Claudio Meller - cp's OEIS frontend

A350570 a(n) is the smallest multiple of 7 that can be formed by concatenating the first n positive integers in some order.

21, 231, 3241, 12453, 123564, 1234576, 12347685, 123456879, 10123456798, 1011123456798, 101111223456789, 10111121323457896, 1011112131423457896, 101111213141523456879, 10111121314151623456789, 1011112131415161723456789, 101111213141516171823458796

Offset: 2

Claudio Meller, Jan 06 2022

			(2, 1)   21
(2, 3, 1)   231
(3, 2, 4, 1)   3241
(1, 2, 4, 5, 3)   12453
(1, 2, 3, 5, 6, 4)   123564
(1, 2, 3, 4, 5, 7, 6)   1234576
(1, 2, 3, 4, 7, 6, 8, 5)   12347685
(1, 2, 3, 4, 5, 6, 8, 7, 9)   123456879
From _Jon E. Schoenfield_, Jan 07 2022: (Start)
Observation: for each term from a(2) through a(18), if the positive integers are originally arranged in a string of digits in order from 1 through 9 (or 1 through n for n < 9), left to right, and then (for n > 9) the remaining numbers are incorporated into the string by placing '10' to the left of the initial '1' and then placing all remaining numbers in order from left to right between the '1' and the '2' (e.g., so as to yield the concatenation 10/1/11/12/13/14/15/16/17/18/2/3/4/5/6/7/8/9 for n=18), then at most 4 digits at the right end of the concatenated string need to be rearranged to obtain a(n), as shown here, where a space is inserted to the left of the digits (if any) that need to be rearranged:
                            21
                           231
                          3241
                        12 453
                       123 564
                      12345 76
                     1234 7685
                    123456 879
                  101234567 98
                10111234567 98
              101111223456789
            1011112132345 7896
          101111213142345 7896
        101111213141523456 879
      10111121314151623456789
    1011112131415161723456789
  10111121314151617182345 8796
(End)

Cf. A008589.

a(12)-a(18) from Jon E. Schoenfield, Jan 06 2022

A339753 Base-ten n whose English number-word expression contains a letter making its n-th appearance in the list of consecutive positive integer number-words.

Original entry on oeis.org

1, 2, 3, 11, 23, 24, 29, 31, 108, 109, 198, 199, 240, 241, 243, 244, 245, 246, 247, 248, 249, 250, 251, 453, 454, 559, 1174, 1716, 5556, 5557, 6956, 6957, 15756, 17155, 24998, 24999, 43568, 43569, 735759, 1105805, 1105806, 1105807, 1107784, 1107785, 1584503, 1584504

Offset: 1

Claudio Meller and Hans Havermann, Dec 15 2020

Conjecture: a(494) = 1001001001001998 (for the letter 'a').

			In the list of English consecutive positive integers (one, two, three, ...)
the first 'o', the first 'n', and the first 'e' are in 'one',
the 2nd 'o' is the only such in 'two',
the 3rd 'e' is the second of two such in 'three',
the 11th 'e' is the second of three such in 'eleven',
the 23rd 't' is the third of three such in 'twenty-three',
the 24th 't' is the first of two such in 'twenty-four',
the 29th 'n' is the second of three such in 'twenty-nine',
the 31st 'n' is the only such in 'thirty-one',
the 108th 'n' is the second of two such in 'one hundred eight', ...

Hans Havermann, Table of n, a(n) for n = 1..493
Hans Havermann, On target (revisited)

Cf. A339752 (digits variant).

A317423 a(n) is the smallest number such that with the letters of the name of that number we can spell the name of n numbers smaller than a(n) in Spanish.

Original entry on oeis.org

16, 18, 23, 34, 44, 45, 42, 84, 128, 116, 54, 133, 132, 159, 196, 134, 124, 156, 145, 144, 225, 149, 261, 153, 143, 148, 147, 142, 229, 263

Offset: 1

Claudio Meller, Jul 27 2018

			a(4) = 34 because with the letters of 34 (treinta y cuatro) we can spell these numbers: 1 (uno), 4 (cuatro), 11 (once) and 30 (treinta). Note that we can also write 40 cuarenta, but 40 is bigger than 34 so it does not count.

Cf. A317422 (English).

A317422 a(n) is the smallest positive integer such that with the letters of the name of that number we can spell the name of exactly n smaller positive integers.

