A064380 -id:A064380 - cp's OEIS frontend

A301866 Numbers k such that iphi(k) = iphi(k+1), where iphi is the infinitary totient function (A064380).

1, 21, 143, 208, 314, 459, 957, 1652, 2685, 5091, 20155, 38180, 41265, 45716, 54722, 116937, 161001, 186794, 230390, 274533, 338547, 416577, 430137, 495187

Offset: 1

Amiram Eldar, Mar 28 2018

a(16) > 10^5. - Robert Price, May 22 2018

			iphi(21) = iphi(22) = 14, thus 21 is in the sequence.

Cf. A064379, A064380.

irelprime[n_] := Select[temp = iDivisors[n]; Range[n], Intersection[iDivisors[#], temp] === {1} &]; bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ {0, #1} & /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]];
  iDivisors[k_Integer] := Sort[(Times @@ (First[it]^(#1 /. z -> List)) &) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]]; iDivisors[1] := {1}; iphi[n_] := Length[irelprime[n]]; iphiQ[n_] := iphi[n] == iphi[n + 1]; Select[Range[10^3], iphiQ](* after Wouter Meeussen at A064379 *)

a(11)-a(15) from Robert Price, May 22 2018

a(16)-a(24) from Amiram Eldar, Mar 26 2023

A362024 The number of iterations of the infinitary totient function iphi (A064380) required to reach from n to 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 4, 5, 5, 6, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 6, 7, 6, 7, 8, 8, 9, 10, 7, 8, 8, 7, 7, 8, 9, 10, 7, 9, 8, 9, 9, 10, 9, 8, 11, 12, 8, 9, 9, 10, 10, 11, 7, 10, 9, 9, 11, 12, 8, 9, 9, 10, 9, 10, 8, 9, 11, 10, 9, 10, 8, 9, 9, 8, 10, 10, 10, 11, 11, 12

Offset: 2

Amiram Eldar, Apr 05 2023

nonn

			a(6) = 3 since there are 3 iterations from 6 to 1: iphi(6) = 3, iphi(3) = 2 and iphi(2) = 1.

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

Cf. A064380, A362025 (indices of records).

Similar sequences: A003434, A049865, A333609.

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
iphi[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
a[n_] := Length@ NestWhileList[iphi, n, # > 1 &] - 1;
Array[a, 100, 2]

isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
iphi(n) = sum(j = 1, n-1, isinfcoprime(j, n));
a(n) = if(n==2, 1, a(iphi(n)) + 1);

a(n) = a(A064380(n)) + 1 for n > 2.

A373057 Numbers k such that iphi(k) divides k, where iphi is the infinitary Euler phi function (A064380).

Original entry on oeis.org

2, 6, 8, 10, 60, 70, 120, 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

Offset: 1

Amiram Eldar, May 21 2024

nonn

Numbers k such that the number of numbers less than k that are infinitarily relatively prime to k is a divisor of k.

			2 is a term since ipghi(2) = 1 divides 2.
6 is a term since ipghi(6) = 6 divides 6.
60 is a term since ipghi(60) = 30 divides 60.

Amiram Eldar, Table of n, a(n) for n = 1..218

Cf. A064380.

Similar sequences: A007694, A097296, A319481, A335327.

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]]; q[n_] := Divisible[n, Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]]; Select[Range[2, 200], q]

isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
is(n) = if(n < 2, 0, !(n % sum(j = 1, n-1, isinfcoprime(j, n))));

A186781 Integer solutions x to the equation A064380(x)-A000010(x)=5.

Original entry on oeis.org

18, 20, 30, 34, 42, 69, 152, 155, 259, 287, 351, 459, 513, 649, 671, 871, 923, 949, 1513, 1649, 1717, 1843, 1919, 1957, 2033, 2071, 2147, 2921, 3013, 3151, 4321, 4379, 4553, 4727, 4843, 4867, 5017, 5053, 5177, 5363, 5549, 5611, 7067, 7141, 7289, 7363, 7807, 8651

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions, see A176472.

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..200

Cf. A000010, A064380, A176472, A176509, A186777, A186778, A186779, A186780.

More terms from Amiram Eldar, Sep 14 2019

A362025 a(n) is the least number that reaches 1 after n iterations of the infinitary totient function A064380.

Original entry on oeis.org

2, 3, 4, 5, 9, 11, 16, 17, 28, 29, 46, 47, 99, 145, 167, 205, 314, 397, 437, 793, 851, 1137, 1693, 2453, 2771, 2989, 3701, 5099, 6801, 9299, 12031, 15811, 16816, 21520, 21521, 29547, 39685, 62077, 83191, 103473, 112117, 149535, 157159, 196049, 200267, 303022

Offset: 1

Amiram Eldar, Apr 05 2023

nonn

Cf. A064380.
Indices of records of A362024.
Similar sequences: A003271, A007755, A333610.

