A280867 -id:A280867 - cp's OEIS frontend

A281687 Number of partitions of 2*n into the sum of two totient numbers (A002202).

1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 6, 7, 7, 9, 8, 9, 8, 9, 9, 11, 10, 12, 10, 11, 10, 12, 11, 13, 10, 11, 12, 13, 12, 15, 13, 12, 13, 13, 12, 15, 14, 14, 14, 16, 15, 19, 16, 16, 16, 17, 15, 19, 15, 18, 16, 19, 16, 20, 18, 19, 18, 20, 17, 22, 19, 21, 18, 21, 19, 22

Offset: 1

Altug Alkan, Jan 27 2017

See also graph of A045917 ("Goldbach's comet"). - Altug Alkan, Jan 30 2017

			a(6) = 3 because 2 * 6 = 12 = 2 + 10 = 4 + 8 = 6 + 6 and 2, 4, 6, 8, 10 are in A002202.

Robert Israel, Table of n, a(n) for n = 1..10000
Altug Alkan, Illustration Of Residue Classes Modulo 6

Cf. A000010, A002202, A045917, A280867.

N:= 1000: V:= Vector(2*N):
V[1]:= 1:
for n from 2 to 2*N by 2 do
  if nops(numtheory:-invphi(n))>1 then V[n]:= 1 fi
od:
C:= map(round,SignalProcessing:-Convolution(V,V)):
seq((C[2*i-1]+V[i])/2,i=1..N); # Robert Israel, Jan 27 2017

a(n) = sum(k=1, n, istotient(k) && istotient(2*n-k));

A281816 Least k such that phi(k) is the sum of two totient numbers (A002202) in exactly n ways, or 0 if no such k exists.

Original entry on oeis.org

1, 3, 11, 13, 23, 29, 37, 41, 81, 53, 67, 61, 73, 97, 103, 89, 109, 143, 139, 113, 137, 157, 149

Offset: 0

Altug Alkan, Jan 30 2017

For the first 10000 terms of A281687, only A281687(93) = 23 and 2 * 93 = 186 is not a totient number. With this observation if we consider the scatterplot of A281687, a(23) is probably equal to 0, but this is still unproved at this moment. So this sequence has keyword "more".

a(24) - a(71) are 173, 181, 193, 235, 247, 301, 271, 229, 253, 289, 233, 519, 269, 281, 293, 337, 317, 439, 349, 397, 373, 353, 409, 575, 535, 433, 401, 571, 389, 449, 551, 461, 879, 623, 577, 743, 521, 509, 557, 685, 689, 569, 661, 593, 767, 709, 653, 641.

			a(3) = 13 because phi(13) = 12 = 2 + 10 = 4 + 8 = 6 + 6; 2, 4, 6, 8, 10 are in A002202 and 13 is the least number with this property.

Cf. A000010, A002202, A280867, A281687, A281875.

c(n) = sum(k=1, n\2, istotient(k) && istotient(n-k));
a(n) = my(k=1); while(c(eulerphi(k)) != n, k++); k;

a(0) = 1 prepended by Chai Wah Wu, Feb 03 2017

cp's OEIS Frontend

A281687 Number of partitions of 2*n into the sum of two totient numbers (A002202).

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