A048050 -id:A048050 - cp's OEIS frontend

A062825 Ch(n-th nonprime) where Ch(n) is Chowla's function, cf. A048050.

0, 2, 5, 6, 3, 7, 15, 9, 8, 14, 20, 21, 10, 13, 35, 5, 15, 12, 27, 41, 30, 14, 19, 12, 54, 21, 16, 49, 53, 39, 32, 25, 75, 7, 42, 20, 45, 65, 16, 63, 22, 31, 107, 33, 40, 62, 18, 77, 57, 26, 73, 122, 39, 48, 63, 18, 89, 105, 39, 43, 139, 22, 45, 32, 91, 143, 20, 75, 34, 49, 24, 155, 72, 56, 116, 113, 105, 86, 55, 171, 105, 40, 135

Offset: 1

Jason Earls, Jul 20 2001

a(n) = A048050(A018252(n)).

a(n+1) = sum of nontrivial divisors of n-th composite number, or row sums in table A163870. - Juri-Stepan Gerasimov, Aug 06 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A048050, A002808.

a062825 1 = 0
a062825 n = sum $ a163870_row (n - 1) -- Reinhard Zumkeller, Mar 29 2014

with(numtheory): a_list := proc(n); {$1..n} minus select(isprime,{$1..n}); sort(convert(%, list)); map(f->add(d,d=(divisors(f) minus {1,f})),%) end: a_list(113); # Peter Luschny, Mar 29 2014

Reap[Do[If[!PrimeQ[k], Sow[If[k == 1, 0, DivisorSigma[1, k] - k - 1 ]]], {k, 1, 120}]][[2, 1]] (* Jean-François Alcover, Feb 12 2018 *)

j=[0]; for(n=2,200, if(isprime(n), n+1,j=concat(j, sigma(n)-n-1))); j

Definition revised and a(1) corrected by Reinhard Zumkeller, Mar 29 2014

A063534 Numbers k such that C(k) = H(k) + d(k), where C(k) is Chowla's function A048050, H(k) is the half-totient function A023022 and d(k) is the number of divisors function A000005.

Original entry on oeis.org

6, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753

Offset: 1

Jason Earls, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048050, A023022, A000005, A001748.

Select[Range[1000], DivisorSigma[1, #] - 1 - # == EulerPhi[#]/2 + DivisorSigma[0, #] &] (* Paolo Xausa, Apr 17 2024 *)

C(n)=sigma(n)-n-1;
H(n)=eulerphi(n)/2;
j=[]; for(n=1,1200, if(C(n)==H(n)+numdiv(n),j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (sigma(m) - m - 1 == eulerphi(m)/2 + numdiv(m), write("b063534.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

is(n) = {my(f = factor(n)); sigma(f) - n - 1 == eulerphi(f) / 2 + numdiv(f);} \\ Amiram Eldar, Apr 15 2024

Conjecture: a(n) = A001748(n), n <> 2. - R. J. Mathar, Dec 15 2008

The conjecture is false. The least counterexample is a(11546) = 368335 = 5 * 11 * 37 * 181. The next counterexample is 4922335, and there are no more below 10^10. - Amiram Eldar, Apr 15 2024

A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

Original entry on oeis.org

0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56, 117

Offset: 1

Labos Elemer, Apr 12 2002

nonn

Remark that A070016(n)=A070015(n+1) in accordance with A048995(k)+1=A005114(k).

			n=127: a(n)=16129, divisors={1,127,16129}, 127=sigma[n]-n-1=127 and 16129 is the smallest.

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

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

f1[x_] := DivisorSigma[1, x]-x-1; t=Table[0, {128}]; Do[b=f1[n]; If[b<129&&t[[b]]==0, t[[b]]=n], {n, 1, 1000000}]; t

a(n)=Min{x; A048050(x)=n} or a(n)=0 if n is from A048995.

A153023 If n is 1 or prime then a(n) = n. Otherwise, start with n and iterate the map k -> A048050(k) until we reach a prime p; then a(n) = p. If we never reach a prime, a(n) = -1. A048050 gives the sum of proper divisors of k, excluding both 1 and n from the sum.

