Michael S. Branicky - cp's OEIS frontend

A386964 a(1) = prime(1) = 2, a(n) = 10*a(n-1) + (prime(n) mod 10).

2, 23, 235, 2357, 23571, 235713, 2357137, 23571379, 235713793, 2357137939, 23571379391, 235713793917, 2357137939171, 23571379391713, 235713793917137, 2357137939171373, 23571379391713739, 235713793917137391, 2357137939171373917, 23571379391713739171, 235713793917137391713

Offset: 1

Michael S. Branicky, Aug 11 2025

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A007652, A276481.

a:= proc(n) option remember; `if`(n<1, 0, a(n-1)*10+irem(ithprime(n), 10)) end:
seq(a(n), n=1..21);  # Alois P. Heinz, Aug 12 2025

a[1]=2;a[n_]:=10a[n-1]+Mod[Prime[n],10];Array[a,21] (* James C. McMahon, Aug 12 2025 *)

from sympy import nextprime
from itertools import islice
def A386964(): # generator of terms
    an = pn = 2
    while True:
        yield an
        an = 10*an + (pn:=nextprime(pn))%10
print(list(islice(A386964(), 21)))

a(n) = concatenation of A007652(1)..A007652(n).

A383927 Binary echo numbers: positive integers k such that the gpf(k-1) is a suffix of k when gpf(k-1) and k are written in binary.

Original entry on oeis.org

7, 15, 19, 21, 55, 61, 63, 71, 101, 115, 127, 155, 157, 163, 181, 255, 273, 295, 301, 331, 349, 351, 365, 487, 501, 541, 573, 585, 599, 631, 687, 711, 723, 741, 781, 817, 827, 901, 1055, 1135, 1211, 1277, 1331, 1361, 1387, 1405, 1459, 1471, 1475, 1501, 1621, 1641, 1751

Offset: 1

Michael S. Branicky, May 15 2025

No term may be even, since if k were even, then k-1 would be odd and have only odd prime factors, none of which could be a suffix of k.

			7 is a term since 7 = 111_2, the gpf(6) = 3 = 11_2, and 11 is a suffix of 111.
21 is a term since 21 = 10101_2, the gpf(20) = 5 = 101_2, and 101 is a suffix of 10101.

Vincenzo Librandi, Table of n, a(n) for n = 1..3158

Binary analog of A383896 (and of A383296).

Cf. A006530.

Select[Range@2000,(f=IntegerDigits[FactorInteger[#-1][[-1,1]],2])==IntegerDigits[#,2][[-Length@f;;]]&] (* Giorgos Kalogeropoulos, May 15 2025 *)

from sympy import factorint
def ok(n): return n > 2 and bin(n)[2:].endswith(bin(max(factorint(n-1)))[2:])
print([k for k in range(1800) if ok(k)]) # Michael S. Branicky, May 15 2025

A383272 Positions of records in A383271.

Original entry on oeis.org

0, 2, 6, 15, 21, 45, 111, 261, 1605, 1995, 4935, 8295, 69825, 268155, 550725, 4574955, 12024855, 39867135, 398467245, 1698754365, 16351800465

Offset: 1

Michael S. Branicky, Apr 21 2025

Using A322743, a(22) <= 72026408685 and, if it is equal, a(23) <= 120554434875.

Base-2 analog of A276694.

Cf. A383271, A322743.

A383271 Number of primes (excluding n) that may be generated by replacing any binary digit of n with a digit from 0 to 1.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 0, 2, 2, 1, 1, 1, 0, 3, 1, 1, 2, 3, 0, 4, 1, 3, 0, 2, 0, 3, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1, 1, 1, 4, 0, 5, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 0, 3, 1, 1, 1, 2, 0, 4, 0, 3, 2, 3, 0, 3, 1, 4, 1, 1, 0, 5, 0, 4, 1, 1, 0, 4, 1, 2, 0, 0, 0, 3, 1, 1

Offset: 0

Michael S. Branicky, Apr 21 2025

Also, the number of prime neighbors of n in H(A070939(n), 2), where H(k, b) is the Hamming graph whose vertices are the sequences of length k over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1 (see A145667).

Prepending 1 is not allowed.

			a(3) = 1 since 3 = 11_2 can be changed to 10_2 = 2, which is prime.
a(5) = 1 since 5 = 101_2 can be changed to 001_2 = 1, 111_2 = 7 (prime), or 100_2 = 4.
a(6) = 2 since 6 = 110_2 can be changed to 010_2 = 2 (prime), 100_2 = 4, or 111_2 = 7 (prime).
a(7) = 2 since 7 = 111_2 can be changed to 011_2 = 3 (prime), 101_2 = 5 (prime), or 110_2 = 6.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Base-2 analog of A209252.

