A091732 -id:A091732 - cp's OEIS frontend

A361922 Infinitary phi-practical numbers: numbers m such that each k <= m is a subsum of a the multiset {iphi(d) : d infinitary divisor of m}, where iphi is an infinitary analog of Euler's phi function (A091732).

1, 2, 3, 6, 8, 12, 15, 24, 30, 40, 42, 56, 60, 72, 84, 105, 108, 120, 132, 135, 156, 165, 168, 195, 210, 216, 240, 255, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 420, 440, 456, 462, 480, 504, 510, 520, 540, 546, 552, 570, 600, 616, 640, 660, 672, 680, 690

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

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

Cf. A077609, A091732.

Similar sequences: A260653, A286906, A334901.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
iPhiPracticalQ[n_] := Module[{s = Sort@ Map[iphi, idivs[n]], ans = True}, Do[If[s[[j]] > Sum[s[[i]], {i, 1, j - 1}] + 1, ans = False; Break[]], {j, 1, Length[s]}]; ans]; Select[Range[700], iPhiPracticalQ]

A361923 Number of distinct values obtained when the infinitary totient function (A091732) is applied to the infinitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 2, 7, 4, 2, 4

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

First differs from A348001 at n = 27.

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

Cf. A077609, A091732.

Similar sequences: A319696, A348001.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
a[n_] := Length @ Union[iphi /@ idivs[n]]; Array[a, 100]

iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
a(n) = {my(d = idivs(n)); #Set(apply(x->iphi(x), d));}

A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

First differs from A003159 at n = 57.
Numbers k such that A361923(k) = A037445(k).
Since Sum_{d infinitary divisor of k} iphi(d) = k, these are numbers k such that the multiset {iphi(d) | d infinitary divisor of k} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345) and all the powers of 4 (A000302).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 66, 651, 6497, 64894, 648641, 6485605, 64851632, 648506213, 6485025363, ... . Apparently, this sequence has an asymptotic density 0.6485...

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

Cf. A000302, A003159, A037445, A061345, A077609, A091732, A361923.

Similar sequences: A326835, A348004.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
q[n_] := Length @ Union[iphi /@ (d = idivs[n])] == Length[d]; Select[Range[100], q]

iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
is(k) = {my(d = idivs(k)); #Set(apply(x->iphi(x), d)) == #d;}

A362664 Numbers k with exactly two solutions x to the equation iphi(x) = k, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

1, 2, 3, 4, 10, 15, 20, 22, 28, 42, 44, 45, 46, 52, 54, 56, 58, 70, 78, 82, 92, 100, 102, 104, 106, 116, 130, 136, 140, 148, 162, 164, 166, 172, 174, 178, 184, 190, 196, 200, 204, 208, 212, 220, 222, 226, 228, 234, 238, 246, 250, 255, 260, 262, 268, 272, 282, 292, 296

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

Numbers k such that A362485(k) = 2.

There are no numbers k with a single solution to iphi(x) = k, because if iphi(x) = k, and A007814(x) is even, then 2*x is also a solution, i.e., iphi(2*x) = k.

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

Cf. A007814, A091732, A362484, A362485.

Similar sequences: A361969, A362185.

Select[Range[300], Length[invIPhi[#]] == 2 &] (* using the function invIPhi from A362484 *)

A332971 Infinitary phibonacci numbers: solutions k of the equation iphi(k) = iphi(k-1) + iphi(k-2) where iphi(k) is an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

3, 4, 7, 23, 121, 2857, 5699, 6377, 9179, 46537, 63209, 244967, 654497, 1067873, 1112009, 3435929, 3831257, 6441593, 7589737, 7784507, 8149751, 14307856, 22434089, 24007727, 24571871, 44503417, 44926463, 56732729, 128199059, 140830367, 190145936, 401767631, 403152737

Offset: 1

Amiram Eldar, Mar 04 2020

nonn

			7 is a term since iphi(7) = 6 and iphi(5) + iphi(6) = 4 + 2 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..70

An infinitary version of A065557.

