A034448 -id:A034448 - cp's OEIS frontend

A327159 Size of the cycle containing n in the map x -> usigma(x)-x or 0 if n is not a member of any finite cycle. Here usigma is the sum of unitary divisors of n (A034448).

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

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

			Because A034460(6) = 6, a(6) = 1.
Because A034460(30) = 42, A034460(42) = 54, A034460(54) = 30, a(30) = a(42) = a(54) = 3.
Because A034460(90) = 90, a(90) = 1. Because A034460(78) = 90, a(78) = 0, as even though 78 ends into a cycle of one, it itself is not a part of that cycle.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A002827 (positions of ones), A063991 (of 2's), A319902 (of 4's), A097024 (of 5's), A319917 (of 6's), A319937 (of 10's), A097030 (of 14's), A327157 (of all nonzero terms).

Cf. also A034448, A034460, A097031, A318880, A318882, A327162.

a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#] == 1 &]] - n /; n > 0
cycleL[k_] := Module[{nL=NestWhileList[a034460, k, UnsameQ, All]}, If[k==Last[nL], Length[nL]-1, 0]]
a327159[n_] := Map[cycleL, Range[n]]
a327159[120] (* Hartmut F. W. Hoft, Feb 04 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A327159(n,orgn=n,xs=Set([])) = if(1==n,0,if(vecsearch(xs,n), if(n==orgn,length(xs),0), xs = setunion([n],xs); A327159(A034460(n),orgn,xs)));

A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 3, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 3, 26, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 50, 1, 1, 1, 3, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1, 1, 5, 1, 1, 1, 3, 1, 10, 1, 5, 1, 1, 1, 11, 1, 50, 10, 130

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 6 != 3*10 = a(8) * a(9).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).

Cf. also A344697, A348049.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (usigma = Times @@ f1 @@@ (fct = FactorInteger[n])) / GCD[usigma, Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };

a(n) = A034448(n) / A348503(n) = A034448(n) / gcd(A000203(n), A034448(n)).

A348947 a(n) = A348944(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 23, 1, 1, 1, 1, 46, 1, 1, 1, 97, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 23, 1, 6, 1, 1, 1, 1, 1, 1, 1, 397, 1, 1, 1, 1, 1, 1, 1, 87, 1, 1, 1, 1, 1, 1, 1, 49, 169, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 46, 1, 1, 1, 227

Offset: 1

Antti Karttunen, Nov 05 2021

Numerator of ratio A348944(n) / A000203(n).

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 97 <> 1 = a(4)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348948 (denominators).

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (s = (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2) / GCD[Times @@ f1 @@@ f, s]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348947(n) = { my(u=A348944(n)); (u/gcd(sigma(n),u)); };

a(n) = A348944(n) / A348946(n) = A348944(n) / gcd(A000203(n), A348944(n)).

A348948 a(n) = sigma(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 20, 1, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 20, 1, 5, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 31, 121, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 217

Offset: 1

Antti Karttunen, Nov 05 2021

Denominator of ratio A348944(n) / A000203(n).

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 91 <> 1 = a(4)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348947 (numerators).

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, (Times @@ f2 @@@ f + Times @@ f3 @@@ f) / 2]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348948(n) = { my(s=sigma(n)); (s/gcd(s,A348944(n))); };

a(n) = A000203(n) / A348946(n) = A000203(n) / gcd(A000203(n), A348944(n)).

A348996 a(n) = usigma(A276086(n)), where usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 10, 30, 6, 18, 24, 72, 60, 180, 26, 78, 104, 312, 260, 780, 126, 378, 504, 1512, 1260, 3780, 626, 1878, 2504, 7512, 6260, 18780, 8, 24, 32, 96, 80, 240, 48, 144, 192, 576, 480, 1440, 208, 624, 832, 2496, 2080, 6240, 1008, 3024, 4032, 12096, 10080, 30240, 5008, 15024, 20032, 60096, 50080, 150240, 50, 150

Offset: 0

Antti Karttunen, Nov 07 2021

Index entries for sequences related to primorial base

Cf. A034448, A276086, A348997, A349000.

Cf. also A324653, A346470, A348949.

A348996(n) = { my(m=1, p=2); while(n, if(n%p, m *= (1+(p^(n%p)))); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A034448(A276086(n)).

