A008475 -id:A008475 - cp's OEIS frontend

A082081 a(n) = fixed point when the pseudo-log function A008475[ ] is iterated.

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 7, 13, 23, 11, 25, 8, 27, 11, 29, 7, 31, 32, 9, 19, 7, 13, 37, 7, 16, 13, 41, 7, 43, 8, 9, 25, 47, 19, 49, 27, 9, 17, 53, 29, 16, 8, 13, 31, 59, 7, 61, 9, 16, 64, 11, 16, 67, 7, 8, 9, 71, 17, 73, 16, 11, 23, 11, 11, 79, 7, 81

Offset: 1

Labos Elemer, Apr 08 2003

nonn

Fixed point is always a prime or a power of prime: fixed points are terms of A000961.

			n=10!=3628800:list to fixed point={3628800,369,50,27}.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A008475, A001414, A056239, A029908, A082083.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] sex[x_] := Apply[Plus, ba[x]^ep[x]] Table[FixedPoint[sex, w], {w, 1, 128}]

A081403 a(n) = A008475(n^2).

Original entry on oeis.org

0, 4, 9, 16, 25, 13, 49, 64, 81, 29, 121, 25, 169, 53, 34, 256, 289, 85, 361, 41, 58, 125, 529, 73, 625, 173, 729, 65, 841, 38, 961, 1024, 130, 293, 74, 97, 1369, 365, 178, 89, 1681, 62, 1849, 137, 106, 533, 2209, 265, 2401, 629, 298, 185, 2809, 733, 146, 113

Offset: 1

Labos Elemer, Mar 31 2003

nonn

			a(1) = 0 since 1 has no prime factor; n = p^2: a(p^2) = p^2; n = 6: a(6) = 4+9 = 13; a(u*w) = a(u)+a(w) if gcd(u,w) = 1; a(21) = a(7)+a(3) = 49+9 = 58; additive with respect of unitary prime divisor decompositions.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A008475, A000142, A081403, A081404.

a:= n-> add(i[1]^i[2], i=ifactors(n^2)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 03 2019

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] supo[x_] := Apply[Plus, ba[x]^ep[x]] Table[supo[w], {w, 1, 25}]

f(n) = { my(f=factor(n)); vecsum(vector(#f~, i, f[i, 1]^f[i, 2])); }; \\ A008475
a(n) = f(n^2); \\ Michel Marcus, Sep 03 2019

from sympy import factorint
def A081403(n): return sum(p**(e<<1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 01 2024

Additive with a(p^e) = p^(2e).

A114518 Numbers n such that A008475(n) is prime.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 18, 19, 22, 23, 24, 28, 29, 31, 34, 36, 37, 40, 41, 43, 47, 48, 52, 53, 54, 58, 59, 61, 67, 71, 72, 73, 76, 79, 82, 83, 88, 89, 97, 100, 101, 103, 107, 108, 109, 112, 113, 118, 127, 131, 137, 139, 142, 148, 149, 151, 157, 160, 162, 163

Offset: 1

Leroy Quet, Dec 05 2005

nonn

			24 = 2^3 * 3 and 2^3 + 3 = 11, which is prime. So 24 is included.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A008475, A114519, A114520, A114522.

f[n_] := Plus @@ Power @@@ FactorInteger[n]; Select[Range[165], PrimeQ[f[ # ]] &] (* Ray Chandler, Dec 07 2005 *)

A008475(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])); for(i=1,500,if(isprime(A008475(i)),print1(i,","))) (Herrgesell)

Extended by Ray Chandler and Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

A082083 a(n)=A082081[n! ]=A082081[A000142[n]] Fixed points of iterated A008475 function initiated at factorials as initial values.

