Hans Havermann - cp's OEIS frontend

A383675 Number of n-digit terms in A157711.

0, 0, 0, 0, 1, 0, 2, 2, 1, 4, 9, 5, 8, 3, 9, 9, 12, 6, 14, 4, 5, 9, 8, 10, 13, 10, 8, 19, 17, 15, 20, 16, 27, 16, 26, 14, 23, 18, 26, 22, 40, 23, 21, 18, 32, 24, 29, 15, 33, 21, 25, 33, 34, 25, 25, 22, 47, 30, 40, 25, 37, 29, 38, 33, 47, 30, 41, 37, 45, 41, 46, 33, 42, 36, 52, 39, 48, 28, 49, 37

Offset: 1

Hans Havermann, May 29 2025

The b-file terms are based on probable prime counts that have been verified correct (by counting proven primes) to index 200. Subsequent terms (as they get larger) seemingly face an increasing probability of counting a probable prime that is actually a composite but it is unknown if such probability is ever large enough to impact the intended proven prime count.

			There are two A157711 terms (1011001, 1100101) containing 7 digits, so a(7) = 2.

Hans Havermann, Table of n, a(n) for n = 1..2500
Hans Havermann, Combined point/points-joined plot of 2500 terms

Cf. A157711.

A374834 Start the sequence with a(1) = 3. From now on iterate: if |a(n)| is prime, subtract from a(n) the |a(n)|-th prime of the list of primes A000040 and if |a(n)| is nonprime, add to a(n) the |a(n)|-th nonprime of the list of nonprimes A018252.

Original entry on oeis.org

3, -2, -5, -16, 9, 24, 59, -218, 58, 138, 316, 709, -4672, 708, 1563, 3408, 7364, 15779, 33601, -363046, 33600, 71181, 150103, 315315, 660199, -9268542, 660198, 1378289, 2870170, 5963401, 12365326, 25593793, 52887805, 109127470, 224867446, 462788654, 951362304, 1953684017, -43964452250, 1953684016

Offset: 1

Eric Angelini and Hans Havermann, Jul 21 2024

sign

			As a(1) = 3 has an absolute prime value, we subtract the 3rd prime from 3 to form the next term: 3 - 5 = -2;
as a(2) = -2 has an absolute prime value, we subtract the 2nd prime from -2 to form the next term: -2 - 3 = -5;
as a(3) = -5 has an absolute prime value, we subtract the 5th prime from -5 to form the next term: -5 - 11 = -16;
as a(4) = -16 has an absolute nonprime value, we add the 16th nonprime to -16 to form the next term: -16 + 25 = 9; etc.

Hans Havermann, Table of n, a(n) for n = 1..71 (terms 1..60 from Michael S. Branicky)
Eric Angelini, Do we yo-yo?, personal blog of the author, July 2024.

Cf. A000040, A018252.

np[n_]:=FixedPoint[n+PrimePi[#]&,n+PrimePi[n]]; (* A018252 *) seq[n_]:=If[PrimeQ[Abs[n]],n-Prime[Abs[n]],n+np[Abs[n]]]; NestList[seq,3,52] (* Hans Havermann, Jul 23 2024 *)

from itertools import islice
from sympy import prime, composite, isprime
def agen(): # generator of terms
    an = 3
    while True:
        yield an
        if isprime(abs(an)):
            an = an - prime(abs(an))
        else:
            an += composite(abs(an)-1) if abs(an) > 1 else 1
print(list(islice(agen(), 38))) # Michael S. Branicky, Jul 22 2024

A373391 Integers whose American English names (minus punctuation) can be split into two parts wherein the product of the letter-ranks of one part is equal to that of the other.

