Paul Duckett - cp's OEIS frontend

A380992 Powers of two which produce a prime number when their digits are reversed.

2, 16, 32, 128, 1024, 131072, 16777216, 137438953472, 35184372088832, 36028797018963968, 1180591620717411303424, 151115727451828646838272, 162259276829213363391578010288128, 174224571863520493293247799005065324265472, 1427247692705959881058285969449495136382746624

Offset: 1

Paul Duckett, Feb 11 2025

A057708 is the main entry for this question. - N. J. A. Sloane, Feb 28 2025

			The digits of 1024 (2^10) reverse to form the prime number 4201.

Cf. A057708.

Select[2^Range[200],PrimeQ[IntegerReverse[#]]&] (* James C. McMahon, Feb 28 2025 *)

a(n) = 2^A057708(n). - Pontus von Brömssen, Feb 11 2025

A380759 Number of coincident digits occurring in expression of integers in both base 2 and base 10.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3

Offset: 1

Paul Duckett, Feb 01 2025

Multiple occurrences (e.g., two ones) count as two occurrences.

The first term having n occurrences is a(10^(n-1)).

			a(10) = 2, because 10 in base 10 is 1010 in base 2 (coincident digits 1 and 0).
For n=1002, the following a(1002) = 3 digits coincide,
   n = decimal    1002
   n = binary     1111101010
                         ^^^ same digits

Cf. A000027, A007088.

a[n_] := Total[Min /@ Transpose[(DigitCount[n, #, {0, 1}] & /@ {2, 10})]]; Array[a, 100] (* Amiram Eldar, Feb 04 2025 *)

a(n) = my(b=binary(n), d=digits(n)); min(#select(x->(x==1), b), #select(x->(x==1), d)) + min(#select(x->(x==0), b), #select(x->(x==0), d)); \\ Michel Marcus, Feb 04 2025

More terms from Michel Marcus, Feb 28 2025

A379372 Sphenic numbers k such that 2*k+1 is also a sphenic number.

Original entry on oeis.org

322, 357, 370, 402, 430, 442, 610, 654, 790, 822, 826, 874, 885, 942, 1045, 1054, 1105, 1130, 1182, 1222, 1342, 1358, 1414, 1510, 1578, 1582, 1677, 1702, 1738, 1742, 1767, 1798, 1802, 1810, 1842, 1947, 2014, 2035, 2086, 2185, 2222, 2247, 2282, 2334, 2365, 2397

Offset: 1

Paul Duckett, Dec 21 2024

			322 is a term because 322 = 2*7*23 (sphenic) and 2*322+1 = 645 = 3*5*43 (sphenic).

Cf. A007304, A177221.

sphenicQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1}; Select[Range[3000], And @@ sphenicQ /@ {#, 2*# + 1} &] (* Amiram Eldar, Dec 21 2024 *)

A378297 Squarefree semiprimes k that remain squarefree semiprimes for exactly two iterations of the map k -> 2*k+1.

Original entry on oeis.org

38, 46, 106, 129, 133, 201, 235, 334, 335, 381, 417, 458, 489, 538, 579, 583, 597, 623, 626, 685, 689, 694, 767, 781, 898, 921, 1073, 1082, 1099, 1214, 1226, 1227, 1234, 1285, 1299, 1315, 1385, 1486, 1514, 1517, 1546, 1603, 1631, 1646, 1799, 1817, 1819, 1841

Offset: 1

Paul Duckett, Nov 22 2024

			38 is a term because 38 (semiprime) gives 2*38+1 (77, semiprime) gives 2*77+1 (155, semiprime).  But 155 gives 2*155+1 (311, not semiprime), so the chain has length three (38, 77, 155).
921 is a term because 921 (semiprime) gives 2*921+1 (1843, semiprime) gives 2*1843+1 (3687, semiprime). But 3687 gives 2*3687+1 (7375, not semiprime), so the chain has length three (921, 1843, 3687).

Cf. A177221, A006881.

s[n_] := -1 + Length@ NestWhileList[2*# + 1 &, n, FactorInteger[#][[;; , 2]] == {1, 1} &]; Select[Range[2000], s[#] == 3 &] (* Amiram Eldar, Dec 17 2024 *)

A376864 4-brilliant numbers with distinct prime factors.

