Ray G. Opao - cp's OEIS frontend

A354523 Number of distinct letters in the English word for n that can also be found in the English word for n+1.

2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 3, 4, 3, 5, 6, 5, 6, 6, 6, 6, 5, 6, 3, 5, 6, 5, 5, 6, 5, 6, 6, 6, 3, 5, 5, 5, 5, 5, 6, 6, 6, 7, 4, 4, 5, 4, 5, 4, 4, 5, 5, 5, 3, 5, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 7, 6, 7, 7, 7, 6, 6, 7, 4, 6, 7, 6, 7, 7, 6, 7, 6, 6, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 2, 7

Offset: 0

Ray G. Opao, Aug 16 2022

US English is assumed (i.e., 101 = 'one hundred one' instead of 'one hundred and one').

There are no zero values since min{a(k) | 0 <= k < 1000} = 1 and "nine" and "one" share common letters whenever the initial power name changes. - Michael S. Branicky, Aug 19 2022

			a(0) = 2 since the letters 'e' and 'o' in 'zero' can also be found in 'one'.
a(11)= 3 since the letters 'e', 'l' and 'v' in 'eleven' can also be found in 'twelve'.

a[n_]:= Length[Intersection[Characters[IntegerName[n]], Characters[IntegerName[n+1]], CharacterRange["a","z"]]]; Array[a,101,0] (* Stefano Spezia, Aug 19 2022 *)

from num2words import num2words as n2w
def b(n): return set(c for c in n2w(n).replace(" and", "") if c.isalpha())
def a(n): return len(b(n) & b(n+1))
print([a(n) for n in range(101)]) # Michael S. Branicky, Aug 19 2022

A354521 a(n) is the position of the first letter in the US English name of n that can also be found in the English name of n+1.

Original entry on oeis.org

2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Ray G. Opao, Aug 16 2022

Spaces and punctuation are ignored when determining the position of the letter.

a(n) is well-defined as n and n+1 always share a letter (see formulas). - Michael S. Branicky, Aug 20 2022

			For n = 4, a(4) = 1 since the 1st letter of 'four' can also be found in 'five'.
For n = 59, a(59) = 2 since the 2nd letter of 'fifty-nine' can be found in 'sixty'.

from num2words import num2words as n2w
def a(i):
    w1 = n2w(i).replace(' ','').replace('-','')
    w2 = n2w(i+1).replace(' ','').replace('-','')
    for j in range(len(w1)):
        if w1[j] in w2:
            return j+1

For n > 1000, a(n) = 1 unless n = b + 1000^e - 1 (for e >= 1, 1 <= b <= 999) in which case a(n) = a(b). Subsequently, 1 <= a(n) <= 3. - Michael S. Branicky, Aug 20 2022

A335732 Emirps whose concatenation of adjacent digit differences either form an emirp that also has this characteristic or form a single-digit prime, and whose emirp also has this characteristic.

Original entry on oeis.org

13, 31, 79, 97, 347, 709, 743, 769, 907, 967, 1847, 7481

Offset: 1

Ray G. Opao, Jun 20 2020

			7481 is in the list as the concatenation of adjacent digit differences forms an emirp (i.e., |7-4|=3; |4-8|=4; |8-1|=7; which form 347, which is an emirp as 743 is also prime). Furthermore, for 347, |3-4|=1; |4-7|=3; forms 13, which is an emirp as 31 is also prime. Finally, |1-3| = 2, which is prime. This characteristic is also true for the emirp of 7481 which is 1847 (i.e., 1847 forms 743 which forms 31 which finally forms 2).

A subset of A006567.

from sympy.ntheory import isprime as isp
i = []
for a in range(10,1000000):
    if isp(a):
        b = str(a)
        d=[]
        for c in range(0,len(b)-1):
            ee = abs(int(b[c])-int(b[c+1]))
            d.append(str(ee))
        f = ''.join(d)
        g = b[::-1]
        if isp(int(f)) and isp(int(g)):
            if len(b)<3:
                i.append(b)
            else:
                if f in i:
                    i.append(b)
print(','.join(i))

A335731 Bemirps that also interpret 2 and 5 as upside-down forms of each other, assuming a digital font.