Original entry on oeis.org

3960, 13986, 15368, 80547, 85740, 111789, 111987, 386048, 408649, 408946, 410699, 410969, 410996, 486014, 519487, 519784, 609408, 609804, 615430, 619814, 629428, 629824, 639438, 639834, 649448, 649844, 659458, 659854, 669468, 669864, 679478, 679874

Offset: 1

Hans Havermann, Jun 03 2024

By letter-rank we mean a=1, b=2, ..., z=26. If the qualifying split happens beside a letter "a" (as for 408649) there will be two solutions, as moving that "a" to the other side of the split will not affect either product.
There can be no terms involving "million" less than 10^60 ("novemdecillion") since "m" = 13 would otherwise have no counterpart to make the product of all valuations square. - Michael S. Branicky, Jun 03 2024
Similarly, there can be no terms involving "quadrillion" less than 10^18 because "q" = 17. a(84)=1000107588, a(10887)=1000000611455. - Hans Havermann, Jun 15 2024

			3960 = threethousandni|nehundredsixty has the product of the letter-ranks of each side of the vertical pipe equal (to 486491443200000).

Hans Havermann, Equal products in split English integer names

Cf. A372222.

from math import prod, isqrt
from num2words import num2words
def n2w(n): return "".join(c for c in num2words(n).replace(" and", "") if c.isalpha())
def ok(n):
    d = [ord(c) - ord('A') + 1 for c in n2w(n).upper()]
    if isqrt(s:=prod(d))**2 != s: return False
    return any(prod(d[:i]) == prod(d[i:]) for i in range(1, len(d)))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Jun 03 2024

from math import prod, isqrt
from num2words import num2words
def n2w(n):
    return "".join(c for c in num2words(n).replace(" and", "") if c.isalpha())
def ok(n):
    d = [ord(c) - ord("A") + 1 for c in n2w(n).upper()]
    pr = prod(d)
    s = isqrt(pr)
    if s**2 != pr:
        return False
    p = 1
    for i in range(len(d)):
        p *= d[i]
        if p > s:
            return False
        if p == s:
            return True
for n in range(1, 100000):
    if ok(n):
        print(n, end=", ")
# David A. Corneth, Jun 03 2024, adapted from Michael S. Branicky, Jun 03 2024

A372106 A370972 terms composed of nine distinct digits which may repeat.

Original entry on oeis.org

1476395008, 116508327936, 505627938816, 640532803911, 1207460451879, 1429150367744, 1458956660623, 3292564845031, 3820372951296, 5056734498816, 6784304541696, 8090702381056, 9095331446784, 10757095489536, 10973607685048, 13505488366293, 14913065975808, 38203732951296

Offset: 1

Hans Havermann, Apr 18 2024

Each factorization is necessarily composed of multipliers that use only the single missing digit.
The single missing digit cannot be 0, 1, 5, or 6. Terms missing 2, 3, 4, 7, and 8 appear within a(1)-a(6). 52612606387341 = 9^6 * 99 * 999999 is an example of a term missing 9. - Michael S. Branicky, Apr 18 2024
Some terms are equal to the sum of two distinct smaller terms:
a(741) = a(635) + a(673)
a(1202) = a(1081) + a(1144)
a(1273) = a(1110) + a(1169)
a(1493) = a(1335) + a(1374)
a(2753) = a(2478) + a(2528)
a(2793) = a(2512) + a(2583)
a(3581) = a(3234) + a(3317)
a(4199) = a(3808) + a(3921)
a(4803) = a(4510) + a(4607) = a(4557) + a(4568)
a(5756) = a(5256) + a(5362)
a(6083) = a(5718) + a(5847)
a(7262) = a(6761) + a(6779)
a(7331) = a(6786) + a(6904)
a(9204) = a(8723) + a(8886)
a(9364) = a(8858) + a(8982)
a(9453) = a(8972) + a(8983) - Hans Havermann, Apr 21 2024

			10973607685048 = 22222*22222*22222 is in the sequence because it has nine distinct digits and may be factored using only its missing digit.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1016 from Hans Havermann)
Michael S. Branicky and Hans Havermann, Table of n, a(n) for n = 1..10000, fully factored

Cf. A370970, A370972.