A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 2, 6, 6, 18, 24, 72, 12, 36, 2, 6, 8, 24, 4, 12, 18, 54, 72, 216, 36, 108, 2, 6, 8, 24, 4, 12, 8, 24, 32, 96, 16, 48, 48, 144, 192, 576, 96, 288, 16, 48, 64, 192, 32, 96, 144, 432, 576, 1728, 288, 864, 16, 48, 64, 192, 32, 96, 2, 6, 8, 24, 4, 12, 12, 36, 48, 144, 24, 72, 4, 12, 16, 48, 8, 24, 36, 108, 144

Offset: 0

Antti Karttunen, Nov 07 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003959, A034448, A276086, A348733, A348949, A348996.

Cf. also A346471 for similar construction. (Compare the scatter plots).

A348997(n) = { my(m1=1, m2=1, p=2); while(n, if(n%p, m1 *= ((1+p)^(n%p)); m2 *= (1+(p^(n%p)))); n = n\p; p = nextprime(1+p)); gcd(m1, m2); };

a(n) = A348733(A276086(n)) = gcd(A348949(n), A348996(n)).

A379513 Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 4, 19, 107, 39, 61, 259, 817, 853, 97, 301, 307, 2209, 187, 2279, 39583, 121129, 122557, 124699, 126127, 509863, 171541, 173921, 526523, 6930479, 6983519, 7063079, 7118771, 7193027, 802663, 405199, 13495327, 1131701, 30726097, 123670153, 622026437, 11910394103

Offset: 1

Amiram Eldar, Dec 23 2024

			Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 817/360, 853/360, 97/40, 301/120, 307/120, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.

Cf. A034448, A064609, A370898, A379514 (denominators), A379515.

Cf. A001620, A308041.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A034448(k)).

a(n)/A379514(n) = B * log(n) + D + O(log(n)^(5/3) * log(log(n))^(4/3) / n), where B = A308041, D = B * (gamma + A1 - A2), gamma = A001620, A1 = Sum_{p prime} ((p*C(p)*log(p)/(p-1)) * Sum_{k>=1} (k/(p^k*(p^(k+1)+1)))), A2 = Sum_{p prime} ((C(p)*log(p)/p^2) * Sum_{k>=0} (1/(p^k*(p^(k+1)+1)))), and C(p) = 1 - (p/(p-1)) * Sum_{k>=1} (1/(p^k*(p^(k+1)+1))) (Sita Ramaiah and Suryanarayana, 1980).

A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 3, 12, 60, 20, 30, 120, 360, 360, 40, 120, 120, 840, 70, 840, 14280, 42840, 42840, 42840, 42840, 171360, 57120, 57120, 171360, 2227680, 2227680, 2227680, 2227680, 2227680, 247520, 123760, 4084080, 340340, 9189180, 36756720, 183783600, 3491888400, 3491888400

Offset: 1

Amiram Eldar, Dec 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.

Cf. A034448, A064609, A370898, A379513 (numerators), A379516.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A034448(k)).

A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 1791, 2047, 2431, 4095, 8191, 14335, 14847, 16383, 27391, 32767, 44031, 57855, 65535, 114687, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 8978431, 12058623, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A002827, A004767, A034448.

For similar sequences, see A336700, A387410, A387415, A387419.

A000265(n) = (n>>valuation(n,2));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
isA387418(n) = !((1+A000265(A034448(n)))%A000265(1+n));

A064012 usigma(usigma(n))=3*n where usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

10, 30, 288, 660, 720, 2146560

Offset: 1

Felice Russo, Sep 07 2001

nonn

Is this sequence finite?
18 is the only known integer usigma(usigma(n)) = kn for some integer k >= 4 (no other one <= 2^30). - Tomohiro Yamada, Apr 22 2017
a(7) > 2.85*10^11, if it exists. The same bound holds also for any n > 18 such usigma(usigma(n)) = k*n for some integer k >= 4. - Giovanni Resta, Apr 10 2019

			10 belongs to the sequence because usigma(usigma(10)) = 30 = 3*10.

V. Sitaramaiah and M. V. Subbarao, On the equation usigma(usigma(n)) = 2n, Util. Math. 53 (1998), 101--124.

Cf. A034448, A038843.

One more term from Naohiro Nomoto, Oct 21 2001

No other terms < 1070000000, Jud McCranie, Oct 28 2001

cp's OEIS Frontend

A327159 Size of the cycle containing n in the map x -> usigma(x)-x or 0 if n is not a member of any finite cycle. Here usigma is the sum of unitary divisors of n (A034448).

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A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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A348947 a(n) = A348944(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

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A348948 a(n) = sigma(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

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A348996 a(n) = usigma(A276086(n)), where usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

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A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

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A379513 Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

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A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

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