Original entry on oeis.org

1061, 1091, 1601, 1901, 10061, 10091, 16001, 19001, 106861, 109891, 110651, 110921, 120121, 121021, 121921, 129011, 129121, 150151, 151051, 151651, 156011, 156151, 168601, 198901, 1022591, 1026521, 1028011, 1055261, 1058011, 1059251, 1069291, 1096561, 1102891, 1105861, 1106881, 1108201, 1108501, 1109881, 1111651

Offset: 1

Ray G. Opao, Jun 20 2020

			110651 is in the list as its upside-down form 110921, and its emirp 156011, and the emirp of its upside-down form 129011, are all primes and uniquely different numbers.

Normal bemirps are defined in A048895.

from sympy.ntheory import isprime as isp
def ip(pp):
    rr = []
    for qq in pp:
        if qq=="0" or qq=="1" or qq=="8":
            rr.append(qq)
        elif qq=="2":
            rr.append("5")
        elif qq=="5":
            rr.append("2")
        elif qq=="6":
            rr.append("9")
        elif qq=="9":
            rr.append("6")
    return "".join(rr)
for bb in range(1,10000000):
    if isp(bb):
        bb = str(bb)
        if ("7" not in bb) and ("4" not in bb) and ("3" not in bb):
            cc = bb[::-1]
            dd = ip(bb)
            ee = ip(cc)
            if bb!=cc and dd!=ee and bb!=dd and bb!=ee and cc!=dd and cc!=ee and isp(int(cc)) and isp(int(dd)) and isp(int(ee)):
                print(bb)

A247849 Apollo missions that landed humans on the moon.

Original entry on oeis.org

11, 12, 14, 15, 16, 17

Offset: 1

Ray G. Opao, Sep 25 2014

NASA, The Apollo Program

A247846 Initial primes in the prime quadruplets generated by A247845.

Original entry on oeis.org

11, 3895757051, 31975161851, 68129352251, 175683369311, 188170293791, 255637427231, 767561723231, 1400284088891, 1888909389371, 2627368679051, 3908473644191

Offset: 1

Ray G. Opao, Sep 25 2014

Subsequence of A007530. - Michel Marcus, Sep 26 2014

			11 is the lesser member of the prime quadruplet 5^2-2*5-4=11; 5^2-2*5-2=13; 5^2-2*5+2=17; and, 5^2-2*5+4=19.

Cf. A007530 (lesser members of prime quadruplets), A247845.

A247882 Numbers, p, that generate the prime quadruplets p^2-2p+2k (for k = -2, -1, 1, 2).

Original entry on oeis.org

5, 15, 705, 2795, 14105, 18645, 38547, 43485, 53915, 57957, 62417, 76287, 82355, 94445, 96657, 145937, 162605, 178817, 180677, 184877, 193625, 234017, 238887, 256557, 261017, 287835, 297815, 334007, 339525, 346425, 387297, 399387, 407145, 417597, 418845, 419147

Offset: 1

Ray G. Opao, Sep 25 2014

nonn

For a subset of this list, restricted only to primes, see A247845.

All terms == 5 or 7 (mod 10). - Robert Israel, Mar 10 2025

			5 is in the sequence as it generates the prime quadruplet 5^2-2*5-4=11; 5^2-2*5-2=13; 5^2-2*5+2=17; and, 5^2-2*5+4=19.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A247845 (subsequence of primes).

filter:= p -> andmap(isprime, [p^2-2*p-4,p^2-2*p-2,p^2-2*p+2,p^2-2*p+4]):
select(filter, [seq(seq(10*i+j,j=[5,7]),i=0..10^6)]);

lista(nn) = {vk = [-2, -1, 1, 2]; for (p = 2, nn, nb = 0; for (k = 1, 4, nb += isprime(p^2-2*p+2*vk[k]);); if (nb == 4, print1(p, ", ")););} \\ Michel Marcus, Sep 26 2014

More terms from Michel Marcus, Sep 26 2014

A247863 Initial primes in the prime quadruplets generated by A247862.

Original entry on oeis.org

11, 101, 300491, 61820577884902511, 348769459612079711, 427950243745391951, 668482766088635621, 2099202306201150041

Offset: 1

Ray G. Opao, Sep 25 2014

Subsequence of A007530. - Michel Marcus, Sep 26 2014

Cf. A007530 (lesser of prime quadruplets), A247862.