Original entry on oeis.org

15, 13, 14, 21, 24, 72, 76, 74, 113, 115, 121, 171, 122, 150, 131, 142, 127, 147, 124, 129, 159, 138, 135, 153, 137, 156, 126, 125, 128, 165, 168, 157, 158, 467, 289, 265, 267, 487, 275, 392, 278, 754, 692, 492, 257, 857, 572, 524, 674, 428, 1133, 748, 1322, 867

Offset: 1

Claudio Meller, Jul 27 2018

a(2065) > 100000. - Hans Havermann, Aug 10 2018

			a(1) = 15 because with the letters of 'fifteen' we can write only one smaller number: ten. And 15 is the smallest number for which this is so. (We cannot write 'nine' because that requires two letters 'n'.)
a(10) = 115 because with the letters of 'one hundred fifteen' we can write the name of ten smaller numbers: one, three, four, nine, ten, fourteen, fifteen, nineteen, one hundred, one hundred ten.

Sequence Fans Mailing list.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2064 from Hans Havermann, terms 1..100 from Daniel Suteu)

Cf. A317423 (Spanish).

from num2words import num2words as n2w
from collections import Counter
from itertools import count, islice
def key(n):
    return Counter(c for c in n2w(n).replace(" and", "") if c.isalpha())
def included(c1, c2): # in Python 3.10+, use c[i] <= c[k] in agen()
    return all(c1[c] <= c2[c] for c in c1)
def agen():
    n, adict, c = 1, {0: 1}, [None]
    for k in count(1):
        c.append(key(k))
        v = sum(1 for i in range(1, k) if included(c[i], c[k]))
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
        if k%10000 == 0:
            print("...", k)
print(list(islice(agen(), 54))) # Michael S. Branicky, Aug 19 2022

A284919 Even integers E such that there is no prime p < E with E - p and E + p both prime.

Original entry on oeis.org

0, 2, 4, 6, 28, 38, 52, 58, 62, 68, 74, 80, 82, 88, 94, 98, 112, 118, 122, 124, 128, 136, 146, 148, 152, 158, 164, 166, 172, 178, 182, 184, 188, 190, 206, 208, 212, 214, 218, 220, 224, 238, 242, 244, 248, 250, 256, 262, 268, 278, 284, 290, 292, 296, 298

Offset: 1

Claudio Meller, Apr 05 2017

nonn

Or, even integers k such that k + p is composite for all primes p, q with p + q = k.
The two initial terms vacuously satisfy the definition, but all even numbers >= 4 are the sum of two primes, according to the Goldbach conjecture.
All odd numbers except for numbers m such that m-2 and m+2 are prime (= A087679) would satisfy the definition. - M. F. Hasler, Apr 05 2017
Conjecture: except for a(4)=6, all terms are coprime to 3. - Bob Selcoe, Apr 06 2017
If E is an even number not divisible by 3, then E is in the sequence unless E-3 and at least one of E+3 and 2E-3 are prime. - Robert Israel, Apr 10 2017
Consider a subsequence with the additional condition: n+odd part of p-1 is composite (for example, for p=19 it is 9). I found that this subsequence begins 0,2,118 and up to 300000 Peter J. C. Moses found only more one term 868. Is this subsequence finite? - Vladimir Shevelev, Apr 12 2017
One can compare the theoretical maxima with the actual sequence numbers of terms.  Doing this at powers of 10, we see at powers {2,3,4,5,6} the ratio progression {2.33, 1.51, 1.25, 1.15, 1.096}.  This implies that the excluded even numbers become increasingly rare (those coprime to 3). - Bill McEachen, Apr 17 2017
From Robert Israel's comment and the distribution of primes, the proportion of even numbers not divisible by 3 that are in the sequence tends to 1. - Peter Munn, Apr 23 2017
Moreover, If n is not divisible by 3 and 2*n - 3 is composite, then 2*n+p is also composite. Indeed, for these 2*n all primes p such that 2*n-p is prime are in the interval (3, 2*n-3). Then either 2*n-p or 2*n+p should be divisible by 3, but 2*n-p is a prime > 3. So 2*n+p is composite and 2*n is in the sequence. - Vladimir Shevelev, Apr 28 2017

			k=28 is in the sequence because 5+23 = 28 and 11+17 = 28, and 28 + {5,11,17,23} are composite; k=26 is not in the sequence because 3+23 = 26, 7+19 = 26 and 13+13 = 26, but 26+3 = 29 (prime).  - _Bob Selcoe_, Apr 06 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms from M. F. Hasler)
Claudio Meller and others, New sequence, SeqFan list, April 5, 2017. (Click "next" for subsequent contributions.)
Robert G. Wilson v, Graph of all terms < 100000

Cf. A284928 (terms/2), A002375 (number of decompositions p + q = 2k), A020481 (least p: p + q = 2k), A277688 (an analog for decompositions odd k as 2p+q).