import heapq
from itertools import islice
def agen(): # generator of terms
    allowed = [2, 3, 4, 7, 8, 9]
    v, oldt, h, repunits, bigr = 1, 0, list((d, d) for d in allowed), [1], 1
    while True:
        v, d = heapq.heappop(h)
        if (v, d) != oldt:
            s = set(str(v))
            if len(s) == 9 and str(d) not in s:
                yield v
            oldt = (v, d)
            while v > bigr:
                bigr = 10*bigr + 1
                repunits.append(bigr)
                for c in allowed:
                    heapq.heappush(h, (bigr*c, c))
            for r in repunits:
                heapq.heappush(h, (v*d*r, d))
print(list(islice(agen(), 100))) # Michael S. Branicky, Apr 19 2024

A370972 Composite numbers with properties that its digits (which may appear with multiplicity) may not appear in any of its factors (wherein the digits may also appear with multiplicity) and the combined digits of the product and the factors must have at least one of each of the ten digits.

Original entry on oeis.org

8596, 8790, 9360, 9380, 9870, 10752, 10764, 10854, 10968, 11760, 12780, 13608, 13860, 14760, 14780, 14820, 15628, 15678, 16038, 16704, 16920, 17082, 17280, 17340, 17640, 17820, 17920, 18090, 18096, 18690, 18720, 18960, 19068, 19084, 19240, 19440, 19460, 19608, 19740, 19780, 19800, 19980, 20457, 20574, 20748, 20754

Offset: 1

N. J. A. Sloane, Apr 15 2024. Terms were computed by Hans Havermann

See A370970 for another version.

Ed Pegg Jr noted that 1476395008 is the smallest term composed of nine distinct digits. See A372106 for subsequent terms. - Hans Havermann, Apr 19 2024

			996880 = 2*2*4*5*17*733: 8 and 9 appear twice each in the product. 2, 3, and 7 appear twice each in the factors. The digits in the product are distinct from the digits in the factors and, ignoring the duplicates, we have a combined 9680245173, one of each of the ten digits. -  _Hans Havermann_, Apr 15 2024

Ed Pegg Jr, Posting to Math-Fun Mailing List, April 2024.

Hans Havermann, Table of a(n) and corresponding factorization(s) for all terms <= 100000.

Cf. A370970, A372106.

A370970 Numbers k which have a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.

Original entry on oeis.org

8596, 8790, 9360, 9380, 9870, 10752, 12780, 14760, 14820, 15628, 15678, 16038, 16704, 17082, 17820, 17920, 18720, 19084, 19240, 20457, 20574, 20754, 21658, 24056, 24507, 25803, 26180, 26910, 27504, 28156, 28651, 30296, 30576, 30752, 31920, 32760, 32890, 34902, 36508, 47320, 58401, 65128, 65821

Offset: 1

N. J. A. Sloane, Apr 13 2024, following emails from Ed Pegg Jr and Hans Havermann. The terms were computed by Hans Havermann

The total number of digits in k, f1, ..., fr is ten, and they are all distinct.

			The complete list of terms:
 8596 = 2*14*307
 8790 = 2*3*1465
 9360 = 2*4*15*78
 9380 = 2*5*14*67
 9870 = 2*3*1645
10752 = 3*4*896
12780 = 4*5*639
14760 = 5*9*328
14820 = 5*39*76
15628 = 4*3907
15678 = 39*402
16038 = 27*594 = 54*297
16704 = 9*32*58
17082 = 3*5694
17820 = 36*495 = 45*396
17920 = 8*35*64
18720 = 4*5*936
19084 = 52*367
19240 = 8*37*65
20457 = 3*6819
20574 = 6*9*381
20754 = 3*6918
21658 = 7*3094
24056 = 8*31*97
24507 = 3*8169
25803 = 9*47*61
26180 = 4*7*935
26910 = 78*345
27504 = 3*9168
28156 = 4*7039
28651 = 7*4093
30296 = 7*8*541
30576 = 8*42*91
30752 = 4*8*961
31920 = 5*76*84
32760 = 8*45*91
32890 = 46*715
34902 = 6*5817
36508 = 4*9127
47320 = 8*65*91
58401 = 63*927
65128 = 7*9304
65821 = 7*9403

Hans Havermann, Pandigital Products, Apr 13 2024

Cf. A370972, A372106.

