Tejo Vrush - cp's OEIS frontend

A350654 Smallest k such that A349949(k) = n, or -1 if no such k exists.

2, 3, 8, 15, 63, 120, 440, 945, 2079, 4095, 21735, 98175, 133056, 395199, 338625, 1890945, 3501576, 8390304, 35820225, 126775935, 149848335, 879207616, 302464800

Offset: 1

Tejo Vrush, Jan 09 2022

a(25) = 879207615. - Chai Wah Wu, Jan 13 2022

Cf, A000005, A349949.

f(n) = my(sd=setunion(divisors(n-1), divisors(n+1))); sumdiv(n, d, (vecsearch(sd, d-1)>0) || (vecsearch(sd, d+1)>0)); \\ A349949
a(n) = my(k=2); while (f(k) != n, k++); k; \\ Michel Marcus, Jan 10 2022

from itertools import count
from sympy import divisors
def A350654(n):
    for m in count(2):
        c = 0
        for d in divisors(m,generator=True):
            if not (((m-1) % (d-1) if d > 1 else True) and (m-1) % (d+1) and ((m+1) % (d-1) if d > 1 else True) and (m+1) % (d+1)):
                c += 1
                if c > n:
                    break
        if c == n:
            return m # Chai Wah Wu, Jan 12 2022

a(11)-a(19) from Jinyuan Wang, Jan 10 2022
Escape clause value changed to -1 by N. J. A. Sloane, Jan 12 2022
a(20)-a(21) from Chai Wah Wu, Jan 12 2022
a(22)-a(23) from Chai Wah Wu, Jan 13 2022

A350540 a(n) = smallest number x such that x^2 == 17 (mod 2^n).

Original entry on oeis.org

0, 1, 1, 1, 1, 7, 9, 23, 23, 23, 233, 279, 279, 1769, 1769, 6423, 9961, 9961, 55575, 55575, 206569, 206569, 842007, 1255145, 2939159, 2939159, 2939159, 2939159, 64169705, 64169705, 204265751, 204265751, 869476073, 869476073, 3425491223, 3425491223, 13754377961

Offset: 0

Tejo Vrush, Jan 04 2022

nonn

17 is the smallest nonsquare that is congruent to a square mod 2^n for any n.

Any number that is congruent to a square mod 2^n for any n is of the form (4^a)*(8b+1). Such numbers have density 1/6.

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A000290, A000079.

Table[PowerMod[17,1/2,2^k],{k,0,36}] (* Giorgos Kalogeropoulos, Jan 31 2023 *)

a(n) = my(x=0); while (Mod(x, 2^n)^2 != 17, x++); x; \\ Michel Marcus, Jan 04 2022

from sympy.ntheory import sqrt_mod
def A350540(n): return min(sqrt_mod(17,2**n,all_roots=True)) # Chai Wah Wu, Jan 12 2022

a(13)-a(28) from Michel Marcus, Jan 04 2022
a(30)-a(36) from Alois P. Heinz, Jan 04 2022
Edited by N. J. A. Sloane, Jan 12 2022

A349949 a(n) is the number of divisors of n that are 1 above or 1 below a divisor of either n+1 or n-1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 4, 4, 2, 2, 3, 3, 2, 2, 2, 3, 4, 2, 2, 4, 3, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 5, 4, 3, 3, 2, 3, 3, 2, 2, 2, 2, 2, 4, 3, 3, 3, 2, 5, 4, 2, 2, 4, 3, 2, 3, 3

Offset: 2

Tejo Vrush, Dec 06 2021

nonn

			a(2) = 1 because 2 and 0 are not divisors of either 1 or 3, but 3 = 2+1 is a divisor of 3.
a(6) = 2 since the divisors of 6 are 1, 2, 3, and 6; those of 5 are 1 and 5; those of 7 are 1 and 7; and, regarding {1, 5, 7}, neither 1-1 = 0 nor 1+1 = 2 are in the set, neither 3-1 = 2 nor 3+1 = 4 is, but 2-1 = 1 is, and 6-1 = 5 is (as is 6+1 = 7).

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A000005.