Cf. A070939, A145667, A352942.

a:= n-> nops(select(isprime, [seq(Bits[Xor](2^i, n), i=0..ilog2(n))])):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 23 2025

A383271[n_] := Count[BitXor[n, 2^Range[0, BitLength[n] - 1]], _?PrimeQ];
Array[A383271, 100, 0] (* Paolo Xausa, Apr 23 2025 *)

from gmpy2 import is_prime
def a(n):
    if n == 0:
        return 0
    if n&1 == 0:
        return int(is_prime(n + 1)) + int(1<<(n.bit_length()-1)^n == 2)
    mask, c = 1, 0
    for i in range(n.bit_length()):
        if is_prime(mask^n):
            c += 1
        mask <<= 1
    return c
print([a(n) for n in range(90)])

A381040 Numbers k such that the concatenation of 1, k! and 1 is prime.

Original entry on oeis.org

7, 9, 10, 15, 21225

Offset: 1

Michael S. Branicky, Apr 14 2025

Also, numbers k such that 10^(A034886(k)+1) + 10k! + 1 is prime.
For k >= 5, the foregoing requires that 10^(A034886(k)+1) + 1 has no prime factors <= k.
a(6) > 10^5. - Michael S. Branicky, Apr 24 2025

			The concatenation of 1, 7! and 1 is 150401, which is prime, so 7 is a term.

Cf. A000142, A034886.

Corresponding primes are in A262195.

A380788 Numbers with a prime number of binary digits.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Michael S. Branicky, Feb 03 2025

			4 is a term since its binary representation has 3 bits, a prime.
64 is a term since its binary representation has 7 bits, a prime.

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

Cf. A272441, A053738.

Select[Range[200], PrimeQ[BitLength[#]] &] (* Paolo Xausa, Feb 03 2025 *)

from sympy import isprime
def ok(n): return isprime(n.bit_length())
print([k for k in range(150) if ok(k)])

# faster for initial segment of sequence
from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    d = 2
    while True:
        yield from (i for i in range(2**(d-1), 2**d))
        d = nextprime(d)
print(list(islice(agen(), 65)))

from sympy import primerange
def A380788(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(min(x,(1<Chai Wah Wu, Feb 03 2025

A376774 Indices of records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 25, 28, 30, 37, 40, 41, 50, 55, 57, 66, 67, 76, 85, 93, 94, 139, 148, 157, 165, 174, 175, 179, 188, 197, 269, 278, 279, 288, 297, 369, 378

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Cf. A302656, A376772, A376773.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

A376773 Records in A376772.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 21, 25, 44, 48, 52, 53, 58, 75, 77, 84, 96, 135, 146, 300, 317, 401, 452, 478, 1608, 1677, 1679, 1681, 1683, 1703, 1753, 1773, 13649, 13704, 124912, 124925, 125336, 128212, 128221, 128347, 128376, 128529

Offset: 1

N. J. A. Sloane, Nov 05 2024 (with thanks to Michael S. Branicky)

Numbers that are the slowest to appear in A302656.

Cf. A302656, A376772, A376774, A376775.

Summary: the 16 sequences derived from A302656 are A376769-A376776, A377903-A377904, A377906-A377911.

a(37)-a(46) from Dominic McCarty, Nov 08 2024

A376874 a(n) = A376877(n) / p where p is the largest prime factor of A376877(n).

Original entry on oeis.org

6, 20, 28, 88, 104, 272, 304, 550, 650, 368, 464, 496, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 11132, 4288, 4544, 4672, 5056, 5312, 5696, 6208, 6464, 6592, 6848, 6976, 7232, 8128, 16768, 17536, 17792, 19072, 19328, 20096, 20864, 21376, 22144, 22912, 23168, 24448

Offset: 1

Peter Luschny and Michael S. Branicky, Oct 27 2024

nonn

Apparently a subset of A006039 and of A180332.
It seems that the terms are abundant numbers unless p is a Mersenne prime; in that case they are perfect numbers (unproved).
Terms a(1)-a(59) are each divisible by the corresponding p, and many of those quotients are powers of 2.

Peter Luschny and Michael S. Branicky, Table of n, a(n) for n = 1..59

Cf. A376877, A006039, A180332, A023196, A083207.

a := proc(n) A376877(n); % / max(NumberTheory:-PrimeFactors(%)) end:
seq(a(n), n = 1..45);

A376850 Numbers k such that 85^k - 2 is prime.

Original entry on oeis.org

1, 4, 10, 29, 44, 381, 565, 5478, 6423, 15484, 32773

Offset: 1

Michael S. Branicky, Oct 10 2024

Dedicated to N. J. A. Sloane for his 85th birthday!

Cf. A248546, A376910.

from sympy import isprime
def ok(n): return isprime(85**n - 2)
print([k for k in range(600) if ok(k)])

cp's OEIS Frontend

User: Michael S. Branicky

A386964 a(1) = prime(1) = 2, a(n) = 10*a(n-1) + (prime(n) mod 10).

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A383927 Binary echo numbers: positive integers k such that the gpf(k-1) is a suffix of k when gpf(k-1) and k are written in binary.

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A383272 Positions of records in A383271.

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A383271 Number of primes (excluding n) that may be generated by replacing any binary digit of n with a digit from 0 to 1.

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A381040 Numbers k such that the concatenation of 1, k! and 1 is prime.

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A380788 Numbers with a prime number of binary digits.

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A376774 Indices of records in A376772.

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A376773 Records in A376772.

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A376874 a(n) = A376877(n) / p where p is the largest prime factor of A376877(n).

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