A056704 -id:A056704 - cp's OEIS frontend

A372280 Composite numbers k such that the digits of k are in nondecreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nonincreasing order.

4, 8, 9, 16, 22, 25, 27, 33, 44, 49, 55, 77, 88, 99, 125, 128, 155, 256, 279, 1477, 1555, 1688, 1899, 2799, 3479, 3577, 14777, 16888, 18999, 22599, 36799, 444577, 455777, 1112447, 1555555, 2555555, 2799999, 3577777, 3799999, 45577777, 124556677, 155555555555, 279999999999

Offset: 1

Scott R. Shannon, Apr 25 2024

A number 155...555 will be a term if it has two prime factors 5 and 3111...111. Therefore 155555555555 and 1555555555555 are both terms. See A056704.

The next term is greater than 10^11.

			444577 is a term as 444577 = 7 * 7 * 43 * 211, and 444577 has nondecreasing digits while its prime factor concatenation "7743211" has nonincreasing digits.

Michael S. Branicky, Table of n, a(n) for n = 1..64 (all terms <= 20 digits)

Cf. A372308, A372034, A056704, A372029, A372055, A027746, A372249.

from sympy import factorint, isprime
from itertools import count, islice, combinations_with_replacement as mc
def ni(s): return s == "".join(sorted(s, reverse=True))
def bgen(d):
    yield from ("".join(m) for m in mc("0123456789", d) if m[0]!="0")
def agen(): # generator of terms
    for d in count(1):
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni("".join(str(p)*e for p,e in factorint(t).items())):
                yield t
print(list(islice(agen(), 41))) # Michael S. Branicky, Apr 26 2024

a(42)-a(43) from Michael S. Branicky, Apr 26 2024

A372335 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) alternate being larger than and then smaller than the previous digit.

Original entry on oeis.org

14, 15, 78, 161, 591, 1214, 1317, 1318, 1326, 1407, 1418, 1438, 1506, 1509, 1514, 1527, 1538, 1618, 1626, 1646, 1658, 1703, 1714, 1718, 1734, 1739, 1758, 1814, 1834, 1838, 1839, 1857, 1858, 1934, 1938, 2307, 2427, 2509, 2517, 2534, 2535, 2715, 2757, 2758, 2869, 2958, 3419, 3439, 3514, 3523

Offset: 1

Scott R. Shannon, Apr 28 2024

No term can end in 0 or 2; a number ending in 2 would mean the first prime factor is 2, which would disqualify the number, while a number ending in 0 would mean the first 3 distinct prime factors would have to be 2, 3, 5 or 2, 5, either of which would also disqualify the number.

			161 is a term as 161 = 7 * 23 which when concatenated give "161723", the digits of which alternate from being larger than and then smaller than the previous digit.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A372336, A372280, A372308, A372034, A372029, A056704.

A372336 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) alternate being smaller than and then larger than the previous digit.

Original entry on oeis.org

6, 51, 91, 106, 219, 323, 406, 435, 437, 518, 529, 609, 614, 626, 629, 634, 658, 703, 705, 818, 826, 838, 878, 906, 938, 978, 2051, 2093, 2173, 3053, 3241, 4151, 4171, 4281, 5041, 5063, 5141, 5183, 5241, 6251, 6591, 7021, 7081, 7251, 8051, 8121, 8491, 8571, 8781, 9121, 9231, 9291, 9583

Offset: 1

Scott R. Shannon, Apr 28 2024

No term can end in 0 or 2; a number ending in 2 would mean the first prime factor is 2, which would disqualify the number, while a number ending in 0 would mean the first 3 distinct prime factors would have to be 2, 3, 5 or 2, 5, either of which would also disqualify the number.

			106 is a term as 106 = 2 * 53 which when concatenated give "106253", the digits of which alternate from being smaller than and then larger than the previous digit.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A372335, A372280, A372308, A372034, A372029, A056704.

A068813 Primes with a 3 followed by 1's.

Original entry on oeis.org

31, 311, 311111, 31111111111, 311111111111, 31111111111111, 31111111111111111111111111111111111, 311111111111111111111111111111111111111111111111, 31111111111111111111111111111111111111111111111111111

Offset: 1

Amarnath Murthy, Mar 07 2002

			Digit 3 followed by rep-units of length 1,2,5,10,11,13,34,47,52,77,88,...

Makoto Kamada, Prime numbers of the form 311...11.
H. C. Williams, Some primes with interesting digit patterns, Math. Comp. vol 32 no 144 (1978) pp 1306-1310, Table 2, r=3.
Index entries for primes involving repunits

Cf. A002275, A056704.

Select[Table[FromDigits[PadRight[{3},n,1]],{n,2,100}],PrimeQ] (* Harvey P. Dale, Nov 18 2012 *)

for(n=1,200, if(isprime(3*10^n+(10^(n)-1)/9)==1,print1(3*10^n+(10^(n)-1)/9,",")))

More terms from Benoit Cloitre, Mar 09 2002

One additional term from Harvey P. Dale, Nov 18 2012

cp's OEIS Frontend

A372280 Composite numbers k such that the digits of k are in nondecreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nonincreasing order.

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A372335 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) alternate being larger than and then smaller than the previous digit.

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A372336 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) alternate being smaller than and then larger than the previous digit.

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Author

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Comments

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A068813 Primes with a 3 followed by 1's.

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Author

Keywords

Examples

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Mathematica

PARI

Extensions