A064380 -id:A064380 - cp's OEIS frontend

A176472 Smallest m for which A064380(m) - A000010(m) = n.

2, 4, 9, 12, 22, 18, 38, 16, 93, 45, 62, 70, 44, 63, 36, 52, 64, 102, 48, 68, 84, 76, 90, 142, 146, 117, 81, 166, 174, 178, 126, 80, 150, 132, 116, 230, 124, 100, 156, 246, 266, 258, 254, 148, 112

Offset: 0

Vladimir Shevelev, Apr 18 2010

nonn

My 1981 publication studies A064380 with the quite natural convention A064380(1)=1. So a(1) could alternatively be defined as 1. By the definitions, it is clear that A064380(m) >= A000010(m).

Theorem. For every n >= 0, the equation A064380(m) - A000010(m) = n has infinitely many solutions.

V. S. Abramovich (Shevelev), On an analog of the Euler function, Proceeding of the North-Caucasus Center of the Academy of Sciences of the USSR (Rostov na Donu), 2 (1981), 13-17.
Vladimir Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 0..1000
Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A000010, A064380, A050376, A050292.

A176472 := proc(n) local m; for m from 2 do if A064380(m) - numtheory[phi](m) = n then return m; end if; end do: end proc: # R. J. Mathar, Jun 16 2010

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[All, 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0&]]];
A064380[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
a[n_] := a[n] = For[m = 2, True, m++, If[A064380[m] - EulerPhi[m] == n, Return[m]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 05 2023, after Amiram Eldar in A064380 *)

a(2), a(3), a(8) and a(15) corrected and sequence extended by R. J. Mathar, Jun 16 2010

A176509 Composite numbers m for which A064380(m) = A000010(m).

Original entry on oeis.org

8, 27, 125, 128, 343, 1331, 2187, 2197, 4913, 6859, 12167, 24389, 29791, 32768, 50653, 68921, 78125, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 823543, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091

Offset: 1

Vladimir Shevelev, Apr 19 2010

nonn

Theorem. A064380(m) = A000010(m) iff m has the form m=p^(2^k-1), k>=1, p a prime. Eliminating the primes (k=1), the terms of the sequence have this form for k>1. All terms of A030078 (k=2) and A092759 (k=3) and prime powers of A010803 (k=4) are in the sequence, for example.

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

Cf. A000010, A010803, A030078, A050376, A056824, A064380, A092759, A176472.

seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 3]]], PrimeQ], e, k, s = {}}, Do[e = Floor[Log[ps[[i]], max]]; k = Floor[Log2[e + 1]]; s = Join[s, ps[[i]]^(2^Range[2, k] - 1)], {i, 1, Length[ps]}]; Sort[s]]; seq[3*10^6] (* Amiram Eldar, Mar 26 2023 *)

is(n)=my(e=isprimepower(n));e>2 && 2^valuation(e+1,2)==e+1 \\ Charles R Greathouse IV, Feb 19 2013

a(n) ~ n^3 log^3 n. - Charles R Greathouse IV, Feb 19 2013

Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/P(2^k-1) = 0.183077059924063305405..., where P(s) is the prime zeta function. - Amiram Eldar, Jul 11 2024

128 inserted, 1024 deleted, 2187 inserted, 32768 inserted, etc. - R. J. Mathar, Nov 21 2010

More terms from Amiram Eldar, Mar 26 2023

A185373 The numerator of the fraction |n^2/A049417(n)-A064380(n)|.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 4, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 5, 1, 2, 1, 2, 9, 3, 1, 1, 1, 4, 11, 11, 25, 2, 1, 1, 9, 2, 1, 11, 1, 4, 3, 7, 1, 2, 1, 2, 1, 13, 1, 3, 1, 13, 49, 17, 1, 0, 1, 1, 49, 16, 25, 1, 1, 17, 19, 35, 1, 14, 1, 2

Offset: 2

Vladimir Shevelev, Feb 17 2011

n^2/A049417(n) is a multiplicative function, whereas A064380 is not. This sequence here measures the (small) differences n^2/A049417(n)-A064380(n) = 1/3, 1/4, 1/5, 1/6, 0, 1/8, 4/15, 1/10, 5/9, 1/12, 1/5 ...

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A185383 (denominators)

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1);
infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
a[n_] := Abs[Numerator[n^2 / isigma[n] - Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]]]; Array[a, 100, 2] (* Amiram Eldar, Mar 20 2025 *)

A185383 a(n) is the denominator of the fraction |n^2/A049417(n)-A064380(n)|.

Original entry on oeis.org

3, 4, 5, 6, 1, 8, 15, 10, 9, 12, 5, 14, 6, 8, 17, 18, 5, 20, 3, 32, 9, 24, 5, 26, 21, 40, 5, 30, 2, 32, 51, 16, 27, 48, 25, 38, 15, 56, 9, 42, 8, 44, 15, 4, 18, 48, 17, 50, 39, 8, 35, 54, 10, 72, 15, 80, 45, 60, 1, 62, 24, 80, 85, 84, 4, 68, 45, 32, 36, 72, 25, 74, 57

Offset: 2

Vladimir Shevelev, Feb 17 2011

If A185373(n)=0, then we accept a(n)=1.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A185373, A064380, A049417, A050376.

a(n)=n+1 iff n is in A050376.

a(73) corrected by Amiram Eldar, Sep 18 2019

A185079 a(n) = A064380(n) * A049417(n).