Original entry on oeis.org

1, 2, 3, 2, 5, 5, 7, 5, 3, 7, 11, 5, 13, 3, 5, 3, 17, 7, 19, 7, 7, 13, 23, 5, 5, 5, 5, 5, 29, 41, 31, 41, 3, 19, 5, 7, 37, 7, 3, 7, 41, 53, 43, 3, 41, 5, 47, -1, 7, 53, 7, 41, 53, 7, 3, 7, 13, 31, 59, 107, 61, 3, 7, 3, 7, 7, 67, 13, 5, 73, 71, 7, 73, 3, -1, 7, 7, 89, 79, 41, 3, 43, 83, 139, 13

Offset: 1

Andrew Carter (acarter09(AT)newarka.edu), Dec 16 2008

sign

			a(18) -> {2,3,6,9} -> 20 -> {2,4,5,10} -> 21 -> {3,7} -> 10 -> {2,5} -> 7 = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001065, A048050, A120716, A153024.

f := proc(n) L := {} ; a := n ; while not isprime(a) do a := A048050(a) ; if a in L then RETURN(-1) ; fi; L := L union {a} ; od; a ; end:
A048050 := proc(n) numtheory[sigma](n)-n-1 ; end:
A153023 := proc(n) if n =1 then 1; elif isprime(n) then n; else f(n) ; fi; end: # R. J. Mathar, Dec 19 2008

Table[If[! CompositeQ[n], n, NestWhile[DivisorSigma[1, #] - (# + 1) &, n, Nor[PrimeQ@ #, # == 0] &, 1, 100] /. k_ /; CompositeQ@ k -> -1], {n, 85}] (* Michael De Vlieger, Nov 03 2017 *)

(define (A153023 n) (let loop ((n n) (visited (list n))) (let ((next (A048050 n))) (cond ((or (= 1 n) (= 1 (A010051 n))) n) ((member next visited) -1) (else (loop next (cons next visited)))))))
(define (A048050 n) (if (= 1 n) 0 (- (A001065 n) 1)))
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 03 2017

Extended by R. J. Mathar, Dec 19 2008

Description clarified by Antti Karttunen, Nov 03 2017

A153024 a(n) is the number of iterations of the map k -> A048050(k) to reach zero. If we never reach 0, then a(n) = -1. A048050 gives the sum of proper divisors of k, excluding both 1 and n from the sum.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 3, 4, 4, 1, 5, 1, 4, 3, 2, 1, 7, 2, 5, 6, 7, 1, 2, 1, 3, 4, 2, 6, 8, 1, 4, 5, 3, 1, 2, 1, 6, 4, 3, 1, -1, 2, 3, 5, 5, 1, 7, 5, 5, 3, 2, 1, 2, 1, 5, 4, 6, 6, 7, 1, 4, 6, 2, 1, 6, 1, 6, -1, 5, 6, 2, 1, 7, 6, 2, 1, 2, 3, 5, 4, 6, 1, 9, 5, -1, 3, 3, 8, 10, 1, 7, 6, 5, 1, 2, 1, 7, 6

Offset: 1

Andrew Carter (acarter09(AT)newarka.edu), Dec 16 2008

sign

Previous name was: The number of iterations for A153023 to converge when started at n.

			With m(n) = A048050(n) we have: m(18) -> m(20) -> m(21) -> m(10) -> m(7) -> 0, thus a(18) = 5.
On the other hand, m(48) = 75 and m(75) = 48, so we ended in a cycle, thus a(48) = a(75) = -1. - Edited by _Antti Karttunen_, Nov 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048050, A153023.

f := proc(n) L := {} ; a := n ; while not isprime(a) do a := A048050(a) ; if a in L then RETURN(-1) ; fi; L := L union {a} ; od; 1+nops(L) ; end:
A153023 := proc(n) if n =1 then 1; elif isprime(n) then 1; else f(n) ; fi; end: # R. J. Mathar, May 25 2013

With[{nn = 100}, Table[If[! CompositeQ[n], 1, Length@ NestWhileList[DivisorSigma[1, #] - (# + 1) &, n, Nor[PrimeQ@ #, # == 0] &, 1, 100]] /. k_ /; k == nn + 1 -> -1, {n, 104}]] (* Michael De Vlieger, Nov 03 2017 *)

(define (A153024 n) (let loop ((n n) (visited (list n)) (i 0)) (let ((next (A048050 n))) (cond ((zero? n) i) ((member next visited) -1) (else (loop next (cons next visited) (+ 1 i)))))))
(define (A048050 n) (if (= 1 n) 0 (- (A001065 n) 1)))
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 03 2017

Name changed and more terms added by Antti Karttunen, Nov 03 2017

A187087 Positive squares in the order of their appearance in A048050.

