A354361 -id:A354361 - cp's OEIS frontend

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

1, 1, 1, 23, 3, 1, 1, 5, 11, 3, 2, 23, 1, 1, 3, 469, 2, 11, 1, 69, 1, 2, 1, 5, 53, 1, 4, 23, 11, 3, 7, 69, 2, 2, 3, 253, 3, 1, 1, 15, 1, 1, 1, 46, 33, 1, 2, 469, 33, 53, 2, 23, 23, 4, 6, 5, 1, 11, 13, 69, 29, 7, 11, 19507, 3, 2, 1, 46, 1, 3, 2, 55, 2, 3, 53, 23, 2, 1, 3, 1407, 2797, 1, 5, 23, 6, 1, 11, 10, 9, 33

Offset: 1

Antti Karttunen, May 23 2022

Cf. A000203, A000720, A354200, A354201, A354202, A354203, A354205, A354207.
Cf. A354361 (positions of 1's).
Cf. also A326042, A348750, A354088, A354096 for similar constructions.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354206(n) = A354203(sigma(A354202(n)));

Multiplicative with a(p^e) = A354203((q^(e+1)-1)/(q-1)) where q = A354200(A000720(p)).
a(n) = A354203(A354205(n)) = A354203(sigma(A354202(n))).
a(n) = n - A354207(n).

A354357 Numbers k, not divisible by 2 or 3, such that sigma(k) is 3-smooth (has no larger prime factors than 3).

Original entry on oeis.org

1, 5, 7, 11, 17, 23, 31, 35, 47, 53, 55, 71, 77, 85, 107, 115, 119, 127, 155, 161, 187, 191, 217, 235, 253, 265, 329, 341, 355, 371, 383, 385, 391, 431, 497, 517, 527, 535, 583, 595, 635, 647, 713, 749, 781, 799, 805, 863, 889, 901, 935, 955, 971, 1081, 1085, 1151, 1177, 1207, 1219, 1265, 1309, 1337, 1397, 1457, 1633

Offset: 1

Antti Karttunen, May 24 2022

nonn

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

Intersection of A007310 and A354356.
Sequence A354202(A354361(n)), n>=1, sorted into ascending order.
Cf. A000203, A065333.

Select[Flatten @ Outer[Plus, 6 * Range[0, 300], {1, 5}], Max @ FactorInteger[DivisorSigma[1, #]][[;;, 1]] <= 3 &] (* Amiram Eldar, May 25 2022 *)
Select[Range[1,1701,2],Mod[#,3]!=0&&Max[FactorInteger[DivisorSigma[1,#]][[;;,1]]]<4&] (* Harvey P. Dale, Dec 17 2023 *)

A065333(n) = ((3^valuation(n, 3)<A065333
A354355(n) = A065333(sigma(n));
isA354357(n) = ((n%2)&&(n%3)&&A354355(n));

cp's OEIS Frontend

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

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