fQ[n_] :=  Select[Select[Prime@Range@PrimePi@n, PrimeQ[n - #] &],    PrimeQ[n + #] &] == {}; Select[2 Range[0, 150], fQ] (* Robert G. Wilson v, Apr 05 2017 *)

is(n)=!bittest(n,0)&&!forprime(p=2,n\2, isprime(n-p) && (isprime(n+p) || isprime(2*n-p)) && return) \\ Charles R Greathouse IV and M. F. Hasler, Apr 05 2017

a(n) = 2*A284928(n). - M. F. Hasler, Apr 06 2017

A276766 a(n) = smallest nonnegative integer not yet in the sequence with no repeated digits and no digits in common with a(n-1), starting with a(0)=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 14, 20, 13, 24, 15, 26, 17, 25, 16, 27, 18, 29, 30, 12, 34, 19, 28, 31, 40, 21, 35, 41, 32, 45, 36, 42, 37, 46, 38, 47, 39, 48, 50, 43, 51, 49, 52, 60, 53, 61, 54, 62, 57, 63, 58, 64, 59, 67, 80, 56, 70, 65, 71, 68, 72, 69, 73, 81, 74, 82, 75

Offset: 0

Claudio Meller, Sep 17 2016

The author of this sequence is Rodolfo Kurchan, who mentioned this sequence in a Facebook Group "Series", cf. link.

The sequence is finite, with last term a(5274) = 78642. - M. F. Hasler, Sep 17 2016

M. F. Hasler, Table of n, a(n) for n = 0..5274 (All terms)
Rodolfo Kurchan, Post in Facebook Group "Series".

Cf. A054659, A067581, A276633, A276512.

{u=[]; (t(k)=if(#Set(k=digits(k))==#k,k)); a=1; for(n=1, 99, print1(a","); u=setunion(u, [a]); t(u[1])||u[1]++; while(#u>1&&u[2]<=u[1]+1, u=u[^1]); for(k=u[1]+1, 9e9, setsearch(u, k)&&next; (d=t(k))&& !#setintersect(Set(digits(a)), Set(d))&&(a=k)&&next(2))); a} \\ M. F. Hasler, Sep 17 2016

def ok(s, t): return len(set(t)) == len(t) and len(set(s+t)) == len(s+t)
def agen(): # generator of complete sequence of terms
    aset, k, mink, MAX = {0}, 0, 1, 987654321
    while True:
        if k < MAX: yield k
        else: return
        k, s = mink, str(k)
        MAX = 10**(10-len(s))
        while k < MAX and (k in aset or not ok(s, str(k))):
            k += 1
        aset.add(k)
        while mink in aset: mink += 1
print(list(agen())[:73]) # Michael S. Branicky, Jun 30 2022

Edited by M. F. Hasler, Sep 17 2016

A274328 a(n) is the sum of a sequence of multiples of the n-th prime such that it contains each of the digits from 0 to 9 exactly once and with the least sum possible, or 0 if there is no satisfying sequence.

Original entry on oeis.org

270, 135, 38475, 252, 1881, 702, 918, 684, 1656, 2349, 1953, 7326, 2952, 2322, 2961, 3339, 3717, 3843, 3015, 3195, 3285, 5688, 8217, 5607, 4365, 95445, 6489, 4815, 3924, 37629, 35433, 10611, 9864, 5004, 41571, 4077, 39564, 2934, 34569, 42039

Offset: 1

Claudio Meller, Jun 21 2016

From Ryan Hitchman, Sep 15 2017: (Start)
a(172) = 1023847569, prime(172) = 1021 is the first entry with one multiple.
a(1884) = 145953, prime(1884) = 16217 is last with more than one multiple.
a(10545) = 0, prime(10545) = 111119 is the first zero. (End)

			For n = 7, a(7) = 918 because prime(7) = 17, sequence 34, 85, 102, 697, sum 918.

Ryan Hitchman, Table of n, a(n) for n = 1..10544
Claudio Meller Blog, Multiples with all the digits, June 13 2016.

Cf. A180489 for n>1884. Superset of A050288. - Ryan Hitchman, Sep 15 2017

(m = Select[#*Range[10000], Max[DigitCount[#]] == 1 &];
   Total[m*LinearProgramming[m, Thread[DigitCount /@ m],
      ConstantArray[{1, 0}, 10], 0, Integers]]) & /@ Prime[Range[40]] (* Ryan Hitchman, Sep 15 2017 *)

Terms a(9) and beyond, zero case from Ryan Hitchman, Sep 15 2017

A274071 a(n) is the least possible sum of a sequence of distinct terms consisting of exactly prime(i) multiples of prime(i) for i = 1 to n.

