A359501 Prime numbers p such that for some r >= 1 we have phi(p - r) + ... + phi(p - 1) = phi(p + 1) + ... + phi(p + r), where phi(i) = A000010(i).

5, 11, 13, 19, 31, 53, 67, 71, 89, 109, 127, 139, 173, 281, 313, 389, 421, 431, 523, 547, 569, 751, 911, 947, 1117, 1201, 1399, 1531, 1609, 1693, 1823, 1973, 2089, 2389, 2591, 2659, 2789, 3217, 3229, 3323, 3607, 3719, 3967, 4339, 4583, 4793, 5351, 5519, 5563, 5647, 5701

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..500

Example

p = 5: phi(4) = phi(6) = 2, thus 5 is a term;
p = 19: phi(16) + phi(17) + phi(18) = phi(20) + phi(21) + phi(22) = 30, thus 19 is a term.

Maple

N:= 10^4: # for terms <= N
Phi:= map(NumberTheory:-phi, [$1..2*N]):
SP:= ListTools:-PartialSums([0, op(Phi)]):
filter:= proc(p)
   isprime(p) and ormap(r -> SP[p]-SP[p-r] = SP[p+r+1]-SP[p+1], [$1..p-1])
end proc:
select(filter, [seq(i, i=3..N, 2)]); # Robert Israel, Apr 17 2026

Prog

(PARI) isok(p) = if (isprime(p), my(k=primepi(p)); for (i=1, k-1, if (sum(j=1, i, eulerphi(p-j)) == sum(j=1, i, eulerphi(p+j)), return(1)); ); ); \\ Michel Marcus, Jan 03 2023

Crossrefs

Cf. A000010, A000040.

Sequence in context: A019395 A045448 A157750 * A045449 A296859 A313992

Adjacent sequences: A359498 A359499 A359500 * A359502 A359503 A359504

Keyword

nonn

Author

Ctibor O. Zizka, Jan 03 2023

Extensions

More terms from Michel Marcus, Jan 03 2023

Status

approved