Added a(8). - Wolfdieter Lang, Oct 16 2014

A247862 Primes p that generate the prime quadruplets p^3-4p+2k (for k = -2, -1, 1, 2).

Original entry on oeis.org

3, 5, 67, 395407, 703903, 753583, 874373, 1280417, 1386977, 2920543, 3459487, 3697927, 3905527, 4384543, 4524427, 5630503, 6289343, 6379517, 7882873, 8599993, 8805653

Offset: 1

Ray G. Opao, Sep 25 2014

			5^3-4*5-4=101, 5^3-4*5-2=103, 5^3-4*5+2=107, 5^3-4*5+4=109 is a prime quadruplet, so 5 is in the sequence.

Cf. A247863.

Select[Prime[Range[600000]],AllTrue[#^3-4#+2{-2,-1,1,2},PrimeQ]&] (* Harvey P. Dale, Feb 16 2024 *)

lista(nn) = {forprime(p=2, nn, pp = p^3-4*p; if (isprime(pp-4) && isprime(pp-2) && isprime(pp+2) && isprime(pp+4), print1(p, ", ")););} \\ Michel Marcus, Oct 10 2014

a(8)-a(21) from Michel Marcus, Oct 10 2014

A247845 Primes, p, that generate the prime quadruplets, p^2-2p+2k (for k = -2, -1, 1, 2).

Original entry on oeis.org

5, 62417, 178817, 261017, 419147, 433787, 505607, 876107, 1183337, 1374377, 1620917, 1976987, 3619607, 4146377, 5260487, 5622047, 6399677, 7166147, 7213847, 7743647, 8055167, 10615967, 13277717, 14042117, 14080277, 15331397, 17433407, 17889587, 17963867

Offset: 1

Ray G. Opao, Sep 25 2014

nonn

Except for a(1), all other terms in the sequence end in 7.

For a similar list not restricted to primes, see A247882.

			5 is in the sequence as it generates the prime quadruplet 5^2-2*5-4=11; 5^2-2*5-2=13; 5^2-2*5+2=17; and, 5^2-2*5+4=19.

Cf. A247846 (lesser of prime quadruplets), A247882 (similar but not restricted to primes).

[p: p in PrimesUpTo(10^7) |IsPrime(p^2-2*p-4) and IsPrime(p^2-2*p-2)and IsPrime(p^2-2*p+2)and IsPrime(p^2-2*p+4)]; // Vincenzo Librandi, Oct 14 2014

lista(nn) = {vk = [-2, -1, 1, 2]; forprime (p=2, nn, nb = 0; for (k=1, 4, nb += isprime(p^2-2*p+2*vk[k]);); if (nb == 4, print1(p, ", ")););} \\ Michel Marcus, Sep 26 2014

More terms from Michel Marcus, Oct 10 2014

cp's OEIS Frontend

User: Ray G. Opao

A354523 Number of distinct letters in the English word for n that can also be found in the English word for n+1.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

Python

A354521 a(n) is the position of the first letter in the US English name of n that can also be found in the English name of n+1.

Views

Author

Keywords

Comments

Examples

Programs

Python

Formula

A335732 Emirps whose concatenation of adjacent digit differences either form an emirp that also has this characteristic or form a single-digit prime, and whose emirp also has this characteristic.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

A335731 Bemirps that also interpret 2 and 5 as upside-down forms of each other, assuming a digital font.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

A247849 Apollo missions that landed humans on the moon.

Views

Author

Keywords

Links

A247846 Initial primes in the prime quadruplets generated by A247845.

Views

Author

Keywords

Comments

Examples

Crossrefs

A247882 Numbers, p, that generate the prime quadruplets p^2-2p+2k (for k = -2, -1, 1, 2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A247863 Initial primes in the prime quadruplets generated by A247862.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A247862 Primes p that generate the prime quadruplets p^3-4p+2k (for k = -2, -1, 1, 2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A247845 Primes, p, that generate the prime quadruplets, p^2-2p+2k (for k = -2, -1, 1, 2).

Views

Author