A355594 -id:A355594 - cp's OEIS frontend

A355593 a(n) is the number of alternating integers that divide n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 6, 1, 4, 3, 5, 1, 6, 1, 5, 4, 2, 2, 7, 3, 2, 4, 5, 2, 7, 1, 6, 2, 3, 3, 9, 1, 3, 2, 6, 2, 7, 2, 3, 5, 3, 2, 8, 3, 6, 2, 4, 1, 8, 2, 7, 2, 4, 1, 9, 2, 2, 6, 6, 3, 4, 2, 4, 4, 7, 1, 11, 1, 3, 4, 5, 2, 5, 1, 7, 5, 3, 2, 9, 3, 3, 4, 4, 2, 11, 2, 5, 2, 4, 2, 10, 1, 6, 3, 7

Offset: 1

Bernard Schott, Jul 08 2022

This sequence first differs from A355302 at index 13, where a(13) = 1 while A355302(13) = 2.

This sequence first differs from A332268 at index 14, where a(14) = 4 while A332268(14) = 3.

			40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 2 are not alternating integers: {20, 40}, hence a(40) = 8 - 2 = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A030141 (alternating integers), A355594, A355595, A355596.

Similar to A332268 (with Niven numbers) and A355302 (with undulating integers).

Alt:= [$1..9, seq(seq(10*i+r - (i mod 2), r=[1,3,5,7,9]),i=1..9)]:
V:= Vector(100):
for t in Alt do J:= [seq(i,i=t..100,t)]; V[J]:= V[J] +~ 1 od:
convert(V,list); # Robert Israel, Nov 26 2023

q[n_] := !MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 120] (* Amiram Eldar, Jul 08 2022 *)

alternate(n,d=digits(n))=for(i=2,#d, if((d[i]-d[i-1])%2==0, return(0))); 1
a(n)=sumdiv(n,d,alternate(d)) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 08 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A030141(n) = 5.1... (the sums up to 10^10, 10^11 and 10^12 are 5.1704..., 5.1727... and 5.1738..., respectively). - Amiram Eldar, Jan 06 2024

A355595 Positions of records in A355593: Integers whose number of alternating divisors sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 72, 144, 180, 360, 504, 630, 1080, 1260, 1890, 2520, 3780, 7560, 15120, 18900, 22680, 30240, 37800, 45360, 75600, 90720, 151200, 162540, 226800, 317520, 325080, 650160, 763560, 1137780, 1243620, 1527120, 2275560, 3054240, 3738420, 4551120, 6826680, 7476840, 14953680, 17445960, 21818160, 26168940, 36363600, 43636320, 52337880

Offset: 1

Bernard Schott, Jul 08 2022

Alternating integers are in A030141.

Corresponding records of number of alternating divisors are 1, 2, 3, 4, 6, 7, 9, 11, ...

			a(6) = 24 is in the sequence because A355593(24) = 7 is larger than any earlier value in A355593.

Cf. A030141, A355593, A355594.

Similar, but with undulating divisors: A355304.

q[n_] := ! MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; f[n_] := DivisorSum[n, 1 &, q[#] &]; fm = 0; s = {}; Do[fn = f[n]; If[fn > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 08 2022 *)

a(21)-a(36) from Amiram Eldar, Jul 08 2022

a(37)-a(50) from Charles R Greathouse IV, Jul 08 2022

A355596 Numbers all of whose divisors are alternating numbers (A030141).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 32, 36, 41, 43, 47, 49, 50, 54, 58, 61, 63, 67, 69, 81, 83, 87, 89, 94, 98, 101, 103, 107, 109, 123, 125, 127, 129, 141, 145, 147, 149, 161, 163, 167, 181, 183, 189, 214, 218, 250, 254, 290, 298

Offset: 1

Bernard Schott, Jul 12 2022

The smallest alternating number that is not a term is 30, because of 15.

