A284597 -id:A284597 - cp's OEIS frontend

A285893 Least number to start a run of exactly n nondecreasing values of sigma (sum of divisors, A000203).

45, 5, 313, 1, 356067536821, 36721681

Offset: 1

M. F. Hasler, May 06 2017

a(6) = 36721681, see also A028965.

The analogous sequence based on tau = A000005 instead of sigma is A284597.

			We have the following values of sigma for n = 1..10:
         n   1   2   3   4   5   6   7   8   9  10  ...
   sigma(n)  0   1   1   2   1   2   1   3   2   2  ...
We see a run of 4 nondecreasing values starting at 1, ending at 4, therefore a(4) = 1. There is a run of 2 nondecreasing values starting at 5, ending at 6, therefore a(2) = 5.
Correspondingly, a run of length 1 corresponds to a number n such that sigma(n-1) > sigma(n) > sigma(n+1). This happens first at a(1) = 45.

Cf. A000005, A000203, A028965, A284597 (analog for sigma_0), A286287 (analog for omega = A001221), A286288 (analog for bigomega = A001222).

Function[s, {45}~Join~Map[Function[r, Select[s, Last@ # == r &][[1, 1]]], Range[2, Max[s[[All, -1]] ] ]]]@ Map[{#[[1, 1]], Length@ # + 1} &, DeleteCases[SplitBy[#, #[[-1]] >= 0 &], k_ /; k[[1, -1]] < 0]] &@ MapIndexed[{First@ #2, #1} &, Differences@ Array[DivisorSigma[1, #] &, 10^6]] (* Michael De Vlieger, May 06 2017 *)

alias(A,A285893);A=vector(19);apply(scan(N,s=1,t=sigma(s))=for(k=s+1,N,t>(t=sigma(k))||next;k-s>#A||A[k-s]||printf("a(%d)=%d,",k-s,s)||A[k-s]=s;s=k);done,[10^8]) \\ Search may be extended using scan(END,START).

a(5)-a(6) from Giovanni Resta, May 07 2017

A286287 Least number to start a run of exactly n nondecreasing values of little omega (A001221).

Original entry on oeis.org

106, 11, 13, 7, 512, 1, 1941, 141, 6847, 211, 195031, 82321, 808083, 534077, 3355906, 526093, 526889774, 127890361, 22529949392, 118968284927, 164159173895, 244022049199, 3022058317713, 585927201061

Offset: 1

N. J. A. Sloane, May 16 2017

a(17) > 10^7.
a(18) = 127890361; a(n) > 4*10^8 for n=17 and for n >= 19. - Jon E. Schoenfield, Jul 16 2017
a(n) > 6*10^12 for n >= 25. - Giovanni Resta, Jul 16 2019

			We have omega(10) = 2, omega(11) = 1, omega(12) = 2, omega(13) = 1. Therefore 11 starts a run of exactly 2 consecutive integers (11, 12) which have nondecreasing (here: strictly increasing) values of omega.
The 6 numbers from 1 through 6 yield values (0, 1, ..., 1, 2) for omega, therefore a(6) = 1. The 4 numbers from 7 through 10 yield values (1, 1, 1, 2) for omega, therefore a(4) = 7.
A run of length 1 is a single number n such that omega(n-1) > omega(n) > omega(n+1). (If we had "<=" in one of the cases, it would be part of a run of at least 2 numbers with nondecreasing omega.) This first happens for a(1) = 106.

M. F. Hasler, Posting to Sequence Fans Mailing List, May 06 2017

Cf. A001221, A284597 (analog for tau = A000005), A285893 (analog for sigma = A000203).

A286288 Least number to start a run of exactly n nondecreasing values of (big) Omega (A001222).

Original entry on oeis.org

46, 5, 43, 1, 2021, 121, 25202, 2521, 162121, 460801, 27268546, 23553169, 244068841, 913535283, 3195380866, 2088087121, 5988790769809, 2601212829601

Offset: 1

N. J. A. Sloane, May 16 2017

a(11) > 10^7. - M. F. Hasler, May 16 2017

a(19) > 7.5*10^12. - Giovanni Resta, Nov 12 2019

			Omega(1..5) = (0, 1, 1, 2, 1), therefore the first run of 4 numbers with nondecreasing Omega (= A001222) starts at a(4) = 1.
Omega(4..7) = (2, 1, 2, 1), so the first run of 2 numbers with nondecreasing Omega starts at a(2) = 5.
A run of subsequent numbers with nondecreasing Omega is of length 1 if it consists of a single number n with Omega(n-1) > Omega(n) > Omega(n+1) (else n belongs to a run of length >= 2). This happens first for a(1) = 46.

M. F. Hasler, Posting to Sequence Fans Mailing List, May 06 2017

Cf. A001222, A284597 (analog for sigma_0 = A000005), A285893 (analog for sigma = A000203), A286287 (analog for omega = A001221).

A303578 List of starts of nondecreasing runs of values of d(n) (the divisor function A000005(n)).

Original entry on oeis.org

1, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 41, 43, 46, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 82, 83, 85, 89, 91, 93, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 121, 125, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 153, 155, 157, 161, 163, 166, 167, 169, 171, 173, 175

Offset: 1

N. J. A. Sloane, Apr 29 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005.
A303577 gives the lengths of the successive runs.
A284597(m) is the smallest number that starts a run of length m.