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
iphi[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
numiter[n_] := Length@ NestWhileList[iphi, n, # > 1 &] - 1;
seq[kmax_] := Module[{v = {}, n = 1}, Do[If[numiter[k] == n, AppendTo[v, k]; n++], {k, 2, kmax}]; v]; seq[1000]

isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
iphi(n) = sum(j = 1, n-1, isinfcoprime(j, n));
numiter(n) = if(n==2, 1, numiter(iphi(n)) + 1);
lista(kmax) = {my(n = 1); for(k = 2, kmax, if(numiter(k) == n, print1(k, ", "); n++)); }

A362024(a(n)) = n, and A362024(k) < n for all k < a(n).

A187033 Places k where the infinitary phi-function A064380(k) divides the infinitary sigma-function A049417(k).

Original entry on oeis.org

2, 3, 6, 14, 24, 40, 48, 51, 54, 57, 60, 120, 440, 450, 1246, 1824, 2280, 8802, 12720, 29019, 36720, 37245, 55428, 74822, 81480, 133950, 169435, 441030

Offset: 1

Vladimir Shevelev, Mar 02 2011

It is the infinitary analog of A020492.

Cf. A020492, A049417, A064380.

a(19)-a(24) from Amiram Eldar, Apr 06 2019

a(25)-a(28) from Amiram Eldar, Mar 26 2023

A030078 Cubes of primes.

Original entry on oeis.org

8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091, 2571353, 2685619, 3307949

Offset: 1

Patrick De Geest

Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001
Intersection of A014612 and A000578. Intersection of A014612 and A030513. - Wesley Ivan Hurt, Sep 10 2013
Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020
There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021

			a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.

Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen über Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Eric Weisstein's World of Mathematics, Prime Power.
Wikipedia, p-group, Classification.
Index to sequences related to prime signature

Other sequences that are k-th powers of primes are: A000040 (k=1), A001248 (k=2), this sequence (k=3), A030514 (k=4), A050997 (k=5), A030516 (k=6), A092759 (k=7), A179645 (k=8), A179665 (k=9), A030629 (k=10), A079395 (k=11), A030631 (k=12), A138031 (k=13), A030635 (k=16), A138032 (k=17), A030637 (k=18).
Cf. A060800, A131991, A000578, subsequence of A046099.
Subsequence of A007422 and of A054397.
Cf. A036966, A085541, A088453, A157289, A258600.

a030078 = a000578 . a000040
a030078_list = map a000578 a000040_list -- Reinhard Zumkeller, May 26 2012

[p^3: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Array[Prime[ # ]^3&, 5! ] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^3 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime, primerange
def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
print(aupton(35)) # Michael S. Branicky, Aug 27 2021

[p**3 for p in prime_range(100)] # Zerinvary Lajos, May 15 2007

n such that A062799(n) = 3. - Benoit Cloitre, Apr 06 2002
a(n) = A000040(n)^3. - Omar E. Pol, Jul 27 2009
A064380(a(n)) = A000010(a(n)). - Vladimir Shevelev, Apr 19 2010
A003415(a(n)) = A079705(n). - Reinhard Zumkeller, Jun 26 2011
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
A000005(a(n)) = 4. - Wesley Ivan Hurt, Sep 10 2013
a(n) = A119959(n) * A008864(n) -1.- R. J. Mathar, Aug 13 2019
Sum_{n>=1} 1/a(n) = P(3) = 0.1747626392... (A085541). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) (A157289).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) (A088453). (End)

A091732 Iphi(n): infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 3, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 6, 24, 12, 16, 18, 28, 8, 30, 15, 20, 16, 24, 24, 36, 18, 24, 12, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 16, 40, 18, 36, 28, 58, 24, 60, 30, 48, 45, 48, 20, 66, 48, 44, 24, 70, 24, 72, 36, 48

Offset: 1

Steven Finch, Mar 05 2004

Not the same as A064380.

With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = (A050376(i)-1) * (A050376(j)-1) * ... * (A050376(k)-1). (Cf. the first formula). - Antti Karttunen, Jan 15 2019

			a(6)=2 since 6=P_1*P_2, where P_1=2^(2^0) and P_2=3^(2^0); hence (P_1-1)*(P_2-1)=2.
12=3*4 (3,4 are in A050376). Therefore, a(12) = 12*(1-1/3)*(1-1/4) = 6. - _Vladimir Shevelev_, Feb 20 2011

Antti Karttunen, Table of n, a(n) for n = 1..21011
Graeme L. Cohen and Peter Hagis, Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (1993) 373-383.
Steven R. Finch, Unitarism and infinitarism, 2004.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A037445, A049417, A050376, A064380, A300841, A323413.