A359482 Lexicographically earliest sequence of distinct terms > 0 such that the sum a(n) + a(n+1) is a substring of the concatenation (a(n), a(n+1)).

Original entry on oeis.org

1, 10, 99, 889, 8009, 1101, 9089, 80718, 100284, 183899, 206021, 396118, 215703, 354632, 108578, 469891, 229021, 61195, 34146, 7321, 13817, 3536, 1825, 749, 167, 508, 324, 2096, 4337, 2958, 2870, 4171, 12941, 16470, 30560, 25465, 21056, 35296, 17665, 35927, 23345, 10106, 548, 279, 516, 1094, 3228, 5302

Offset: 1

Eric Angelini and Hans Havermann, Jul 03 2023

Is this sequence a permutation of the integers > 0?

I conjecture that it isn't, and more specifically, that a(1) = 1 is the only single-digit term, and a(2) = 10 the only multiple of 10 below 100. See Examples for other terms of the form x*10^k, 1 <= x <= 9. - M. F. Hasler, Jul 03 2023

			1 + 10 = 11 and 11 is a substring of concat(1, 10) = 110.
10 + 99 = 109 and 109 is a substring of concat(10, 99) = 1099.
99 + 889 = 988 and 988 is a substring of concat(99, 889) = 99889.
889 + 8009 = 8898 and 8898 is a substring of 8898009.
8009 + 1101 = 9110 and 9110 is a substring of 80091101, etc.
Some examples of terms of the form x*10^k, x < 10: a(2136) = 800, a(4204) = 1000, a(6246) = 900, a(6618) = 100, a(11268) = 400, a(17446) = 10000, a(39292) = 600, a(44989) = 700, a(91359) = 300, ... - _M. F. Hasler_, Jul 03 2023

Eric Angelini, Échecs et Maths, Personal blog, bottom of the page, July 2023.
Hans Havermann, 100000 terms, July 2023

Cf. A300000.

A359482_first(n)={my(ok(a,k)=my(c=a*10^logint(k*10,10)+k); k=10^logint(10*a+=k,10); until(a>c\=10, c%k==a&& return(1)), U=[], a=0); vector(n,n, my(k=1); while(setsearch(U,k)|| !ok(a,k), k++); U=setunion(U,[k]); a=k)} \\ Becomes slow for n > 10. - M. F. Hasler, Jul 03 2023

A362335 Lexicographically earliest sequence of distinct nonnegative terms wherein every digit of a(n) is the absolute difference of two adjacent digits in a(n+1).

Original entry on oeis.org

0, 11, 10, 100, 110, 112, 102, 1002, 1022, 1102, 1120, 1124, 1026, 10028, 10086, 10082, 10866, 10822, 10886, 10882, 11086, 11082, 11208, 11976, 10928, 100913, 10096, 10093, 10966, 10933, 10996, 10993, 11096, 11093, 22309, 11309, 23009, 13009, 23099, 13099, 23309

Offset: 1

Eric Angelini and Hans Havermann, May 27 2023

All terms > a(24) contain at least one 9 with an adjacent 0. All terms > a(25) contain at least one instance of identical adjacent digits.

			a(125) = 9902. The next term is 10097, not 20009, because, in spite of its providing more digit differences than are needed, it is lexicographically earlier.