# Assume that b.txt contains a b-file for A000005
awk ' BEGIN {i = 1; print i}
      {if ($2 < i) print $1; i = $2} ' b.txt >out

A286289 Least number to start a run of exactly n nondecreasing values of the Euler phi function (A000010).

Original entry on oeis.org

314, 6, 315, 14, 1

Offset: 1

N. J. A. Sloane, May 16 2017

a(6) > 10^7. - Michael De Vlieger, May 19 2017

a(6) > 10^13. - Giovanni Resta, Nov 12 2019

			From _Michael De Vlieger_, May 19 2017: (Start)
A run of subsequent numbers with nondecreasing phi is of length 1 if it consists of a single number n with phi(n-1) > phi(n) > phi(n+1) (else n belongs to a run of length >= 2). This happens first for a(1) = 314.
Phi(14..18) = (6, 8, 8, 16, 6), therefore the first run of 4 numbers with nondecreasing phi(= A000010) starts at a(4) = 14. (End)

M. F. Hasler, Posting to Sequence Fans Mailing List, May 06 2017

Cf. A000010, A284597, A285893, A286287, A286288.

Prepend[#, Module[{k = 2}, While[Sign@ Differences@ EulerPhi[k + {-1, 0, 1}] != {-1, -1}, k++];k]] &@ Function[s, Function[r, If[Length@ # > 0, #[[1, 1]], -1] &@ Select[s, Length@ # == r &]] /@ Range@ Max@ Map[Length, s]]@ DeleteCases[SplitBy[MapIndexed[Function[k, (2 Boole[#1 <= #2] - 1) k & @@ #1]@ First@ #2 &, Partition[Array[EulerPhi, 10^7], 2, 1]], Sign], w_ /; First@ w < 0] (* Michael De Vlieger, May 19 2017 *)

A303577 Break up the list of values of the divisor function d(k) into nondecreasing runs; sequence gives lengths of successive runs.

Original entry on oeis.org

4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 1, 1, 2, 4, 2, 2, 4, 4, 2, 3, 1, 2, 2, 2, 2, 3, 3, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 1, 1, 2, 4, 2, 2, 2, 2, 6

Offset: 1

N. J. A. Sloane, Apr 29 2018

nonn

			The initial values of d(k) = A000005(k) for k = 1,2,3,... are
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, ...
Breaking this up into nondecreasing runs we get:
[1, 2, 2, 3], [2, 4], [2, 4], [3, 4], [2, 6], [2, 4, 4, 5], [2, 6], [2, 6], [4, 4], [2, 8], [3, 4, 4, 6], [2, 8], [2, 6], [4, 4, 4, 9], [2, 4, 4, 8], [2, 8], [2, 6, 6], [4], [2, 10], [3, 6], [4, 6], [2, 8], [4, 8], [4, 4], [2, 12], [2, 4, 6, 7], ...
whose successive lengths are
4,2,2,2,2,4,2,2,2,2,4,2,2,4,4,2,3,1,2,...

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005.
A303578(m) gives value of n that starts the m-th run.
A284597(m) is the smallest number that starts a run of length m.

More terms from Seiichi Manyama, Apr 29 2018

A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

Original entry on oeis.org

1, 3, 4, 5, 5, 7, 8, 9, 8, 9, 11, 12, 13, 14, 15, 10, 13, 15, 17, 18, 19, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 16, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 20, 22, 24, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39

Offset: 1

Jon E. Schoenfield, Apr 02 2019

Row n lists the numbers k such that
  0 < |{m : Sum_j={m..m+n-1} tau(j) = k}| < infinity
where tau(j) = A000005(j) is the number of divisors of j.

			There is only one number with exactly 1 divisor (namely, k=1), but there are infinitely many numbers with j divisors for every j >= 2, so row 1 consists only of the single term 1.
The sequence of values tau(k) for k >= 1 is A000005, which begins 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ..., from which the sums of two consecutive terms are 1+2=3, 2+2=4, 2+3=5, 3+2=5, 2+4=6, 4+2=6, 2+4=6, 4+3=7, 3+4=7, ...; no number j < 3 appears as such a sum, every j >= 6 appears infinitely many times as such a sum, and each j in {3,4,5} appears as such a sum only finitely many times, so row 2 is {3, 4, 5}.
Row 3 does not contain 6 as a term because there exists no run of 3 consecutive numbers whose sum of tau values is exactly 6.
The first six rows of the table are as follows:
  row 1: {1};
  row 2: {3, 4, 5};
  row 3: {5, 7, 8, 9};
  row 4: {8, 9, 11, 12, 13, 14, 15};
  row 5: {10, 13, 15, 17, 18, 19};
  row 6: {14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27}.

Cf. A000005, A005237, A006558, A048892, A072507, A100366, A119479, A141621, A284596, A284597, A292580, A319037, A319045, A319046.

cp's OEIS Frontend

A285893 Least number to start a run of exactly n nondecreasing values of sigma (sum of divisors, A000203).

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A286287 Least number to start a run of exactly n nondecreasing values of little omega (A001221).

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A286288 Least number to start a run of exactly n nondecreasing values of (big) Omega (A001222).

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A303578 List of starts of nondecreasing runs of values of d(n) (the divisor function A000005(n)).

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A286289 Least number to start a run of exactly n nondecreasing values of the Euler phi function (A000010).

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Mathematica

A303577 Break up the list of values of the divisor function d(k) into nondecreasing runs; sequence gives lengths of successive runs.

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A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

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