Table[DivisorSum[n, 1 &, If[# == 1, Or[Mod[n - 1, # + 1] == 0, Mod[n + 1, # + 1] == 0], AnyTrue[# + {-1, 1}, Or[Mod[n - 1, #] == 0, Mod[n + 1, #] == 0] &]] &], {n, 2, 88}] (* Michael De Vlieger, Dec 06 2021 *)

a(n) = my(sd=setunion(divisors(n-1), divisors(n+1))); sumdiv(n, d, (vecsearch(sd, d-1)>0) || (vecsearch(sd, d+1)>0)); \\ Michel Marcus, Dec 07 2021

from sympy import divisors
def aupton(nn):
    alst, prevdivs, divs, nextdivs = [], set(), {1}, {1, 2}
    for n in range(2, nn+1):
        prevdivs, divs, nextdivs = divs, nextdivs, set(divisors(n+1))
        neighdivs = prevdivs | nextdivs
        an = sum(1 for d in divs if {d-1, d+1} & neighdivs != set())
        alst.append(an)
    return alst
print(aupton(88)) # Michael S. Branicky, Dec 06 2021

def A349949(n): return sum(1 for m in filter(lambda d:not (((n-1) % (d-1) if d > 1 else True) and (n-1) % (d+1) and ((n+1) % (d-1) if d > 1 else True) and (n+1) % (d+1)), divisors(n,generator=True))) # Chai Wah Wu, Dec 30 2021

a(p) = 2 for odd prime p. - Chai Wah Wu, Dec 30 2021

a(6), a(12), a(14), a(18) corrected and a(31) and beyond from Michael S. Branicky, Dec 06 2021

A349521 Numbers k such that A348172(k) = 1.

Original entry on oeis.org

4, 16, 20, 28, 32, 36, 44, 48, 52, 64, 68, 72, 76, 92, 100, 108, 112, 116, 124, 140, 144, 148, 160, 164, 172, 176, 180, 188, 192, 196, 208, 212, 216, 220, 224, 236, 240, 244, 252, 256, 260, 268, 272, 284, 292, 304, 308, 316, 320, 332, 336, 340, 352, 356, 360, 364, 368, 380, 388, 396, 400

Offset: 1

Tejo Vrush, Nov 20 2021

nonn

Cf. A348172.

Similar sequences: A349467.

Block[{nn = 9, m, s}, m = 2^(2 nn); s = KeySort@ PositionIndex[Array[DivisorSigma[0, #]/# &, m]]; s = Reverse@ KeyDrop[s, TakeWhile[Keys@ s, 4/#^2 > m &]]; Position[Length /@ Array[Lookup[s, DivisorSigma[0, #]/#] &, 2^nn], 1][[All, 1]]] (* Michael De Vlieger, Dec 06 2021 *)

isok(k) = my(q=numdiv(k)/k); sum(i=1, 4/q^2, numdiv(i)/i == q) == 1; \\ Michel Marcus, Nov 20 2021

A349467 Numbers k such that A349410(k) = 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 17, 19, 21, 23, 25, 28, 29, 31, 32, 33, 36, 37, 39, 40, 41, 43, 47, 48, 49, 51, 53, 57, 59, 61, 67, 69, 70, 71, 73, 75, 79, 81, 83, 87, 89, 90, 93, 96, 97, 98, 101, 103, 107, 109, 110, 111, 113, 120, 121, 123, 126, 127, 128, 129, 130

Offset: 1

Tejo Vrush, Nov 18 2021

nonn

Does this sequence have density 1/3? This sequence has infinitely many terms because every prime number is a term.

The numbers of terms not exceeding 10^k for k = 1, 2, ... are 8, 50, 396, 3566, 33943, 332042, 3297317, 32983277, ... Apparently this sequence has an asymptotic density of about 0.33. - Amiram Eldar, Nov 18 2021

Cf. A349410.

a[n_] := Module[{s = NestWhileList[n*DivisorSigma[0, #] &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; Select[Range[130], a[#] == 1 &] (* Amiram Eldar, Nov 18 2021 *)

A349483 Length of cycle reached when iterating the mapping x-> n*A035116(x) on 1.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 5, 2, 2, 7, 2, 1, 2, 5, 6, 1, 2, 2, 2, 3, 2, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 3, 2, 4, 2, 1, 3, 2, 2, 4, 3, 6, 1, 1, 2, 2, 3, 3, 1, 2, 2, 4, 2, 2, 1, 3, 3, 3, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2, 8, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 1, 1, 1, 2, 3, 1, 2, 4, 1, 2

Offset: 1

Tejo Vrush, Nov 19 2021

nonn

The terms 1-25 all appear below 10^8; the last of these are a(12545280) = 21, a(12684672) = 24, and a(96940800) = 25. - Charles R Greathouse IV, Nov 23 2021

			For n = 2, 1 --> 2 --> 8 --> 32 --> 72 --> 288 --> 648 --> 800 --> 648. The cycle reached has just two terms: 648 and 800. Therefore, a(2) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A035116.