Cf. A065900, A075565, A076136, A076251, A091732, A145469, A291126, A291176, A292033, A294995.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], 1])); iphi[1] = 1; iphi[n_] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) - 1); Select[Range[3, 10^5], iphi[#] == iphi[# - 1] + iphi[# - 2] &]

A344772 Ordinal transform of infinitary phi, A091732.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 2, 3, 2, 1, 4, 1, 4, 2, 2, 1, 4, 1, 2, 1, 3, 2, 3, 1, 3, 4, 5, 1, 6, 1, 2, 1, 2, 1, 3, 1, 5, 2, 2, 1, 4, 2, 4, 3, 2, 1, 6, 1, 4, 2, 1, 3, 2, 1, 4, 1, 7, 1, 8, 1, 4, 5, 1, 2, 9, 1, 3, 1, 3, 1, 5, 1, 2, 1, 5, 1, 3, 2, 2, 4, 2, 3, 6, 1, 6, 2, 4, 1, 4, 1, 6, 7

Offset: 1

Antti Karttunen, May 31 2021

Number of values of k, 1 <= k <= n, with A091732(k) = A091732(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A091732, A050376, A302777.

Cf. also A081373 (ordinal transform of phi), A303756 (of Carmichael's lambda), A330739 (of unitary phi).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1);
b[_] = 0;
a[n_] := a[n] = With[{t = iphi[n]}, b[t] = b[t] + 1];
Array[a, 105] (* Jean-François Alcover, Dec 27 2021, after Amiram Eldar in A091732 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
v344772 = ordinal_transform(vector(up_to,n,A091732(n)));
A344772(n) = v344772[n];

A361925 Infinitary phi-practical (A361922) whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

Original entry on oeis.org

1, 3, 12, 15, 60, 105, 108, 132, 156, 165, 195, 240, 255, 660, 960, 1020, 1140, 1155, 1188, 1380, 1680, 1716, 1728, 1740, 1785, 1860, 1995, 2052, 2145, 2220, 2244, 2415, 2460, 2484, 2496, 2508, 2580, 2640, 2652, 2805, 2820, 2940, 3036, 3045, 3120, 3132, 3135, 3180

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

An infinitary phi-practical number k is a number k such that each number in the range 1..k is a subsum of the multiset {iphi(d) | d infinitary divisor of k}. This sequence is restricted to cases in which all the values in this multiset are distinct.

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

Intersection of A361922 and A361924.
Cf. A091732.
Similar sequences: A359417, A359418.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
iPhiPracticalQ[n_] := Module[{s = Sort@Map[iphi, idivs[n]], ans = True}, Do[If[s[[j]] > Sum[s[[i]], {i, 1, j - 1}] + 1, ans = False; Break[]], {j, 1, Length[s]}]; ans];
Select[Range[3200], UnsameQ @@ iphi /@ idivs[#] && iPhiPracticalQ[#] &]

A362489 a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

5, 1, 6, 12, 36, 24, 396, 48, 216, 96, 528, 144, 384, 2784, 432, 240, 1296, 288, 1584, 1800, 480, 1680, 1080, 864, 576, 3240, 2016, 960, 6624, 720, 1152, 7776, 12000, 8448, 5280, 1728, 10752, 2304, 4032, 4800, 6048, 3840, 2160, 5184, 4608, 6336, 1440, 10560, 29568

Offset: 0

Amiram Eldar, Apr 22 2023

nonn

a(n) is the least number k such that A362485(k) = 2*n. Odd values of A362485 are impossible.

Is there any n for which a(n) = -1?

Amiram Eldar, Table of n, a(n) for n = 0..300

Cf. A091732, A362484, A362485.

Similar sequences: A007374, A063507, A361970, A362186.

solnum[n_] := Length[invIPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k]/2 + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invIPhi from A362484 *)

A362665 a(n) is the smaller of the two solutions to A091732(x) = A362664(n).

Original entry on oeis.org

1, 3, 4, 5, 11, 16, 33, 23, 29, 43, 69, 64, 47, 53, 76, 87, 59, 71, 79, 83, 141, 101, 103, 159, 107, 177, 131, 137, 213, 149, 163, 249, 167, 173, 236, 179, 235, 191, 197, 303, 309, 265, 321, 253, 223, 227, 229, 316, 239, 332, 251, 256, 393, 263, 269, 411, 283

Offset: 1

Amiram Eldar, Apr 29 2023

nonn

The larger solution is 2*a(n).

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

Cf. A091732, A362484, A362664.