Original entry on oeis.org

210, 46189, 55913, 62491, 70499, 75361, 78793, 81719, 84227, 89947, 95381, 96577, 99671, 100529, 101959, 103037, 104533, 110143, 111397, 114257, 116831, 121693, 121771, 124729, 127699, 128557, 128843, 130169, 131461, 133331, 134849, 139403, 141427, 143429

Offset: 1

Paul Duckett, Oct 07 2024

			210 = 2*3*5*7 is a term.
130169 = 13*17*19*31 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A046386 and A376704.

from sympy import factorint
def ok(n):
    f = factorint(n)
    return len(f) == sum(f.values()) == 4 and len(set([len(str(p)) for p in f])) == 1
print([k for k in range(144000) if ok(k)]) # Michael S. Branicky, Oct 08 2024

from math import prod
from sympy import primerange
from itertools import count, combinations, islice
def bgen(d): # generator of terms that are products of d-digit primes
    primes, out = list(primerange(10**(d-1), 10**d)), set()
    for t in combinations(primes, 4): out.add(prod(t))
    yield from sorted(out)
def agen(): # generator of terms
    for d in count(1): yield from bgen(d)
print(list(islice(agen(), 34))) # Michael S. Branicky, Oct 08 2024

Terms corrected by Michael S. Branicky, Oct 08 2024

A376800 3-brilliant numbers with distinct prime factors.

Original entry on oeis.org

30, 42, 70, 105, 2431, 2717, 3289, 3553, 4147, 4199, 4301, 4433, 4807, 5083, 5291, 5423, 5681, 5797, 5863, 6061, 6149, 6409, 6479, 6721, 6851, 6919, 7163, 7337, 7429, 7579, 7657, 7667, 7733, 7843, 8041, 8177, 8437, 8569, 8671, 8723, 8789, 8987, 9061, 9139, 9269

Offset: 1

Paul Duckett, Oct 04 2024

			30 = 2*3*5 is a term.
2431 = 11*13*17 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A376703 and A007304.

from sympy import factorint
def ok(n):
    f = factorint(n)
    return len(f) == sum(f.values()) == 3 and len(set([len(str(p)) for p in f])) == 1
print([k for k in range(9300) if ok(k)]) # Michael S. Branicky, Oct 05 2024

from math import prod
from sympy import primerange
from itertools import count, combinations, islice
def bgen(d): # generator of terms that are products of d-digit primes
    primes, out = list(primerange(10**(d-1), 10**d)), set()
    for t in combinations(primes, 3): out.add(prod(t))
    yield from sorted(out)
def agen(): # generator of terms
    for d in count(1): yield from bgen(d)
print(list(islice(agen(), 45))) # Michael S. Branicky, Oct 05 2024

a(6) and beyond from Michael S. Branicky, Oct 05 2024

A375733 Integers of the form p^(q^r) where p, q, r are distinct primes.

Original entry on oeis.org

390625, 1953125, 5764801, 40353607, 214358881, 815730721, 2357947691, 6975757441, 10604499373, 16983563041, 78310985281, 118587876497, 322687697779, 500246412961, 847288609443, 852891037441, 1801152661463, 3512479453921, 7984925229121, 11688200277601, 14507145975869

Offset: 1

Paul Duckett, Aug 25 2024

a(15) is the first term not an eighth or ninth power.

			390625 is a term because it equals 5^(2^3).
118587876497 is a term because it equals 17^(3^2).

Subsequence of A217709.

N:= 10^20: # for terms <= N
with(priqueue):
initialize(pq):
if 2^(3^5) < N then insert([-2^(3^5),2,3,5],pq) fi:
if 2^(5^3) < N then insert([-2^(5^3),2,5,3],pq) fi:
if 3^(2^5) < N then insert([-3^(2^5),3,2,5],pq) fi:
if 3^(5^2) < N then insert([-3^(5^2),3,5,2],pq) fi:
if 5^(2^3) < N then insert([-5^(2^3),5,2,3],pq) fi:
if 5^(3^2) < N then insert([-5^(3^2),5,3,2],pq) fi:
Res:= NULL:
while pq[0] > 0 do
  t:= extract(pq);
  Res:= Res, -t[1];
  q:= nextprime(t[2]);
  while member(q,{t[3],t[4]}) do q:= nextprime(q) od;
  v:= q^(t[3]^t[4]);
  if v <= N then insert([-v,q,t[3],t[4]],pq) fi;
  if t[2] = 2 then
    q:= nextprime(t[3]);
    if q = t[4] then q:= nextprime(q) fi;
    v:= 2^(q^t[4]);
    if v <= N then insert([-v,2,q,t[4]],pq) fi;
  fi;
  if {t[2],t[3]} = {2,3} then
    q:= nextprime(t[4]);
    v:= t[2]^(t[3]^q);
    if v <= N then insert([-v,t[2],t[3],q],pq) fi;
  fi
od:
Res; # Robert Israel, Aug 27 2024

seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 8]]], PrimeQ], t}, t = Select[Tuples[ps, {3}], UnsameQ @@ # &]; t = Select[t, #[[1]] < Surd[max, (#[[2]]^#[[3]])] &]; Sort[Select[(#[[1]]^(#[[2]]^#[[3]])) & /@ t, # <= max &]]]; seq[10^14] (* Amiram Eldar, Aug 27 2024 *)

A374561 Integers which are palindromes when expressed in more than one base 2 to 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18, 20, 21, 24, 26, 27, 28, 31, 33, 36, 40, 45, 46, 50, 51, 52, 55, 57, 63, 65, 67, 73, 78, 80, 82, 85, 88, 91, 92, 93, 98, 99, 100, 104, 105, 107, 109, 111, 114, 119, 121, 127, 129, 130, 135, 141, 142, 150, 151, 154, 160, 164, 170, 171, 173, 178

Offset: 1

Paul Duckett, Jul 11 2024

Sequence is infinite because all integers of the form 4^n-1 are palindromic in bases 2 and 4.