Original entry on oeis.org

6, 20, 87, 304, 1398, 3582, 9218, 18270, 34873, 70451

Offset: 1

Claudio Meller, Jun 09 2016

From R. J. Mathar, Jun 23 2016: (Start)
a(5) <= 1398 with [5 7 11 22 25 33 35 49 55 77 91 99 119 121 143 154 165 187].
a(6) <= 3582 with [7 11 13 26 33 35 39 49 55 65 77 91 117 119 121 143 169 187 209 221 247 253 286 299 325 385].
a(7) <= 9372 with [7 11 13 17 34 49 51 55 65 68 77 85 91 119 121 143 153 169 187 209 221 247 253 289 299 319 323 341 377 391 403 425 481 493 527 595 663 1001].
a(8) <= 19649 with [11 13 17 19 38 57 76 77 85 91 95 119 121 133 143 169 171 187 209 221 247 253 285 289 299 319 323 341 361 377 391 403 407 437 475 481 493 527 533 551 559 589 611 629 665 697 703 731 779 799 833 901 1309].
a(9) <= 37439. a(10) <= 74605. a(11) <= 128595. a(12) <= 215047. a(13) <= 345639. a(14) <= 506980. a(15) <= 724064. (End)
a(11-20) <= (118833, 202546, 322763, 470583, 668392, 956378, 1363577, 1830468, 2461758, 3229840). - Lars Blomberg, Oct 03 2016

			a(1) = 6 from the sequence (2,4).
a(3) = 87 (3,9,5,10,15,20,25): this sequence has only two multiples of 2, only three multiples of 3 and only five multiples of 5.

Claudio Meller Blog, Problem 1443, April 28 2016.
Lars Blomberg, Details of solutions (1-10) and estimates (11-20)

a(4) from Zak Seidov, Jun 09 2016
"Distinct terms" added to definition by N. J. A. Sloane, Jun 21 2016
a(5)-a(10) from Lars Blomberg, Oct 03 2016

A238047 a(n) is the least number larger than the total number of letters in the Spanish names for all terms up to and including a(n).

Original entry on oeis.org

6, 10, 15, 26, 40, 48, 60, 70, 76, 90, 98, 108, 126, 140, 160, 170, 190, 200, 208, 228, 250, 270, 290, 300, 311, 330, 358, 380, 400, 408, 430, 460, 478, 500, 502, 520, 540, 560, 580, 600, 601, 620, 640, 660, 680, 700, 702, 720, 740, 760, 780, 800, 806, 828, 850, 870, 890, 900, 915, 940, 960, 980, 1000

Offset: 1

Claudio Meller, Feb 17 2014

Spanish version of A233184.

			The total length up to a(n), followed by the name of a(n), is as follows:
4 (seis), 8 (diez), 14 (quince), 24 (veintiseis), 32 (cuarenta), 47 (cuarenta y ocho), 54 (sesenta), 61 (setenta), 75 (setenta y seis), 82 (noventa), 96 (noventa y  ocho), 107 (ciento ocho),...

E. Angelini, Cumulative quantity of letters used in a list of English number-names, SeqFan list, Dec 05 2013

Cf. A233184.

A237263 Smallest prime such that if up to n copies of the final digit are appended to the number, it remains prime, but it does not for n+1 copies.

Original entry on oeis.org

2, 31, 19, 23, 139, 6089, 40949, 13153513, 748105003, 11307204673, 202073177599, 14740352606699

Offset: 0

Claudio Meller, Feb 05 2014

			23 is prime, so are 233, 2333, 23333, but not 233333 = 353 * 661.  No prime smaller than 23 has this property for n = 3, so a(3) = 23.

D. J. Broadhurst et al., Prime Numbers and Primality Testing, Topic 25454, Yahoo Groups, 2014
David Broadhurst, Kevin Acres, Mark Underwood, Base-10 puzzle, digest of 6 messages in primenumbers Yahoo group, Jan 16 - Feb 1, 2014.

a(1)-a(10) from David Broadhurst, Feb 01 2014

a(11) from Giovanni Resta, Feb 06 2014

cp's OEIS Frontend

User: Claudio Meller

A350570 a(n) is the smallest multiple of 7 that can be formed by concatenating the first n positive integers in some order.

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A339753 Base-ten n whose English number-word expression contains a letter making its n-th appearance in the list of consecutive positive integer number-words.

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A317423 a(n) is the smallest number such that with the letters of the name of that number we can spell the name of n numbers smaller than a(n) in Spanish.

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A317422 a(n) is the smallest positive integer such that with the letters of the name of that number we can spell the name of exactly n smaller positive integers.

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A284919 Even integers E such that there is no prime p < E with E - p and E + p both prime.

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A276766 a(n) = smallest nonnegative integer not yet in the sequence with no repeated digits and no digits in common with a(n-1), starting with a(0)=0.

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A274328 a(n) is the sum of a sequence of multiples of the n-th prime such that it contains each of the digits from 0 to 9 exactly once and with the least sum possible, or 0 if there is no satisfying sequence.

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A274071 a(n) is the least possible sum of a sequence of distinct terms consisting of exactly prime(i) multiples of prime(i) for i = 1 to n.

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A238047 a(n) is the least number larger than the total number of letters in the Spanish names for all terms up to and including a(n).

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