A091732 := proc(n) local f,a,e,p,b; a :=1 ; for f in ifactors(n)[2] do e := op(2,f) ; p := op(1,f) ; b := convert(e,base,2) ; for i from 1 to nops(b) do if op(i,b) > 0 then a := a*(p^(2^(i-1))-1) ; end if; end do: end do: a ; end proc:
seq(A091732(n),n=1..20) ; # R. J. Mathar, Apr 11 2011

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); Array[a, 100] (* Amiram Eldar, Feb 28 2020 *)

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); }; \\ Antti Karttunen, Jan 15 2019

Consider the set, I, of integers of the form p^(2^j), where p is any prime and j >= 0. Let n > 1. From the fundamental theorem of arithmetic and the fact that the binary representation of any integer is unique, it follows that n can be uniquely factored as a product of distinct elements of I. If n = P_1*P_2*...*P_t, where each P_j is in I, then iphi(n) = Product_{j=1..t} (P_j - 1).
From Vladimir Shevelev, Feb 20 2011: (Start)
Thus we have the following analog of the formula phi(n) = n*Product_{p prime divisors of n} (1-1/p): if the factorization of n over distinct terms of A050376 is n = Product(q) (this factorization is unique), then a(n) = n*Product(1-1/q). Thus a(n) is infinitary multiplicative, i.e., if n_1 and n_2 have no common i-divisors, then a(n_1*n_2) = a(n_1)*a(n_2). Now we see that this property is stronger than the usual multiplicativity, therefore a(n) is a multiplicative arithmetic function.
Add that Sum_{d runs i-divisors of n} a(d)=n and a(n) = n*Sum_{d runs i-divisors of n} A064179(d)/d. The latter formulas are analogs of the corresponding formulas for phi(n): Sum_{d|n} phi(d) = n and phi(n) = n*Sum_{d|n} mu(d)/d. (End).
a(n) = n - A323413(n). - Antti Karttunen, Jan 15 2019
a(n) <= A064380(n), with equality if and only if n is in A050376. - Amiram Eldar, Feb 18 2023

A177329 Number of factors in the representation of n! as a product of distinct terms of A050376.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 6, 6, 4, 5, 7, 8, 9, 10, 11, 12, 8, 9, 9, 11, 12, 13, 13, 14, 15, 16, 14, 15, 16, 17, 19, 21, 17, 16, 15, 16, 17, 18, 19, 20, 22, 23, 21, 21, 21, 22, 23, 22, 23, 25, 22, 23, 22, 24, 26, 28, 28, 29, 27, 28, 29, 30, 32, 34, 30, 31, 31, 28, 27, 28, 29, 30, 31, 33, 31, 31, 30

Offset: 2

Vladimir Shevelev, May 06 2010

nonn

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

Chai Wah Wu, Table of n, a(n) for n = 2..10000 (terms 2..1000 from Amiram Eldar)
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. A000120, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A176525.

read("transforms") ; A064547 := proc(n) f := ifactors(n)[2] ; a := 0 ; for p in f do a := a+wt(op(2,p)) ; end do: a ; end proc:
A177329 := proc(n) A064547(n!) ; end proc: seq(A177329(n),n=2..80) ; # R. J. Mathar, May 28 2010

f[p_, e_] := DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n!]; Array[a, 100, 2] (* Amiram Eldar, Aug 24 2024 *)

a(n) = vecsum(apply(x -> hammingweight(x), factor(n!)[,2])); \\ Amiram Eldar, Aug 24 2024

from collections import Counter
from sympy import factorint
def A177329(n): return sum(map(int.bit_count,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())) # Chai Wah Wu, Jul 18 2024

a(n) = Sum_{i} A000120(e_i), where n! = Product_{i} p_i^e_i is the prime factorization of n!.

a(n) = A064547(n!). - R. J. Mathar, May 28 2010

a(20)=10 inserted by Vladimir Shevelev, May 08 2010

Terms from a(14) onwards replaced according to the formula - R. J. Mathar, May 28 2010

A384247 The number of integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 8, 24, 12, 18, 18, 28, 8, 30, 16, 20, 16, 24, 24, 36, 18, 24, 16, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 18, 40, 24, 36, 28, 58, 24, 60, 30, 48, 48, 48, 20, 66, 48, 44, 24

Offset: 1

Amiram Eldar, May 23 2025

Analogous to A047994, as A064380 is analogous to A116550.

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

Row lengths of A384246.

Cf. A000010, A001146, A006519, A047994, A064380, A091732, A116550, A138302, A268335, A384248.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i,1]^(1 << valuation(f[i,2], 2))));}

Multiplicative with a(p^e) = p^e * (1 - 1/p^A006519(e)).
a(n) >= A091732(n), with equality if and only if n is in A138302.
a(n) <= A047994(n), with equality if and only if n is in A138302.
a(n) >= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) is odd if and only if n = 1 or 2^(2^k) for k >= 0 (A001146). a(2^(2^k)) = 2^(2^k)-1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.66718130416373472394..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).

cp's OEIS Frontend

A301866 Numbers k such that iphi(k) = iphi(k+1), where iphi is the infinitary totient function (A064380).

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A362024 The number of iterations of the infinitary totient function iphi (A064380) required to reach from n to 1.

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A373057 Numbers k such that iphi(k) divides k, where iphi is the infinitary Euler phi function (A064380).

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A186781 Integer solutions x to the equation A064380(x)-A000010(x)=5.

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A362025 a(n) is the least number that reaches 1 after n iterations of the infinitary totient function A064380.

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A187033 Places k where the infinitary phi-function A064380(k) divides the infinitary sigma-function A049417(k).

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A030078 Cubes of primes.

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A091732 Iphi(n): infinitary analog of Euler's phi function.

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