A065387 -id:A065387 - cp's OEIS frontend

A062784 Numbers k such that sigma(k) + phi(k) is a perfect square.

2, 4, 56, 110, 125, 161, 287, 391, 418, 423, 511, 588, 609, 675, 721, 799, 910, 935, 1048, 1057, 1102, 1130, 1281, 1351, 1457, 1485, 1630, 1716, 1799, 1826, 1921, 2047, 2060, 2177, 2255, 2378, 2403, 2449, 2457, 2472, 3199, 3266, 3915, 4010, 4376, 4417

Offset: 1

Jason Earls, Jul 18 2001

nonn

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

Cf. A065387.

Select[Range[4500],IntegerQ[Sqrt[DivisorSigma[1,#]+EulerPhi[#]]]&] (* Harvey P. Dale, Jan 18 2012 *)

select(k->issquare(sigma(k) + eulerphi(k)), [1..5000])

A065388 Record values for sigma(m) + phi(m): sum of sigma and totient is larger than for any previous number.

Original entry on oeis.org

2, 4, 6, 9, 10, 14, 19, 22, 32, 39, 45, 50, 68, 80, 103, 106, 108, 140, 144, 184, 219, 248, 258, 284, 316, 392, 451, 528, 594, 624, 672, 808, 816, 915, 948, 955, 1088, 1266, 1440, 1640, 1704, 1824, 1843, 2020, 2031, 2176, 2208, 2610, 3072, 3304, 3512, 3888

Offset: 1

Labos Elemer, Nov 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1288 (terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A065387, A015702.

{ n=r=0; for (m=1, 10^9, x=sigma(m) + eulerphi(m); if (x > r, r=x; write("b065388.txt", n++, " ", x); if (n==500, return)) ) } \\ Harry J. Smith, Oct 17 2009

a(n) = sigma(A015702(n)) + phi(A015702(n));
a(n) = A000203(A015702(n)) + A000010(A015702(n)).
a(n) = A065387(A015702(n)). - Amiram Eldar, Mar 22 2025

A071391 Least number m such that sigma(m) + phi(m) = n or 0 if no such number exists.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 0, 4, 5, 0, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, 10, 0, 0, 0, 13, 0, 0, 0, 14, 0, 12, 0, 17, 0, 0, 0, 19, 16, 0, 0, 0, 0, 21, 18, 22, 0, 0, 0, 20, 25, 0, 0, 26, 0, 0, 0, 27, 0, 0, 0, 31, 0, 0, 0, 0, 0, 24, 0, 34, 0, 35, 0, 37, 0, 0, 0, 38, 32, 30, 0, 41, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, May 23 2002

			n=256: a(256) = 110, sigma(110) + phi(110) = 216 + 40 = 256 = n and no positive integer k < 110 has sigma(k) + phi(k) = 256.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A065387, A051612, A071390.

f[x_] := DivisorSigma[1, x]+EulerPhi[x] t=Table[0, {100}]; Do[c=f[n]; If[c<101&&t[[c]]==0, t[[c]]=n], {n, 1, 1000000}]; t

a(n) = for(m=1, n, if(sigma(m)+eulerphi(m)==n, return(m))); 0; \\ Jinyuan Wang, Jul 29 2020

first(n) = { my(v = vector(n)); for(i = 1, n, c = sigma(i) + eulerphi(i); if(c <= n, if(v[c] == 0, v[c] = i ) ) ); v } \\ David A. Corneth, Jul 30 2020

a(n) = Min{x; A000203(x)+A000010(x)=n} or a(n) = 0 if no solution exists.

A072779 a(n) = sigma_2(n) + phi(n) * sigma(n).

Original entry on oeis.org

2, 8, 18, 35, 50, 74, 98, 145, 169, 202, 242, 322, 338, 394, 452, 589, 578, 689, 722, 882, 884, 970, 1058, 1330, 1271, 1354, 1540, 1722, 1682, 1876, 1922, 2373, 2180, 2314, 2452, 3003, 2738, 2890, 3044, 3650, 3362, 3652, 3698, 4242, 4238, 4234, 4418

Offset: 1

T. D. Noe, Jul 15 2002

This sequence is interesting because (1) a(n) >= 2 n^2, with equality only when n is prime (or 1) and (2) a(n) = 2 + 2*n^2 if and only if n is the product of two distinct primes. Note for twin primes: let n = m^2 - 1, then m-1 and m+1 are twin primes if and only if a(n) = 2 + 2*n^2. Note for the Goldbach conjecture: let n = m^2 - r^2, then m-r and m+r are primes that add to 2m if and only if a(n) = 2 + 2*n^2. See A072780 for a(n) - 2*n^2.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A000203, A001157, A065387, A072780.

