A246167 -id:A246167 - cp's OEIS frontend

A246271 Starting from A003961(n), the number of additional iterations of A003961 required for the result to be of the form 4k+1.

0, 1, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 6, 1, 0, 2, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 7, 0, 0, 0, 1, 1, 0, 1, 2, 5, 0, 2, 0, 0, 1, 2, 1, 1, 0, 1, 3, 1, 2, 0, 3, 0, 1, 2, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 6, 0, 0, 6, 1, 0, 0, 4, 0, 0, 3, 0, 2, 0, 1, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

			a(5) = 2, because exactly two additional iterations of A003961 are needed before A003961(5) = 7 is of the form 4k+1; as A003961(7) = 11 and A003961(11) = 13. (We have 7 = 3 mod 4, 11 = 3 mod 4 and 13 = 1 mod 4.)

Antti Karttunen, Table of n, a(n) for n = 1..10001

A246261 gives the positions of zeros, A246263 the positions of nonzeros.
A246280 the positions where n occurs for the first time, A246167 the positions of new distinct values.
Cf. A003961, A246260, A246272, A246259, A246277.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A246271(n) = {my(i); i=0; n = A003961(n); while(((n%4)!=1), i++; n = A003961(n)); i};
for(n=1, 10001, write("b246271.txt", n, " ", A246271(n)));
(Scheme, two different variants, the second one employing memoizing definec-macro)
(define (A246271 n) (let loop ((i 0) (n n)) (let ((next (A003961 n))) (if (= 1 (modulo next 4)) i (loop (+ i 1) next)))))
(definec (A246271 n) (if (= 1 (A246260 n)) 0 (+ 1 (A246271 (A003961 n)))))

If A246260(n) = 1, a(n) = 0, otherwise 1 + a(A003961(n)).

A246280 a(n) = the smallest starting value for k such that n will be the exact number of additional iterations of A003961 needed when we start from A003961(k) before we first reach a number of the form 4k+1; the least k such that A246271(k) = n.

Original entry on oeis.org

1, 2, 5, 66, 91, 55, 21, 46, 1362, 1654, 1419, 574, 6463, 5263, 4607, 3497, 589843, 430261, 574823, 567583, 554111, 545869, 20490043, 14635735, 8781429

Offset: 0

Antti Karttunen, Aug 22 2014

nonn

The sequence is infinite (well-defined for all n), provided that A246271 is not bounded.
All the terms are squarefree (see the comment in A246259).
From 2 onward their prime factorizations are: 2, 5, 2*3*11, 7*13, 5*11, 3*7, 2*23, 2*3*227, 2*827, 3*11*43, 2*7*41, 23*281, 19*277, 17*271, 13*269, 571*1033, 13*23*1439, 563*1021, 557*1019, 547*1013, 541*1009, 7*2927149, 5*2927147, 3*2927143. (Note that 2927149 = A000040(211943)).
Note how A003961(21) = 55 and A003961(55) = 91. Also A003961(545869) = 554111, A003961(554111) = 567583, A003961(567583) = 574823.
Similarly: A003961(8781429) = 14635735 and A003961(14635735) = 20490043.
Apart from those descending subsections, the growth rate of the sequence should be roughly a(n) ~ 2^n, assuming that the distribution of 4k+1 (A002144) and 4k+3 primes (A002145) among the primes is even and essentially random.

			a(0) = 1, because 1 is the first such number >= 1 that no iterations are needed before A003961(1) (= 1) is of the form 4k+1.
a(1) = 2, because 2 is the first such number >= 1 that exactly one additional iteration of A003961 is needed before A003961(2) = 3 is of the form 4k+1; as A003961(3) = 5.
a(2) = 5, because 5 is the first such number that exactly two additional iterations of A003961 are needed before A003961(5) = 7 is of the form 4k+1; as A003961(7) = 11 and A003961(11) = 13.

Leftmost column of array A246259.
A246167 gives the same sequence sorted into ascending order.
Cf. A003961, A002144, A002145, A246271, A048672.

;; Other code as in A246259:
(define (A246280 n)  (A246259bi n 1))

cp's OEIS Frontend

A246271 Starting from A003961(n), the number of additional iterations of A003961 required for the result to be of the form 4k+1.

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