A163139 -id:A163139 - cp's OEIS frontend

A377423 Distinct values of the number of integers between consecutive self numbers (A163139), in order of occurrence.

1, 10, 14, 27, 40, 53, 66, 79, 92, 105, 118, 100, 117, 130, 143, 156, 169, 182, 195, 208, 23, 89, 203, 220, 233, 246, 259, 272, 285, 298, 34, 78, 293, 306, 323, 336, 349, 362, 375, 388, 45, 67, 383, 396, 409, 426, 439, 452, 465, 478, 56, 473, 486, 499

Offset: 1

Daniel Mondot, Oct 27 2024

Each new value is typically found between self numbers located around 10^k, for some k.

This sequences exhibits interesting patterns, for instance, many new numbers are 13 apart.

			Between the first 2 self numbers 1 and 3, there is 1 integer. So 1 is in the sequence
The next new gap is between 9 and 20, with 10 integers, so 10 is in the sequence.
The next new gap is between 1006 and 1021, with 14 integers, so 14 is in the sequence.

Daniel Mondot, Table of n, a(n) for n = 1..10000
Daniel Mondot, First self number found after a new gap value

Cf. A003052, A163128, A163139, A225793, A230093, A230094, A377422, A377472, A377473.

a(n) = A377473(n)-1. - Daniel Mondot, Apr 17 2025

A377472 First differences of Colombian or self numbers A003052.

Original entry on oeis.org

2, 2, 2, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 11, 11, 11, 11, 11, 11, 11, 11, 15, 11

Offset: 1

M. F. Hasler, Oct 29 2024

nonn

Most terms are equal to 11.
Up to a(103) = 15, the only exceptions are a(n) = 2 for n = 1, 2, 3, 4, 14, 24, ..., 94.
Then a(n) = 2 for n = 112, 122, ..., 192, and a(201) = 15.
Then a(n) = 2 for n = 210, 220, ..., 290, and a(299) = 15.
This pattern repeats 8 times, with a(n) = 15 for n = 397, 495, ..., 887.
Then a(984) = 28, after which the previous pattern resumes, with a(n) = 2 for n = 992, 1002, ..., 1072, then a(n) = 15 for n = 1081, 1179, ..., 1865.
This larger pattern continues with a(n) = 28 for n = 1962, 2940, 3918, ..., 8808.
A still larger pattern starts at a(9785) = 41, after which the previous pattern repeats with a(9881) = 15, later a(10762) = 28 etc.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A003052 (Colombian or self numbers), A377473 (range of this sequence), A377474 (indices where new terms appear), A163128, A163139.

A377472_upto(N=99)={my(o,L=List()); for(n=1+o=1,oo,if(is_A003052(n), listput(L,n-o); o=n; #L

a(n) = A003052(n+1) - A003052(n).

a(n) = A163139(n) + 1 = A163128(n+1) - A163128(n) + 1. - Max Alekseyev, Jan 02 2025

A377473 Distinct first differences of Colombian or self numbers (A377472), listed in the order they appear.

Original entry on oeis.org

2, 11, 15, 28, 41, 54, 67, 80, 93, 106, 119, 101, 118, 131, 144, 157, 170, 183, 196, 209, 24, 90, 204, 221, 234, 247, 260, 273, 286, 299, 35, 79, 294, 307, 324, 337, 350, 363, 376, 389, 46, 68, 384, 397, 410, 427, 440, 453, 466, 479, 57, 474, 487, 500

Offset: 1

M. F. Hasler, Oct 30 2024

See A377474 for the indices where these first differences appear for the first time.

			A377472(n) = 2 = a(1) for all n <= 4. Then, A377472(n) = 11 = a(2) up to n = 13.
Then again, A377472(14..23) = (2, 11, ..., 11) and similarly up to n = 94.
But A377472(103) = 15 = a(3). Then the previous pattern repeats, with A377472(n) = 2 for n = 112, 122, ..., 192, followed by A377472(n) = 15 at n = 201, 299, 397, ..., 887.
Then A377472(984) = 28 = a(4), and it goes on with A377472(n) = 2 at n = 992, 1002, ..., 1072, and so on, with A377472(n) = 28 at n = 1962, 2940, 3918, ..., 8808.
Then A377472(9785) = 41 = a(5), and the whole previous pattern repeats, with A377472(9881) = 15, then A377472(10762) = 28 etc.
At n = 97786, we find A377472(n) = 54 = a(6), and again the whole previous pattern repeats again 8 more times, each time separated by a 54, until we have, at n = 977787, A377472(n) = 67 = a(7). And so on.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A003052 (Colombian numbers), A377472 (1st differences of Colombian numbers), A163139 (= A377472 - 1), A377423.

A377473_upto(N=9, show=1)={my(o, c, d, L=List()); for(n=1+o=1, oo, is_A003052(n)||next; c++; if(!setsearch(L, d=n-o), show && printf("%d, ",[c,d]); listput(L,d); #L

a(n) = A377423(n) + 1.

Terms a(9) onward computed from A377423 by Max Alekseyev, Dec 31 2024

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A377423 Distinct values of the number of integers between consecutive self numbers (A163139), in order of occurrence.

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A377472 First differences of Colombian or self numbers A003052.

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A377473 Distinct first differences of Colombian or self numbers (A377472), listed in the order they appear.

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