Cf. A002117, A065465.

a072779 n = a001157 n + (a000203 n) * (a000010 n)
-- Reinhard Zumkeller, Jan 15 2013

Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n], {n, 100}]

a(n)=sigma(n,2)+eulerphi(n)*sigma(n) \\ Charles R Greathouse IV, May 15 2013

a(n) = A001157(n) + A000203(n)*A000010(n). - Reinhard Zumkeller, Jan 15 2013

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) + Product_{p prime} (1 - 1/(p^2*(p+1))) = A002117 + A065465 = 2.083570742884... . - Amiram Eldar, Dec 03 2023

A077081 Fixed point when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 6, 864, 864, 10, 10, 864, 864, 864, 864, 864, 864, 864, 864, 20, 20, 22, 22, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 864, 48, 864, 46, 46, 48, 864, 864, 48, 864, 864, 48, 48, 48, 48, 58, 58

Offset: 1

Labos Elemer, Oct 28 2002

nonn

A065387 when iterated seems to converge [tested for initial values below 1024]. On the other hand iterating A051682 often ends in cycle.
Iteration of phi(A065387())=phi(sigma()+phi()) seems to converge. Tested below n=1024. Critical values however arise. For example: n=534,556,557,580,624,702,710, etc. These initial values generate very large terms and i was unable to decide if they converge.
For n=1..1024 no more but 27 distinct fixed points arised:{1,2,6,10,..,3552,570240}

			n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, a[25]=570240.

Cf. A000010, A000203, A065387, A077080, A077082, A077083, A077084, A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[FixedPoint[f, w], {w, 1, 256}]

a(n) = FixedPoint[A077080, n].

A077082 Largest value arising when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

Original entry on oeis.org

1, 2, 3, 6, 6, 6, 7, 1044, 1044, 10, 11, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 20, 21, 22, 23, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 1044, 48, 1044, 46, 47, 48, 1044

Offset: 1

Labos Elemer, Oct 28 2002

nonn

			n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, largest is 653312=a(225).

Cf. A000010, A000203, A065387, A077080-A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[Max[FixedPointList[f, w]], {w, 1, 1024}]

a(n) = Max[FixedPointList[A077080, n]]. See program below. Seems convergent. [tested for initial values below 1024.]

A077083 Length of iteration until a fixed point is reached when phi(sigma[n]+phi(n)) = A077080(n) is iterated with initial value of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 2, 16, 16, 1, 2, 16, 17, 17, 16, 15, 16, 15, 16, 1, 2, 1, 2, 14, 14, 16, 15, 14, 15, 14, 15, 13, 14, 15, 15, 10, 11, 15, 14, 13, 14, 11, 12, 2, 14, 1, 2, 1, 12, 6, 2, 12, 13, 3, 2, 2, 3, 1, 2, 11, 12, 11, 2, 11, 14, 10, 11, 13, 2, 2, 3, 5, 6, 14, 10, 10, 2, 12, 13, 9

Offset: 1

Labos Elemer, Oct 28 2002

nonn

			n=1,2,6,10,..864: a[n]=1, n is fixed point for some n; n=225: results in iteration sequence of 44 terms: {225,522,444,...,471744,653312,570240}, 44=a[225].

Cf. A000010, A000203, A065387, A077080-A077085.

f[x_] := EulerPhi[DivisorSigma[1, x]+EulerPhi[x]] Table[Length[FixedPointList[f, w]]-1, {w, 1, 1024}]

a(n)=Length[FixedPointList[A077080, n]]-1.

A077088 a(n) = phi(sigma(n) - phi(n)), where phi is Euler's totient function and sigma is the sum of divisors function, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 10, 6, 6, 1, 8, 1, 6, 8, 22, 1, 20, 1, 16, 8, 12, 1, 24, 10, 8, 10, 20, 1, 32, 1, 46, 12, 18, 8, 78, 1, 12, 16, 36, 1, 24, 1, 32, 18, 20, 1, 36, 8, 72, 16, 36, 1, 32, 16, 32, 20, 30, 1, 72, 1, 20, 32, 72, 12, 60, 1, 46, 24, 32, 1, 108, 1, 24, 24, 48, 12, 48, 1, 60

Offset: 1

Labos Elemer, Nov 04 2002

a(p) = 1 for p prime. Otherwise a(n) is even.

			a(10) = 6 because sigma(10) = 18 and phi(10) = 4, and so phi(18 - 4) = phi(14) = 6.
a(11) = 1 because sigma(11) = 12 and phi(11) = 10, so phi(12 - 10) = phi(2) = 1.
a(12) = 8 because sigma(12) = 28 and phi(12) = 4, so phi(28 - 4) = phi(24) = 8.