Similar sequences: A349410.

a[n_] := Module[{s = NestWhileList[n*DivisorSigma[0, #]^2 &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

brent(f,x)=my(pow=1,lam=1,tortoise=x,hare=f(x)); while(tortoise!=hare, if(pow==lam, tortoise=hare; pow<<=1; lam=0); hare=f(hare); lam++); lam
a(n)=brent(k->n*numdiv(k)^2,1) \\ Charles R Greathouse IV, Nov 19 2021

A349428 Smallest k such that A349410(k) = n or -1 if no such number exists.

Original entry on oeis.org

1, 4, 15, 30, 42, 360, 196, 525, 2080, 320, 7168, 123200, 35200, 150920, 196000, 1232000, 61236, 466560, 106831872, 49787136, 14580000, 155648000, 94058496, 123561984, 47385000

Offset: 1

Tejo Vrush, Nov 17 2021

Cf. A349410.

Similar sequences: A005179, A348184.

f[n_] := Module[{s = NestWhileList[n * DivisorSigma[0, #] &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && v[[i]] == 0, c++; v[[i]] = n]; n++]; TakeWhile[v, # > 0 &]]; seq[15, 10^6] (* Amiram Eldar, Nov 17 2021 *)

Escape clause value changed to -1. - N. J. A. Sloane, Jan 14 2022

A349410 Length of cycle reached when iterating the mapping x-> n*A000005(x) on 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 4, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2

Offset: 1

Tejo Vrush, Nov 16 2021

nonn

			For n = 9, 1 --> 9 --> 27 --> 36 --> 81 --> 45 --> 54 --> 72 --> 108 --> 108. The cycle reached has just one term: 108. Therefore, a(9) = 1.

Cf. A000005, A000040, A062011.

a[n_] := Module[{s = NestWhileList[n * DivisorSigma[0, #] &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)

f(n, x) = n*numdiv(x);
find(nm, v) = {forstep (n=#v-1, 1, -1, if (v[#v] == v[n], return(#v-n);););}
a(n) = {my(list = List(), found=0, m=n); listput(list, m); while (! found, my(nm = f(n, m)); listput(list, nm); found = find(nm, list); m = nm;); found;} \\ Michel Marcus, Nov 17 2021

from sympy import divisor_count
terms = []
for n in range(1, 101):
    s, t = [1], True
    while t:
        for i in range(2, len(s)):
            if s[-i] == s[-1]:
                t = False
                terms.append(i - 1)
                break
        s.append(n*divisor_count(s[-1]))
print(terms) # Gleb Ivanov, Nov 17 2021

a(A000040(n)) = 1.

A348568 Highly composite numbers (A002182) such that the exponents of 2 and 3 in their prime factorization are equal.

Original entry on oeis.org

1, 6, 36, 180, 1260, 7560, 45360, 83160, 498960, 1081080, 6486480, 32432400, 110270160, 551350800, 2095133040, 10475665200, 73329656400, 240940299600, 1686582097200, 48910880818800, 1516237305382800

Offset: 1

Tejo Vrush, Oct 27 2021

These are all the terms in the sequence because for a number x that has exponents of 2 and 3 equal and >= 5 in its prime factorization, 8x/9 is a smaller number with at least the same number of divisors. Since 1516237305382800 is the greatest highly composite number that is not a multiple of 32, A134592(32), it is the last term of this sequence.

			1516237305382800 is highly composite and its prime factorization is 2^4 * 3^4 * 5^2 * 7^2 * 11 * 13 * 17 * 19 * 23 * 29 * 31. Since the exponents of 2 and 3 are both 4, 1516237305382800 is in this sequence.

Cf. A000005, A002182, A064615, A134592.

HCN = Import["https://oeis.org/A002182/b002182.txt", "Table"][[;; , 2]]; Select[HCN, IntegerExponent[#, 2] == IntegerExponent[#, 3] &] (* Amiram Eldar, Oct 27 2021 *)

Intersection of A002182 and A064615.

A348532 a(n) is the number of multisets of integers that are possible to reach by starting with n occurrences of 0 and by splitting and reverse splitting.

Original entry on oeis.org

1, 1, 2, 2, 7, 9, 43, 59, 338, 490, 3097, 4639, 31283, 48107, 338553, 531469, 3857036, 6157068, 45713546, 73996100

Offset: 0

Tejo Vrush, Oct 21 2021

Splitting is taking 2 occurrences of the same integer and incrementing one of them by 1 and decrementing the other occurrence by 1.