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

{upto(N) = my(U=[], a=0); vector(N,n, if(n>1, my(da=Set(if(a,digits(a)))); a=10^#da; while( setsearch(U,a) || #setminus(da, Set(abs((n=digits(a))[^1]-n[^-1]))), a++)); U=setunion(U,[a]); a)} \\ M. F. Hasler, May 27 2023

from itertools import count, islice
def c(k, d):
    dk = list(map(int, str(k)))
    return set(abs(dk[i+1]-dk[i]) for i in range(len(dk)-1)) >= d
def agen(): # generator of terms
    an, aset = 0, {0}
    while True:
        yield an
        d = set(map(int, set(str(an))))
        an = next(k for k in count(10**len(d)) if k not in aset and c(k, d))
        aset.add(an)
print(list(islice(agen(), 41))) # Michael S. Branicky, May 27 2023
def A362335(n, A=[0]):
    while len(A) <= n:
        z = lambda a: zip(d := tuple(int(d) for d in str(a)), d[1:])
        D = set(str(A[-1])) ; a = 10**len(D)
        while a in A or D - set(str(abs(x-y)) for x,y in z(a)): a += 1
        A . append(a)
    return A[n] # M. F. Hasler, May 27 2023

a(27) and beyond from Michael S. Branicky, May 27 2023

A360656 Least k such that the decimal representation of 2^k contains all possible n-digit strings.

Original entry on oeis.org

68, 975, 16963, 239697, 2994863

Offset: 1

Hans Havermann, Feb 15 2023

			2^68 = 295147905179352825856 is the least power of 2 containing all ten decimal digits, so a(1) = 68 = A171744(1).
2^975 is the least power of 2 containing all 100 two-digit strings, so a(2) = 975.

Cf. A171744, A360623.

def a(n, starte=0):
    e, p2, t = starte, 2**starte, 10**n
    while True:
        s2, ss = str(p2), set()
        for i in range(len(s2)-n+1):
            ss.add(s2[i:i+n])
            if len(ss) == t:
                return e
        e += 1
        p2 *= 2
print([a(n) for n in range(1, 4)]) # Michael S. Branicky, Feb 22 2023

A360623 Largest k such that the decimal representation of 2^k is missing any n-digit string.

Original entry on oeis.org

168, 3499, 53992, 653060

Offset: 1

Hans Havermann, Feb 14 2023

All terms are conjectural, checked up to 10^6.

			168 is the largest base-ten power of 2 that is missing any of the 10 length-1 digit-strings (missing '2').
3499 is the largest base-ten power of 2 that is missing any of the 100 length-2 digit-strings (missing '95').
53992 is the largest base-ten power of 2 that is missing any of the 1000 length-3 digit-strings (missing '661').
653060 is the largest base-ten power of 2 that is missing any of the 10000 length-4 digit-strings (missing '6164').

Hans Havermann, Small-string final non-appearance coincidences in base-ten powers of two

Cf. A094776, A360656.

cp's OEIS Frontend

User: Hans Havermann

A383675 Number of n-digit terms in A157711.

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A374834 Start the sequence with a(1) = 3. From now on iterate: if |a(n)| is prime, subtract from a(n) the |a(n)|-th prime of the list of primes A000040 and if |a(n)| is nonprime, add to a(n) the |a(n)|-th nonprime of the list of nonprimes A018252.

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A373391 Integers whose American English names (minus punctuation) can be split into two parts wherein the product of the letter-ranks of one part is equal to that of the other.

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A372106 A370972 terms composed of nine distinct digits which may repeat.

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A370972 Composite numbers with properties that its digits (which may appear with multiplicity) may not appear in any of its factors (wherein the digits may also appear with multiplicity) and the combined digits of the product and the factors must have at least one of each of the ten digits.

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A370970 Numbers k which have a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.

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A359482 Lexicographically earliest sequence of distinct terms > 0 such that the sum a(n) + a(n+1) is a substring of the concatenation (a(n), a(n+1)).

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A362335 Lexicographically earliest sequence of distinct nonnegative terms wherein every digit of a(n) is the absolute difference of two adjacent digits in a(n+1).

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A360656 Least k such that the decimal representation of 2^k contains all possible n-digit strings.

A370970 Numbers k which have a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together give 0,1,...,9 exactly once.