s(phi(k)), where j(k) = A033831(k), s(k) = sigma(k) - k.

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A033856 Numbers k such that j(k)*phi(k) = sigma(phi(k)), j(k) = A033831(k).

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A033858 Numbers k such that j(k) + ud(k) = d(k), where j(k) = A033831, ud(k) = number of unitary divisors of k (A034444), d(k) = number of divisors of k (A000005).

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A033859 Numbers k such that A033831(k) = A034444(k) where A034444(k) = number of unitary divisors of k.

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A033872 Numbers k such that A033831(k) is prime.

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PARI

A033852 Integers k such that j(k)*phi(k) = sigma(k), where j(n) = A033831(n).

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PARI

A033853 Integers k such that j(k)*d(k)=phi(k), where j = A033831.

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A033854 Numbers k such that j(k)*ud(k)=phi(k), j(n) = A033831, ud(n) = number of unitary divisors of n (A034444).

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A033857 Numbers n such that j(n)*phi(n) = usigma(n), where j(n) = A033831, usigma(n) = sum of unitary divisors of n (A034448).

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A100563 Number of bases less than sqrt(n) in which n is a palindrome.

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