Original entry on oeis.org

3, 8, 15, 24, 36, 48, 60, 80, 90, 120, 140, 168, 192, 216, 255, 288, 330, 360, 390, 448, 504, 528, 600, 624, 672, 720, 760, 840, 936, 960, 1020, 1056, 1134, 1200, 1300, 1368, 1440, 1512, 1620, 1680, 1632, 1848, 1920, 1980, 2088, 2208, 2312, 2400, 2496, 2592, 2730, 2808, 2880, 3024, 3240, 3200, 3330

Offset: 2

Vladimir Shevelev, Feb 18 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A007357, A185078, A185373, A185383.

a(n) = n^2 + o(n^(1+eps)).

Corrected and extended by T. D. Noe, Feb 18 2011

A186777 Solutions x of the equation A064380(x)-A000010(x)=1 in integers x>=2.

Original entry on oeis.org

4, 6, 10, 15, 35, 77, 91, 143, 187, 209, 221, 247, 299, 323, 391, 437, 493, 527, 551, 589, 667, 703, 713, 851, 899, 943, 989, 1073, 1147, 1189, 1247, 1271, 1333, 1363, 1457, 1517, 1537, 1591, 1643, 1739, 1763, 1829, 1891, 1927, 1961, 2021, 2173, 2183, 2257, 2279

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

Equals {4, 6, 10} UNION A082663. - Eric M. Schmidt, Oct 04 2013

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric M. Schmidt, The Near Equivalence of A082663 and A186777

Cf. A000010, A064380, A082663, A176472, A176509.

More terms from Amiram Eldar, Sep 14 2019

A185078 Numbers k for which A064380(k) = k/2.

Original entry on oeis.org

2, 6, 8, 10, 60, 70, 128, 136, 9822, 18632, 32768, 32896, 36720, 69726, 73662, 73686, 73734, 85962, 86046, 87114, 87198, 87222, 87258, 87294, 87306, 87342, 87366, 87546, 87558, 88014, 88278, 88302, 88338, 88386, 127326, 128046, 128082, 128382, 128406, 128598, 128802

Offset: 1

Vladimir Shevelev, Feb 18 2011

nonn

Note that, if there exist infinitely many infinitary perfect numbers (A007357), then, as k tends to infinity over such numbers, A064380(k)/k = 1/2 + o(k^(-1+eps)). We conjecture that here A064380(k)/k = 1/2 infinitely many times, and thus the sequence contains infinitely many infinitary perfect numbers.

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A064380, A007357, A064380, A185373, A185383.

a(7)-a(13) from Amiram Eldar, Sep 13 2019

a(14)-a(41) from Amiram Eldar, Mar 26 2023

A186778 Solutions n of the equation A064380(n)-A000010(n)=2 in integers n>=2.

Original entry on oeis.org

9, 14, 21, 24, 33, 40, 55, 65, 119, 133, 253, 319, 341, 377, 403, 481, 629, 697, 731, 779, 799, 817, 893, 1007, 1081, 1219, 1357, 1403, 1541, 1711, 1769, 1943, 2059, 2077, 2117, 2201, 2263, 2291, 2407, 2449, 2573, 2759, 2923, 3071, 3293, 3403, 3589, 3649, 3737

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..420

Cf. A000010, A064380, A176472, A176509, A186777.

More terms from Amiram Eldar, Sep 14 2019

A186779 Solutions n of the equation A064380(n)-A000010(n)=3 in integers n>=2.

Original entry on oeis.org

12, 39, 56, 85, 88, 95, 104, 161, 407, 451, 473, 533, 559, 611, 901, 1003, 1037, 1121, 1139, 1159, 1273, 1349, 1387, 1633, 1679, 1817, 1909, 2047, 2581, 2813, 2929, 2987, 3007, 3103, 3131, 3161, 3193, 3277, 3317, 3379, 3503, 4181, 4699, 4847, 5069, 5143, 5207

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A000010, A064380, A176472, A176509.

More terms from Amiram Eldar, Sep 14 2019

A186780 Solutions n of equation A064380(n)-A000010(n)=4 in integers n>=2.

Original entry on oeis.org

22, 25, 26, 32, 51, 57, 115, 135, 136, 145, 189, 203, 217, 297, 517, 583, 689, 767, 793, 1207, 1241, 1343, 1411, 1501, 1577, 1691, 2231, 2323, 2369, 2461, 2507, 2599, 3683, 3799, 3937, 3973, 4031, 4061, 4247, 4309, 4619, 4681, 5513, 5587, 5809, 6031, 6179, 6401

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The sequence is infinite.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A000010, A064380, A176472, A176509, A186777, A186778, A186779.

More terms from Amiram Eldar, Sep 14 2019

cp's OEIS Frontend

A176472 Smallest m for which A064380(m) - A000010(m) = n.

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A176509 Composite numbers m for which A064380(m) = A000010(m).

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A185373 The numerator of the fraction |n^2/A049417(n)-A064380(n)|.

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A185383 a(n) is the denominator of the fraction |n^2/A049417(n)-A064380(n)|.

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A185079 a(n) = A064380(n) * A049417(n).

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A186777 Solutions x of the equation A064380(x)-A000010(x)=1 in integers x>=2.

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A185078 Numbers k for which A064380(k) = k/2.

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A186778 Solutions n of the equation A064380(n)-A000010(n)=2 in integers n>=2.

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A186779 Solutions n of the equation A064380(n)-A000010(n)=3 in integers n>=2.

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