Original entry on oeis.org

0, 2, 5, 11, 16, 7, 37, 149, 7, 27, 11, 11, 23, 2389, 49, 11, 31, 19, 67, 109, 13, 8, 25, 8, 461, 179, 1319, 9, 193, 16, 7, 4931, 121, 7, 9, 8, 7, 8, 2895630887, 25, 19, 13, 19, 41, 2209493509721, 32, 5939, 23, 43, 11

Offset: 1

Labos Elemer, Apr 08 2003

nonn

			Fixed point is always a prime or a true power of prime:
a term from A000961.
n=20!=2432902008176640000, a(20)=109 because
fixed point list={2432902008176640000,269439,214,109}}

Cf. A001414, A056239, A008475, A082081-A082084, A000961, A000142.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] sex[x_] := Apply[Plus, ba[x]^ep[x]] Table[FixedPoint[sex, w! ], {w, 1, 128}]

A081404 a(n) = A008475(prime(n)-1).

Original entry on oeis.org

0, 2, 4, 5, 7, 7, 16, 11, 13, 11, 10, 13, 13, 12, 25, 17, 31, 12, 16, 14, 17, 18, 43, 19, 35, 29, 22, 55, 31, 23, 18, 20, 25, 28, 41, 30, 20, 83, 85, 47, 91, 18, 26, 67, 53, 22, 17, 42, 115, 26, 37, 26, 24, 127, 256, 133, 71, 34, 30, 20, 52, 77, 28, 38, 24, 83, 21, 26, 175, 36

Offset: 1

Labos Elemer, Mar 31 2003

nonn

			a(1) = 0 since 1 has no prime factor.
a(100) = A008475(541-1) = A008475(4*27*5) = 4+27+5 = 36.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A008475, A000142, A081402, A081403.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] supo[x_] := Apply[Plus, ba[x]^ep[x]] Table[supo[w], {w, 1, 25}]

A114519 a(n) = A008475(A114518(n)).

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 7, 13, 17, 11, 19, 13, 23, 11, 11, 29, 31, 19, 13, 37, 13, 41, 43, 47, 19, 17, 53, 29, 31, 59, 61, 67, 71, 17, 73, 23, 79, 43, 83, 19, 89, 97, 29, 101, 103, 107, 31, 109, 23, 113, 61, 127, 131, 137, 139, 73, 41, 149, 151, 157, 37, 83, 163, 19, 167, 47

Offset: 1

Leroy Quet, Dec 05 2005

nonn

			A114518(15) = 24 = 2^3 * 3 and 2^3 + 3 = 11 (which is prime). So a(15) = 11.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A008475, A114518.

f[n_] := Plus @@ Power @@@ FactorInteger[n]; Select[f /@ Range[175], PrimeQ[ # ] &] (* Ray Chandler, Dec 07 2005 *)

A008475(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])); for(i=1,500,if(isprime(A008475(i)),print1(A008475(i),","))) (Herrgesell)

Extended by Ray Chandler and Lambert Herrgesell (zero815(AT)googlemail.com), Dec 07 2005

A159077 a(n) = A008475(n) + 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 6, 8, 9, 10, 8, 12, 8, 14, 10, 9, 17, 18, 12, 20, 10, 11, 14, 24, 12, 26, 16, 28, 12, 30, 11, 32, 33, 15, 20, 13, 14, 38, 22, 17, 14, 42, 13, 44, 16, 15, 26, 48, 20, 50, 28, 21, 18, 54, 30, 17, 16, 23, 32, 60, 13, 62, 34, 17, 65, 19, 17, 68, 22, 27, 15, 72, 18, 74

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

If n = Product (p_i^k_i) for i = 1, …, j then a(n) is sum of divisor d from set of divisors{1, p_1^k_1, p_2^k_2, …, p_j^k_j}.

			For n = 12, set of divisors {1, p_1^k_1, p_2^k_2, …, p_j^k_j}: {1, 3, 4}. a(12) = 1+3+4=8.

Cf. A008475, A023888.

f[n_] := 1 + Plus @@ Power @@@ FactorInteger@ n; f[1] = 1; Array[f, 60]

a(n)=local(t); if(n<1, 0, t=factor(n); 1+sum(k=1, matsize(t)[1], t[k, 1]^t[k, 2])) /* Anton Mosunov, Jan 05 2017 */

a(n) = [Sum_(i=1,…, j) p_i^k_i] + 1 = A000203(n) - A178636(n).

a(1) = 1, a(p) = p+1, a(pq) = p+q+1, a(pq...z) = p+q+...+z+1, a(p^k) = p^k+1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Edited by N. J. A. Sloane, Apr 07 2009

A331000 Unitary Ruth-Aaron numbers: numbers k such that A008475(k) = A008475(k+1).