			5 is a term since it's palindromic in more than one base: base 2 (101) and base 4 (11).
121 is a term since it's palindromic in base 3 (11111) and base 7 (232), and also in fact in bases 8 and 10.

Cf. A050812, A253594.

q[n_] := Count[Range[2, 10], ?(PalindromeQ[IntegerDigits[n, #]] &)] > 1; Select[Range[180], q] (* _Amiram Eldar, Jul 20 2024 *)

isok(k) = sum(b=2, 10, my(v=digits(k, b)); v==Vecrev(v)) > 1; \\ Michel Marcus, Aug 03 2024

from sympy.ntheory import is_palindromic
def ok(n):
    c = 0
    for b in range(2, 11):
        c += int(is_palindromic(n, b))
        if c > 1: return True
    return False
print([k for k in range(1, 180) if ok(k)]) # Michael S. Branicky, Aug 02 2024

A050812(a(n)) >= 2. - Michael S. Branicky, Aug 02 2024

A367334 Integers using the same number of letters in U.S. English and French.

Original entry on oeis.org

0, 3, 5, 6, 9, 10, 14, 21, 22, 30, 33, 35, 36, 39, 154, 180, 182, 191, 193, 204, 234, 247, 248, 267, 268, 273, 340, 343, 345, 346, 349, 357, 358, 360, 363, 365, 366, 369, 372, 375, 376, 377, 378, 400, 402, 424, 431, 432, 540, 543, 545, 546, 549, 557, 558, 560, 563, 565, 566, 569

Offset: 1

Paul Duckett, Nov 14 2023

Hyphens are not counted.

Numbers k such that A005589(k) = A167507(k).

			"Three" and "trois" have the same number of letters, so 3 is a term.
"One hundred eighty two" and "cent quatre vingt deux" have the same number of letters, so 182 is a term.

Cf. A005589, A167507.

A355847 Irregular table read by rows, in which the rows list integers formed in the process in A180301, but generalized to other starting integers. A row ends when reaching a term in A180301.

Original entry on oeis.org

1, 2, 3, 12, 20, 21, 22, 200, 4, 6, 10, 12, 20, 21, 22, 200, 5, 6, 10, 12, 20, 21, 22, 200, 6, 10, 12, 20, 21, 22, 200, 7, 10, 12, 20, 21, 22, 200, 8, 9, 10, 12, 20, 21, 22, 200, 9, 10, 12, 20, 21, 22, 200, 10, 12, 20, 21, 22, 200

Offset: 1

Paul Duckett, Jul 18 2022

			a(2) = 2 (a term in A180301);
a(3) = 3 (three), so a(4) = 12 (twelve).
The triangle starts:
   1:   1
   2:   2
   3:   3   12   20   21   22  200
   4:   4    6   10   12   20   21   22  200
   5:   5    6   10   12   20   21   22  200
   6:   6   10   12   20   21   22  200
   7:   7   10   12   20   21   22  200
   8:   8    9   10   12   20   21   22  200
   9:   9   10   12   20   21   22  200
  10:  10   12   20   21   22  200
  11:  11   12   20   21   22  200
  12:  12   20   21   22  200
  13:  13   20   21   22  200
  14:  14   16   20   21   22  200
  15:  15   16   20   21   22  200
  16:  16   20   21   22  200
  17:  17   20   21   22  200
  18:  18   19   20   21   22  200
  19:  19   20   21   22  200
  20:  20   21   22  200

cp's OEIS Frontend

User: Paul Duckett

A380992 Powers of two which produce a prime number when their digits are reversed.

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Formula

A380759 Number of coincident digits occurring in expression of integers in both base 2 and base 10.

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PARI

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A379372 Sphenic numbers k such that 2*k+1 is also a sphenic number.

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Mathematica

A378297 Squarefree semiprimes k that remain squarefree semiprimes for exactly two iterations of the map k -> 2*k+1.

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Mathematica

A376864 4-brilliant numbers with distinct prime factors.

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Python

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A376800 3-brilliant numbers with distinct prime factors.

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Python

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A375733 Integers of the form p^(q^r) where p, q, r are distinct primes.

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Maple

Mathematica

A374561 Integers which are palindromes when expressed in more than one base 2 to 10.

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Mathematica