Original entry on oeis.org

9, 16, 49, 25, 16, 49, 64, 121, 36, 81, 64, 169, 36, 225, 100, 225, 64, 36, 441, 36, 169, 361, 225, 144, 441, 441, 144, 256, 400, 196, 64, 441, 144, 361, 64, 400, 441, 729, 961, 64, 196, 144, 729, 100, 841, 729, 400, 256, 1225, 100, 729, 1225, 961, 900, 841

Offset: 1

Zak Seidov, Mar 04 2011

nonn

Corresponding values of n are in A187086. A048050 is Chowla's function: sum of divisors of n except 1 and n.

By the Goldbach conjecture, every even square appears; take two odd primes p and q such that p+q = k^2, then Chowla function of p*q is k^2. It appears that 17^2 is the first odd square not in A048050.

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

Cf. A048050, A187086.

A048050:=func< n | n eq 1 or IsPrime(n) select 0 else &+[ a: a in Divisors(n) | a ne 1 and a ne n ] >; [ a: n in [1..2500] | a gt 0 and IsSquare(a) where a is A048050(n) ]; // Klaus Brockhaus, Mar 04 2011

chowla[n_] := DivisorSigma[1, n] - n - 1; s = {}; Do[c = chowla[n]; If[c > 0 && IntegerQ@Sqrt[c], AppendTo[s, c]], {n, 1, 10^3}]; s (* Amiram Eldar, Aug 28 2019 *)

{for(n=1,2000,spf=sumdiv(n,x,x)-1-n;if(spf>0&&issquare(spf),print1(spf",")))}

A049030 Sum of sigma(j) for 1<=j<10^n, where sigma(j) = A048050(j) is the sum of the proper divisors >1 of j (excluding 1 and n).

Original entry on oeis.org

16, 3034, 320243, 32226805, 3224444759, 322465138002, 32246681892518, 3224670122682648, 322467031114802292, 32246703322412473945, 3224670334023621455211, 322467033422357645316809, 32246703342390510922780778, 3224670334240928188556405242

Offset: 1

Enoch Haga and Jud McCranie

			For n = 1, the sum of sigma(j), for j < 10 is 0 + 0 + 0 + 2 + 0 + 5 + 0 + 6 + 3 = 16, so a(1) = 16.

Amiram Eldar, Table of n, a(n) for n = 1..36 (calculated from the b-file at A049000)
Carlos Rivera, Problem 23.- Divisors (V) Aliquot sequences, Sociable Numbers, Prime Puzzles and Problems Connection.

Cf. A001065, A046915, A048050, A048995, A049000.

Cf. A072691 (Pi^2/12).

At a(3) = 320243, for example, take a(3) from A049000: 820741 - 500498 = 320243. Compute 500498 from 999*1000/2 = 499500, split evenly and reverse to 500499 - 1 = 500498. Add a 9 and 0 for each successive term.

a(n) = A049000(n) - 10^n * (10^n + 1) / 2 + 2 ~ (Pi^2/12 - 1/2) * 10^(2*n). - Amiram Eldar, Feb 16 2020

More terms from Amiram Eldar, Feb 16 2020

A062950 C(H(n)), where C(n) is Chowla's function (A048050) and H(n) is the half-totient function (A023022).

Original entry on oeis.org

-1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 5, 0, 2, 2, 6, 0, 3, 2, 5, 0, 0, 2, 7, 5, 3, 5, 9, 2, 8, 6, 7, 6, 15, 5, 20, 3, 15, 6, 21, 5, 10, 7, 15, 0, 0, 6, 10, 7, 14, 15, 15, 3, 21, 15, 20, 9, 0, 6, 41, 8, 20, 14, 35, 7, 14, 14, 13, 15, 12, 15, 54, 20, 21, 20, 41, 15, 16, 14, 12, 21, 0, 15, 30, 10, 27, 21, 39, 15, 54, 13, 41, 0, 54, 14, 75, 10, 41, 21, 42, 14

Offset: 3

Jason Earls, Jul 21 2001

sign

Cf. A048050, A023022.