Original entry on oeis.org

5, 77, 714, 948, 2491, 2996, 3450, 4293, 5405, 6669, 9125, 10807, 13869, 14587, 16932, 17346, 19511, 19967, 23323, 26642, 27104, 31931, 33019, 37925, 41124, 43616, 48635, 52554, 55499, 58077, 58695, 79248, 80837, 86088, 89979, 95709, 98644, 99163, 108458, 117467

Offset: 1

Amiram Eldar, Jan 05 2020

nonn

A variation of Ruth-Aaron numbers with unitary prime-power divisors instead of prime divisors.

			5 is a term since A008475(5) = A008475(6) = 5.

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

Cf. A006145, A039752, A008475, A330999.

s[1] = 0; s[n_] := Plus @@ (Power @@@ FactorInteger[n]); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A078771 a(n) = A008475(n) - A001414(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0, 0, 0, 0, 2, 15, 0, 18, 0, 0, 0, 0, 22, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 35, 15, 0, 0, 0, 18, 0, 2, 0, 0, 0, 0, 0, 0, 3, 52, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 15, 0, 0, 0, 0, 8, 69, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 22, 0, 35, 3, 15

Offset: 1

Benoit Cloitre, Jan 11 2003

nonn

a(n) is not zero if n is in A046790.

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

Cf. A001414, A008475, A046790.

Array[Total[Power @@@ #] - Total[Times @@@ #] &@ FactorInteger@ # &, 100] (* Michael De Vlieger, Nov 17 2017, after Olivier Gérard at A008475 and Ray Chandler at A001414 *)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ This function from M. F. Hasler, Feb 07 2009
A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); };
A078771(n) = (A008475(n) - A001414(n)); \\ Antti Karttunen, Nov 17 2017

a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i,1]^f[i,2] - f[i,1]*f[i,2]);} \\ Amiram Eldar, May 03 2025

Additive with a(p^e) = p^e - p*e. - Amiram Eldar, May 03 2025

A081402 a(n) = A008475(n!).

Original entry on oeis.org

0, 2, 5, 11, 16, 30, 37, 149, 221, 369, 380, 1310, 1323, 2389, 2975, 33695, 33712, 72312, 72331, 269439, 282855, 545109, 545132, 4254514, 4269514, 8463974, 9999248, 35167130, 35167159, 71972737, 71972768, 2152347552, 2161914700

Offset: 1

Labos Elemer, Mar 31 2003

nonn

			a(1) = 0 since 1! = 1 has no prime factor.
a(8) = 2^7 + 3^2 + 5 + 7 = 149 since 8! = 2^7*3^2*5*7.

Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A008475, A000120, A000142, A081403, A081404.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}]; supo[x_] := Apply[Plus, ba[x]^ep[x]]; Table[supo[w], {w, 1, 25}]

a(n) = my(f=factor(n!)); sum(k=1, #f~, f[k,1]^f[k,2]); \\ Michel Marcus, Jul 09 2018

From Amiram Eldar, Dec 10 2024: (Start)
a(n) = 2^(n-s_2(n)) + O(sqrt(3)^n), where s_2(n) = A000120(n).
Sum_{k=1..n} a(k) = 2^(n+O(log(n))).
Both formulas from De Koninck and Verreault (2024, pp. 51-52, eq. (3.10) and (3.16)). (End)

cp's OEIS Frontend

A082081 a(n) = fixed point when the pseudo-log function A008475[ ] is iterated.

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A081403 a(n) = A008475(n^2).

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A114518 Numbers n such that A008475(n) is prime.

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A082083 a(n)=A082081[n! ]=A082081[A000142[n]] Fixed points of iterated A008475 function initiated at factorials as initial values.

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A081404 a(n) = A008475(prime(n)-1).

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A114519 a(n) = A008475(A114518(n)).

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A159077 a(n) = A008475(n) + 1.

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A331000 Unitary Ruth-Aaron numbers: numbers k such that A008475(k) = A008475(k+1).

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