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

Cf. A000010, A000203, A051612, A065387. See iterations in A077090-A077100.

List([1..100],n->Phi(Sigma(n)-Phi(n))); # Muniru A Asiru, Mar 04 2018

with(numtheory); A077088:=n->phi(sigma(n)-phi(n)); seq(A077088(n), n=1..100); # Wesley Ivan Hurt, Dec 02 2013

Table[EulerPhi[DivisorSigma[1, n] - EulerPhi[n]], {n, 100}] (* Alonso del Arte, Nov 29 2013 *)

A077088(n) = if(1==n,0,eulerphi(sigma(n) - eulerphi(n))); \\ Antti Karttunen, Mar 04 2018

a(1) = 0; and for n > 1, a(n) = A000010(A051612(n)).

Value of a(1) clarified by Antti Karttunen, Mar 04 2018

A077102 Smallest number m such that GCD(a+b,a-b) = n, where a = sigma(m) and b = phi(m).

Original entry on oeis.org

4, 1, 18, 21, 200, 14, 3364, 12, 722, 328, 9801, 42, 25281, 116, 1800, 15, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 284, 98942809, 488, 1547536, 364, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 440, 3150464641

Offset: 1

Labos Elemer, Nov 12 2002

nonn

			For n = 10, a(10) = 328, sigma(328) = 630, phi(328) = 160, sigma(328) + phi(328) = 790, sigma(328) - phi(328) = 470, GCD(790,470) = 10.
For n = odd number, a(n) should be either a square or twice a square and so faster search for large values is possible, like e.g., for n = 97: a(97) = 435979^2 is the smallest solution.

Amiram Eldar, Table of n, a(n) for n = 1..82

Cf. A000010, A000203, A051612, A065387, A077099, A077100, A077101.

f[x_] := Apply[GCD, {DivisorSigma[1, x]+EulerPhi[x], DivisorSigma[1, x]-EulerPhi[x]}]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}]; t

lista(len) = {my(v = vector(len), c = 0, k = 1, a, b, i); while(c < len, f = factor(k); a = sigma(f); b = eulerphi(f); i = gcd(a+b,a-b); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 14 2024

a(n) = Min{x; A077099(x) = n}.

A156775 Number of iterations of x->(sigma(x)+phi(x))/2 until a non-integer or a previous term is reached, starting with x=n; a(n)=0 if this never happens.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Feb 15 2009

nonn

In [Guy 1997] the iteration is said to fracture when sigma(x)+phi(x) becomes odd. It is not known if a(n)=0 for some n.

A156776(n) gives the number of iterations until the sequence fractures, resp. 0 if this never happens.

			Let f(x)=(sigma(x)+phi(x))/2. For x=1 we have f(x) = (1+1)/2 = 1, i.e. after a(1)=1 iterations, the initial term 1 is encountered. For x=2 we have f(x) = (3+1)/2 = 2, so a(2)=1 for the same reason; idem for x=3 and x=5. For x=4 we have f(x) = (7+2)/2 = 9/2, the sequence "fractures" after a(4)=1 iterations. For x=6 we have f(x) = (12+2)/2 = 7, f(7) = (8+6)/2 = 7: after a(6)=2 iterations, there's a value already seen before.

Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A156776, A065387(n) = A000203(n) + A000010(n).

f[n_] := If[IntegerQ[n], n, 0]; g[n_] := f[(DivisorSigma[1, n] + EulerPhi[n])/2]; a[n_] := Module[{s = NestWhileList[g, n, UnsameQ, All]}, Length[s] - If[s[[-1]] == 0, 2, 1]]; Array[a, 105] (* Amiram Eldar, Apr 01 2024 *)

A156775(n,u=[])={ until( denominator( n=(sigma(n)+eulerphi(n))/2)>1 || setsearch(u,n), u=setunion(u,Set(n)));#u }

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A062784 Numbers k such that sigma(k) + phi(k) is a perfect square.

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A065388 Record values for sigma(m) + phi(m): sum of sigma and totient is larger than for any previous number.

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A071391 Least number m such that sigma(m) + phi(m) = n or 0 if no such number exists.

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A072779 a(n) = sigma_2(n) + phi(n) * sigma(n).

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A077081 Fixed point when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

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A077082 Largest value arising when phi(sigma(n)+phi(n))=A077080 is iterated with initial value of n.

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A077083 Length of iteration until a fixed point is reached when phi(sigma[n]+phi(n)) = A077080(n) is iterated with initial value of n.

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A077088 a(n) = phi(sigma(n) - phi(n)), where phi is Euler's totient function and sigma is the sum of divisors function, with a(1) = 0.

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