Reverse splitting is taking two elements with a difference of 2 and incrementing the smaller one by 1 and decrementing the larger one by 1. It is the opposite of splitting.

			For n = 5, the multisets are as follows:
  {{0,0,0,0,0}}   {{-1,0,0,0,1}}   {{-1,-1,0,1,1}}
  {{-1,-1,0,0,2}} {{-1,-1,-1,1,2}} {{-2,0,0,1,1}}
  {{-2,0,0,0,2}}  {{-2,-1,1,1,1}}  {{-2,-1,0,1,2}}.
  Therefore, a(5) = 9.
For n = 6, the multisets are as follows:
  {{0,0,0,0,0,0}}     {{-1,0,0,0,0,1}}     {{-1,-1,0,0,1,1}}
  {{-1,-1,0,0,0,2}}   {{-1,-1,-1,1,1,1}}   {{-1,-1,-1,0,1,2}}
  {{-1,-1,-1,0,0,3}}* {{-1,-1,-1,-1,2,2}}* {{-1,-1,-1,-1,1,3}}*
  {{-2,0,0,0,1,1}}    {{-2,0,0,0,0,2}}     {{-2,-1,0,1,1,1}}
  {{-2,-1,0,0,1,2}}   {{-2,-1,0,0,0,3}}*   {{-2,-1,-1,1,1,2}}
  {{-2,-1,-1,0,2,2}}  {{-2,-1,-1,0,1,3}}   {{-2,-1,-1,-1,2,3}}*
  {{-2,-2,1,1,1,1}}*  {{-2,-2,0,1,1,2}}    {{-2,-2,0,0,2,2}}
  {{-2,-2,0,0,1,3}}   {{-2,-2,-1,1,2,2}}   {{-2,-2,-1,1,1,3}}
  {{-2,-2,-1,0,2,3}}  {{-2,-2,-2,2,2,2}}*  {{-2,-2,-2,1,2,3}}*
  {{-3,0,0,0,0,3}}*   {{-3,0,0,0,1,2}}*    {{-3,0,0,1,1,1}}*
  {{-3,-1,1,1,1,1}}*  {{-3,-1,0,1,1,2}}    {{-3,-1,0,0,2,2}}
  {{-3,-1,0,0,1,3}}   {{-3,-1,-1,1,2,2}}   {{-3,-1,-1,1,1,3}}
  {{-3,-1,-1,0,2,3}}  {{-3,-2,1,1,1,2}}*   {{-3,-2,0,1,2,2}}
  {{-3,-2,0,1,1,3}}   {{-3,-2,0,0,2,3}}    {{-3,-2,-1,2,2,2}}*
  {{-3,-2,-1,1,2,3}}.
  Therefore, a(6) = 43.
The ones marked with an asterisk are the ones that need reverse splitting
to be reached. They are not produced using the rules of A347913.

Cf. A000571, A347913.

def nextq(q):
    used, used2 = set(), set()
    for i in range(len(q)-1):
        for j in range(i+1, len(q)):
            if q[i] == q[j]:
                if q[i] in used: continue
                used.add(q[i])
                qc = list(q); qc[i] -= 1; qc[j] += 1
                yield tuple(sorted(qc))
            elif q[j] - q[i] == 2:  # assumes q is sorted
                if q[i] in used2: continue
                used2.add(q[i])
                qc = list(q); qc[i] += 1; qc[j] -= 1
                yield tuple(sorted(qc))
def a(n):
    s = tuple(0 for i in range(n)); reach = {s}; expand = list(reach)
    while len(expand) > 0:
        q = expand.pop()
        for qq in nextq(q):
            if qq not in reach:
                reach.add(qq)
                expand.append(qq)
    return len(reach)
print([a(n) for n in range(13)]) # Michael S. Branicky, Oct 21 2021

It appears that a(n) = A000571(n) for odd n.

a(6)-a(19) from Michael S. Branicky, Oct 21 2021

cp's OEIS Frontend

User: Tejo Vrush

A350654 Smallest k such that A349949(k) = n, or -1 if no such k exists.

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A350540 a(n) = smallest number x such that x^2 == 17 (mod 2^n).

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A349949 a(n) is the number of divisors of n that are 1 above or 1 below a divisor of either n+1 or n-1.

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A349521 Numbers k such that A348172(k) = 1.

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A349467 Numbers k such that A349410(k) = 1.

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A349483 Length of cycle reached when iterating the mapping x-> n*A035116(x) on 1.

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A349428 Smallest k such that A349410(k) = n or -1 if no such number exists.

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A349410 Length of cycle reached when iterating the mapping x-> n*A000005(x) on 1.

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