A051612 -id:A051612 - cp's OEIS frontend

A271440 a(n) = sigma(prime(n)^n) - phi(prime(n)^n).

2, 7, 56, 743, 30746, 773527, 49783736, 1837403019, 160181560802, 29532404308019, 1666577516860962, 360777399719461393, 45691067858241526814, 3477439299142731351087, 518913689466371066697746, 147680787468230866751370317, 43490064769447225534580532962

Offset: 1

Wesley Ivan Hurt, Apr 07 2016

Harvey P. Dale, Table of n, a(n) for n = 1..303

Cf. A000010 (phi), A000040 (primes), A000203 (sigma), A051612, A062457.

with(numtheory): A271440:=n->sigma(ithprime(n)^n)-phi(ithprime(n)^n): seq(A271440(n), n=1..30);

Table[DivisorSigma[1, Prime[n]^n] - EulerPhi[Prime[n]^n], {n, 20}]
DivisorSigma[1,#]-EulerPhi[#]&/@Table[Prime[n]^n,{n,20}] (* Harvey P. Dale, Feb 07 2025 *)

a(n) = sigma(prime(n)^n) - eulerphi(prime(n)^n); \\ Altug Alkan, Apr 08 2016

a(n) = (2*prime(n)^n-prime(n)^(n-1)-1) / (prime(n)-1).
a(n) = (prime(n)^(n+1)-prime(n)^(n-1)*(prime(n)-1)^2-1) / (prime(n)-1).
a(n) = A051612(A062457(n)) = A000203(A062457(n)) - A000010(A062457(n)).

A302172 Distance from sigma(n) to nearest multiple of phi(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 2, 1, 1, 2, 2, 0, 2, 0, 0, 1, 2, 3, 2, 2, 4, 4, 2, 4, 9, 6, 4, 4, 2, 0, 2, 1, 8, 6, 0, 5, 2, 6, 8, 6, 2, 0, 2, 4, 6, 6, 2, 4, 15, 7, 8, 2, 2, 6, 8, 0, 8, 6, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 8, 0, 2, 3, 2, 6, 4, 4, 24, 0, 2, 6, 13, 6, 2, 8, 20, 6, 8, 20, 2, 6, 32, 8, 8, 6, 24, 4, 2, 3, 24, 17

Offset: 1

Altug Alkan, Apr 03 2018

Numbers n such that a(n) <> A063514(n) are 8, 16, 21, 22, 25, 28, 32, 36, 40, 48, 50, 54, 55, 63, 64, 65, 68, 76, 77, 80, ...

Numbers n such that a(n) = 1 are 4, 8, 9, 16, 32, 64, 128, 256, 400, 512, 1024, 2048, 4096, 8192, 16384, ...

			a(21) = 4 because sigma(21) = 32 and phi(21) = 12; 12*3 - 32 = 4 is the smallest corresponding distance.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A020492, A051612, A063514.

dsp[n_]:=Module[{s=DivisorSigma[1,n],p=EulerPhi[n],m},m=Floor[s/p];Abs[ Nearest[ {m*p,(m+1)p},s]-s]]; Array[dsp,100][[All,1]] (* Harvey P. Dale, Apr 29 2018 *)

a(n) = {my(k=0, s=sigma(n), p=eulerphi(n)); while((s+k) % p != 0 && (s-k) % p != 0, k++); k;}

a(A020492(n)) = 0.
a(2^k) = 1 for k > 1.
a(p) = 2 for prime p > 3.

A068419 Odd prime values of sigma(k) - phi(k) taking k in increasing order.

Original entry on oeis.org

5, 11, 7, 23, 11, 47, 79, 73, 67, 23, 191, 283, 383, 739, 47, 1459, 281, 607, 59, 1153, 1069, 3319, 83, 1801, 2671, 3517, 743, 107, 6679, 3529, 6619, 6143, 6271, 4153, 10169, 9817, 167, 6451, 179, 24097, 23539, 10369, 227, 263, 16369, 41203, 20809, 54919

Offset: 1

Benoit Cloitre, Mar 02 2002

			sigma(2312) - phi(2312) = 3517 is prime, so is in the sequence.

Cf. A051612.

lista(nn) = {for (k = 1, nn, if (isprime(p = (sigma(k) - eulerphi(k))) && (p % 2), print1(p, ", ")););} \\ Michel Marcus, Nov 24 2013

Terms 3517, 10169, 9817 added by Michel Marcus, Nov 24 2013

A077103 Numbers n such that gcd(a,b) is not equal to gcd(a+b,a-b), where a=sigma(n)=A000203(n) and b=phi(n)=A000010(n).

Original entry on oeis.org

1, 2, 12, 15, 30, 39, 44, 55, 56, 76, 78, 87, 95, 99, 110, 111, 125, 140, 143, 147, 159, 171, 172, 174, 175, 183, 184, 190, 198, 215, 216, 222, 236, 247, 250, 252, 264, 268, 286, 287, 294, 295, 303, 315, 318, 319, 327, 332, 335, 336, 342, 350, 357, 363, 364

Offset: 1

Labos Elemer, Nov 12 2002

nonn

			n=76: a=sigma(76)=140, b=phi(76)=36, a+b=176, a-b=104, gcd(a,b) = gcd(140,36) = 4 < gcd(a+b,a-b) = gcd(176,104) = 8.

Cf. A000010, A000203, A051612, A065387, A055008, A077099.

Do[s=GCD[a=DivisorSigma[1, n], b=EulerPhi[n]]; s1=GCD[a+b, a-b]; If[ !Equal[s, s1], Print[{n, a, b, a+b, a-b, s, s1, s1/s}]], {n, 1, 1000}]

gcd(A000010(n), A000203(n)) is not equal to gcd(A065387(n), A051612(n)); or A055008(n) is not equal to A077099(n).

cp's OEIS Frontend

A271440 a(n) = sigma(prime(n)^n) - phi(prime(n)^n).

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A302172 Distance from sigma(n) to nearest multiple of phi(n).

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A068419 Odd prime values of sigma(k) - phi(k) taking k in increasing order.

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A077103 Numbers n such that gcd(a,b) is not equal to gcd(a+b,a-b), where a=sigma(n)=A000203(n) and b=phi(n)=A000010(n).

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