Similar sequences: A131826, A362211, A362212.

invIPhi[#][[1]]& /@ Select[Range[300], Length[invIPhi[#]] == 2 &] (* using the function invIPhi from A362484 *)

A049417 a(n) = isigma(n): sum of infinitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90

Offset: 1

Yasutoshi Kohmoto, Dec 11 1999

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = Product_{q in Q_n}(q+1) =  n*Product_{q in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014
1/a(n) is the asymptotic density of numbers that are infinitarily divided by n (i.e., numbers whose set of infinitary divisors includes n). - Amiram Eldar, Jul 23 2025

			If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
n = 90 = 2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - _Vladimir Shevelev_, Feb 19 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7417 from R. J. Mathar)
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, pp. 49-56.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, Vol. 47 (2017), pp. 211-218; arXiv preprint, arXiv:1705.10933 [math.NT], 2017.

Cf. A004607, A037445, A050376, A077609, A091732, A127661, A293355.
Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
Cf. A000079, A001615, A027748, A030308, A124010, A126168.

a049417 1 = 1
a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
           (map (subtract 1 . (p ^)) a000079_list)
-- Reinhard Zumkeller, Sep 18 2015

isidiv := proc(d, n)
    local n2, d2, p, j;
    if n mod d <> 0 then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        padic[ordp](n,p) ;
        n2 := convert(%, base, 2) ;
        padic[ordp](d,p) ;
        d2 := convert(%, base, 2) ;
        for j from 1 to nops(d2) do
            if op(j, n2) = 0 and op(j, d2) <> 0 then
                return false;
            end if;
        end do:
    end do;
    return true;
end proc:
idivisors := proc(n)
    local a, d;
    a := {} ;
    for d in numtheory[divisors](n) do
        if isidiv(d, n) then
            a := a union {d} ;
        end if;
    end do:
    a ;
end proc:
A049417 := proc(n)
    local d;
    add(d, d=idivisors(n)) ;
end proc:
seq(A049417(n),n=1..100) ; # R. J. Mathar, Feb 19 2011

bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
(* Second program: *)
a[n_] := If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
Array[a, 100] (* Jean-François Alcover, Mar 23 2020, after Paul Abbott in A077609 *)

A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[,2], b = binary(f[k,2]); prod(j=1, #b, if(b[j], 1+f[k,1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014

isigma(n)=vecprod([vecprod([f[1]^2^k+1|k<-[0..exponent(f[2])], bittest(f[2],k)])|f<-factor(n)~]) \\ M. F. Hasler, Oct 20 2022

from math import prod
from sympy import factorint
def A049417(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

Multiplicative: If e = Sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = Product_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005 [This means there is a factor p^2^k + 1 if d_k = 1, otherwise the factor is 1. - M. F. Hasler, Oct 20 2022]
Let n = Product(q_i) where {q_i} is a set of distinct terms of A050376. Then a(n) = Product(q_i + 1). - Vladimir Shevelev, Feb 19 2011
If n is squarefree, then a(n) = A001615(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{k>=1} A077609(n,k). - R. J. Mathar, Oct 04 2017
a(n) = A126168(n)+n. - R. J. Mathar, Oct 05 2017
Multiplicative with a(p^e) = Product{k >= 0, e_k = 1} p^2^k + 1, where e = Sum e_k 2^k, i.e., e_k is bit k of e. - M. F. Hasler, Oct 20 2022
a(n) = iphi(n^2)/iphi(n), where iphi(n) = A091732(n). - Amiram Eldar, Sep 21 2024

More terms from Wouter Meeussen, Sep 02 2001

cp's OEIS Frontend

A361922 Infinitary phi-practical numbers: numbers m such that each k <= m is a subsum of a the multiset {iphi(d) : d infinitary divisor of m}, where iphi is an infinitary analog of Euler's phi function (A091732).

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A361923 Number of distinct values obtained when the infinitary totient function (A091732) is applied to the infinitary divisors of n.

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A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

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A362664 Numbers k with exactly two solutions x to the equation iphi(x) = k, where iphi is the infinitary totient function A091732.

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A332971 Infinitary phibonacci numbers: solutions k of the equation iphi(k) = iphi(k-1) + iphi(k-2) where iphi(k) is an infinitary analog of Euler's phi function (A091732).

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A344772 Ordinal transform of infinitary phi, A091732.

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A361925 Infinitary phi-practical (A361922) whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

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A362489 a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.

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A362665 a(n) is the smaller of the two solutions to A091732(x) = A362664(n).

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