C(n)=sigma(n)-n-1; H(n)=eulerphi(n)/2; j=[]; for(n=3,150,j=concat(j,C(H(n)))); j

A062984 a(n) = M(C(n)), where M(n) is Mertens's function (A002321) and C(n) is Chowla's function (A048050).

Original entry on oeis.org

0, 0, 0, 0, 0, -2, 0, -1, -1, -2, 0, -1, 0, -2, -2, -2, 0, -3, 0, -2, -1, -3, 0, -1, -2, -1, -2, -1, 0, -1, 0, -3, -2, -3, -2, -3, 0, -2, -1, -3, 0, -3, 0, 0, -4, -2, 0, -3, -2, -2, -3, -3, 0, 0, -1, -1, -1, -4, 0, -3, 0, -3, 0, -1, -2, -2, 0, -1, -1, -4, 0, -2, 0, 0, -3, -1, -2, -2, 0, -3, 0, -3, 0, -4, -1, -3, -4, -1, 0, -1, -3, -3, -2, -3

Offset: 1

Jason Earls, Jul 25 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A002321, A048050.

A062984[n_] := Sum[MoebiusMu[k], {k, DivisorSigma[1, n] - n - 1}];
Array[A062984, 100] (* Paolo Xausa, May 03 2024 *)

M(n)=sum(k=1,n,moebius(k));
C(n)=sigma(n)-n-1;
j=[]; for(n=1,350,j=concat(j,M(C(n)))); j

{ for (n=1, 2000, a=sum(k=1, sigma(n) - n - 1, moebius(k)); write("b062984.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 15 2009

A063900 Numbers k such that sum of proper divisors or aliquot parts of k^2 (excluding 1) is a square, or A048050(k^2) is a square.

Original entry on oeis.org

866, 9271, 18167, 30887, 39959, 114607, 119279, 129911, 153631, 239111, 343207, 517591, 582583, 602159, 607340, 1202282, 1397863, 1729999, 1920647, 2533183, 2547119, 2558183, 5740127, 7122959, 9379871, 10218407, 10891103, 13549399

Offset: 1

Jason Earls, Aug 29 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100 (terms 1..55 from Harry J. Smith)
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Cf. A008847, A048050.

chowla[n_] := DivisorSigma[1, n] - n - 1; aQ[n_] := IntegerQ@Sqrt@chowla[n^2]; Select[Range[10^6], aQ] (* Amiram Eldar, Aug 30 2019 *)

s(n)=sigma(n)-n-1;
for(n=1,10^8, if(issquare(s(n^2)),print(n)))

s(n)=sigma(n) - n - 1
{ n=0; for (m=1, 10^9, if(issquare(s(m^2)), write("b063900.txt", n++, " ", m); if (n==55, break)) ) } \\ Harry J. Smith, Sep 02 2009

a(19)-a(28) from Harry J. Smith, Sep 02 2009

cp's OEIS Frontend

A062825 Ch(n-th nonprime) where Ch(n) is Chowla's function, cf. A048050.

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A063534 Numbers k such that C(k) = H(k) + d(k), where C(k) is Chowla's function A048050, H(k) is the half-totient function A023022 and d(k) is the number of divisors function A000005.

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A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

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A153023 If n is 1 or prime then a(n) = n. Otherwise, start with n and iterate the map k -> A048050(k) until we reach a prime p; then a(n) = p. If we never reach a prime, a(n) = -1. A048050 gives the sum of proper divisors of k, excluding both 1 and n from the sum.

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A153024 a(n) is the number of iterations of the map k -> A048050(k) to reach zero. If we never reach 0, then a(n) = -1. A048050 gives the sum of proper divisors of k, excluding both 1 and n from the sum.

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A187087 Positive squares in the order of their appearance in A048050.

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A049030 Sum of sigma(j) for 1<=j<10^n, where sigma(j) = A048050(j) is the sum of the proper divisors >1 of j (excluding 1 and n).

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