Haines Hoag - cp's OEIS frontend

User: Haines Hoag

Haines Hoag has authored 3 sequences.

A376254 Numbers k such that A376294(k) < k.

32, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 864, 961, 972, 1024, 1152, 1250, 1296, 1331, 1369, 1458, 1536, 1600, 1681, 1728, 1849, 1875, 1944, 2000, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2560, 2592, 2662, 2744, 2809, 2916, 3087, 3125, 3136

Offset: 1

Haines Hoag, Sep 17 2024

There are infinitely many numbers in this sequence, since the growth of powers of small primes far outpaces the growth of their digits when concatenated.

First differs from A195330 at 320 which is a term there but not here.

			32 is a term since 32=2^5 and 25<32.
1152 is a term since 1152=2^7*3^2 and 27*32=864, and 864<1152.

Haines Hoag, Table of n, a(n) for n = 1..20000

Cf. A376294, A195330.

f[p_, e_] := 10^IntegerLength[e]*p + e; q[1] = False; q[k_] := Times @@ f @@@ FactorInteger[k] < k; Select[Range[3200], q] (* Amiram Eldar, Sep 26 2024 *)

from math import prod
from sympy import factorint
def ok(n): return prod(int(str(p)+str(e)) for p, e in factorint(n).items()) < n
print([k for k in range(1, 3200) if ok(k)]) # Michael S. Branicky, Sep 27 2024

A376294 The product of n's prime powers, with the base and exponent concatenated.

Original entry on oeis.org

1, 21, 31, 22, 51, 651, 71, 23, 32, 1071, 111, 682, 131, 1491, 1581, 24, 171, 672, 191, 1122, 2201, 2331, 231, 713, 52, 2751, 33, 1562, 291, 33201, 311, 25, 3441, 3591, 3621, 704, 371, 4011, 4061, 1173, 411, 46221, 431, 2442, 1632, 4851, 471, 744, 72, 1092, 5301

Offset: 1

Haines Hoag, Sep 19 2024

Multiplicative with a(p^e) = the concatenation of p and e.
A prime which occurs just once in the factorization of n is taken as exponent 1 so that for instance n = 7 = 7^1 becomes 71.
It is unknown whether there are any fixed points a(n) = n beyond n=1.

			For n=5, a(5) = 51 since 5=5^1.
For n=20, a(20) = 22*51 = 1122 since 20=2^2*5^1.

Haines Hoag, Table of n, a(n) for n = 1..10000
Combo Class, The Mysterious Pattern I Found Within Prime Factorizations, Sep 12 2024.

Cf. A376254 (indices of a(n) < n), A287483 (indices of local maxima).

a:= n-> mul(parse(cat(i[])), i=ifactors(n)[2]):
seq(a(n), n=1..51);  # Alois P. Heinz, Sep 19 2024

f[p_, e_] := 10^IntegerLength[e] * p + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2024 *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my([b,e]=f[i,]); b*10^(1+logint(e,10))+e)} \\ Andrew Howroyd, Sep 19 2024

from math import prod
from sympy import factorint
def a(n): return prod(int(str(p)+str(e)) for p, e in factorint(n).items())
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Sep 27 2024

For primes p, a(p) = 10p + 1.

A350853 a(1) = 2, a(2) = 3; a(n) is the smallest prime not included earlier such that concatenation of three successive terms is a prime.

Original entry on oeis.org

2, 3, 11, 23, 31, 13, 29, 7, 17, 19, 37, 41, 59, 79, 67, 107, 47, 61, 43, 113, 71, 109, 89, 53, 157, 97, 83, 101, 73, 173, 131, 223, 149, 127, 197, 137, 373, 139, 167, 163, 179, 151, 191, 193, 241, 317, 211, 229, 281, 103, 227, 233, 283, 251

Offset: 1

Haines Hoag, Jan 18 2022

Not a permutation of the primes. 5 never appears, since numbers m mod 10 = 5 are divisible by 5, and concatenation of 2 previous terms and 5 guarantee a composite number. - Michael De Vlieger, Feb 16 2022

			From _Michael De Vlieger_, Feb 16 2022: (Start)
a(3) = 11 since 235 and 237 are composite, but 2311 is prime.
a(4) = 23 since 3115, 3117, 31113, 31117, and 31119 are composite, but 31123 is prime.
a(5) = 31 since 11235, 11237, 112313, 112317, 112319, and 112329 are composite, but 112331 is prime. (End)

Haines Hoag, Table of n, a(n) for n = 1..31499
Michael De Vlieger, Scatterplot of a(n) for n = 1..2^14, showing records in red and local minima (aside from q = 5, which never appears) in blue.

Cf. A073641, A000040.

a[1]=2; a[2]=3; a[n_]:=a[n]=(k=2; While[!PrimeQ[FromDigits@Join[Flatten[IntegerDigits/@{a[n-2],a[n-1]}],IntegerDigits@k]]||MemberQ[Array[a,n-1],k],k=NextPrime@k];k);Array[a,54] (* Giorgos Kalogeropoulos, Jan 19 2022 *)

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User: Haines Hoag

A376254 Numbers k such that A376294(k) < k.

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A376294 The product of n's prime powers, with the base and exponent concatenated.

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A350853 a(1) = 2, a(2) = 3; a(n) is the smallest prime not included earlier such that concatenation of three successive terms is a prime.

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