			32 is a term since all the divisors of 32, i.e., 1, 2, 4, 8, 16 and 32, are alternating numbers

Subsequence of A030141.
Cf. A355593, A355594, A355595.
Similar sequences: A062687, A190217, A329419, A337941.

q[n_] := AllTrue[Divisors[n], !MemberQ[Differences[Mod[IntegerDigits[#], 2]], 0] &]; Select[Range[300], q] (* Amiram Eldar, Jul 12 2022 *)

isokd(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141
isok(m) = sumdiv(m, d, isokd(d)) == numdiv(m); \\ Michel Marcus, Jul 12 2022

from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def ok(n):
    return c(n) and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Jul 12 2022

a(51) and beyond from Michael S. Branicky, Jul 12 2022

A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

Original entry on oeis.org

1, 10, 20, 30, 48, 72, 60, 140, 144, 120, 210, 180, 300, 240, 560, 504, 360, 420, 780, 1764, 900, 960, 720, 1200, 840, 1560, 2640, 1260, 1440, 2400, 3900, 3024, 1680, 3120, 2880, 4800, 7056, 3600, 2520, 3780, 3360, 5460, 6480, 16848, 6300, 8820, 7200, 9240, 6720, 12480, 5040

Offset: 0

Bernard Schott, Jul 14 2022

			48 has 10 divisors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}, only 12, 16, 24 and 48 are nonpalindromic; no positive integer smaller than 48 has four nonpalindromic divisors, hence a(4) = 48.

Michael S. Branicky, Table of n, a(n) for n = 0..710

Cf. A029742, A087991, A093037, A334391.

Similar sequences: A087997, A333456, A355303, A355594.

f[n_] := DivisorSum[n, 1 &, ! PalindromeQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^5] (* Amiram Eldar, Jul 14 2022 *)

isnp(n) = my(d=digits(n)); d!=Vecrev(d); \\ A029742
a(n) = my(k=1); while (sumdiv(k, d, isnp(d)) != n, k++); k; \\ Michel Marcus, Jul 14 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s != s[::-1]
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 0, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 51))) # Michael S. Branicky, Jul 27 2022

More terms from Michel Marcus, Jul 14 2022

A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 54, 24, 36, 48, 72, 180, 144, 360, 336, 468, 504, 936, 1008, 1512, 2520, 3024, 5040, 6552, 7560, 22680, 13104, 19656, 49140, 105840, 39312, 78624, 98280, 248976, 334152, 196560, 393120, 668304, 1244880, 1670760, 1867320, 4520880, 3341520, 3734640

Offset: 1

Bernard Schott, Sep 16 2022

This sequence is finite since A009993 is finite with 511 nonzero terms, hence the last term is a(511) = lcm of the 511 nonzero terms of A009993.

a(511) = 8222356410...6120992000 and has 1036 digits. - Michael S. Branicky, Sep 16 2022

			For n=7, the divisors of 54 are {1, 2, 3, 6, 9, 18, 27, 54} of which 7 have their digits in strictly increasing order (all except 54). No integer < 54 has 7 such divisors, so a(7) = 54.

Cf. A009993, A357171, A357173, A160218.

Similar: A087997 (palindromic), A355303 (undulating), A355594 (alternating).

s[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, v[[i]] = n; c++]; n++]; v]; seq[25, 10^4] (* Amiram Eldar, Sep 16 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
f(n) = sumdiv(n, d, isok(d)); \\ A357171
a(n) = my(k=1); while (f(k) !=n, k++); k; \\ Michel Marcus, Sep 16 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s == "".join(sorted(set(s)))
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 37))) # Michael S. Branicky, Sep 16 2022

More terms from Amiram Eldar, Sep 16 2022

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A355593 a(n) is the number of alternating integers that divide n.

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A355595 Positions of records in A355593: Integers whose number of alternating divisors sets a new record.

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A355596 Numbers all of whose divisors are alternating numbers (A030141).

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A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

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A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.

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